Fabrication, Characterization and Theoretical
Modeling of Te-doped Bi Nanowire Systems for
Thermoelectric Applications
by
F
Yu-Ming Lin
Z NflF~r
1sossi
OOOZ
8311
331Z V18
AUDOOMAJ±J
B.S., National Taiwan University (1996)
Submitted to the Department of Electrical Engineering
and Computer Science
in partial fulfillment of the requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2000
@
Massachusetts Institute of Technology 2000. All rights reserved.
ENG
MASACHUSETTS INSTITUTE
OF TECHNOLOGY
JUN 2 22000
.-..
LIBRARIES
A uthor ..................
Department of Electrical Engineering and Computer Science
ADril 15, 2000
Certified by.
Mildred S. Dresselhaus
Institute Professor of Electrical Engineering and Physics
Thesis Supervisor
Certified by.
V
Jackie Y. Ying
Associate Professor of Chemical Engineering
T4ifi
M11r)ervisor
Accepted by.............(
Arthur C. Smith
Chairman, Departmental Committee on Graduate Students
-A
Fabrication, Characterization and Theoretical Modeling of
Te-doped Bi Nanowire Systems for Thermoelectric
Applications
by
Yu-Ming Lin
Submitted to the Department of Electrical Engineering and Computer Science
on April 15, 2000, in partial fulfillment of the
requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
Abstract
In this thesis, I present a novel fabrication technique for synthesizing Te-doped Bi
nanowire arrays, which represent a unique class of one-dimensional systems. Due to
the small electron effective mass and the highly anisotropic Fermi surface of Bi, Bi
nanowires are especially intriguing for the study of one-dimensional systems, as well as
novel thermoelectric materials. Te, which acts as an electron donor in Bi, is introduced
to synthesize n-type Bi nanowires to optimize the thermoelectric performance. The
fabrication of Te-doped Bi nanowires consists of preparing porous anodic alumina,
followed by the pressure injection of Te-doped liquid Bi into the porous template.
With this technique, nanowires with diameters ranging from 7 nm to 200 nm, and
lengths of ~ 50 pim are achieved. The nanowires are highly crystalline, dense and
continuous, as determined by scanning electron microscopy (SEM) and X-ray diffraction (XRD) studies. From XRD studies, we found that our nanowires possess a
preferred growth orientation along the wire axis, and this orientation is found to be
dependent on the wire diameter.
This thesis also presents an improved theoretical model for Bi nanowire systems,
which takes into account the circular wire boundary conditions, anisotropic carrier
pockets, and non-parabolic dispersion relation for L-point electrons and holes. A
powerful numerical method that can be generalized to other problems has been
designed to help solve the complicated electronic band structure of Bi nanowires.
Theoretical calculations show that Bi nanowires with small diameters (< 10 nm)
would have a thermoelectric performance superior to bulk materials. An even more
significant enhancement in the thermoelectric performance is expected if the T-point
hole pocket can be removed or suppressed.
The transport properties of pure and Te-doped Bi nanowires have been studied
experimentally over a wide range of temperatures (4-300 K) in this thesis. The
temperature dependence of the resistance for Bi nanowires with different diameters
shows good agreement with theoretical modeling, and provides strong evidence for
2
the semimetal-semiconductor transition in Bi nanowires. The effect of Te doping has
been confirmed by resistance and Seebeck coefficient measurements. The experimental results obtained are consistent with the theoretical predictions.
Thesis Supervisor: Mildred S. Dresselhaus
Title: Institute Professor of Electrical Engineering and Physics
Thesis Supervisor: Jackie Y. Ying
Title: Associate Professor of Chemical Engineering
3
Acknowledgments
To Prof. Mildred Dresselhaus and Prof. Jackie Ying I owe my sincere gratitude,
without them this thesis would not be possible. From Prof. Dresselhaus I received
much more than just research training. Her diligence and devotion to the pursuit of
science have been a constant inspiration and stimulation to me. I thank Prof. Jackie
Ying for giving me the opportunity to work in her laboratory, for offering support
and advice when things were going slow, and for encouragement when things were
smooth.
The thesis would not be as successful without Dr. Zhibo Zhang, who patiently
and systematically taught me the fabrication of nanowires and many experimental
skills. Discussions with Dr. Xiangzhong Sun have been valuable in developing the
theoretical part of the thesis. I am grateful to Yinlin Xie and Fang-Cheng Chou who
helped with the preparation of Te-doped Bi alloys.
I was very fortunate to be part of the Dresselhaus group. Takaaki Koga, my officemate, is a very interesting person, and I had a lot of fun conversations with him. His
creativity and mathematical insight always impressed me. Taka, Steve Cronin and
Dr. Dima Gekhtman all gave me a lot of assistance and advice on both the theoretical
and experimental aspects of my thesis. The incredible laughter of Sandra Brown from
three offices away was always helpful for stress relief. The TA experience with Marcie
Black was a joyful one. I enjoyed the salads and cr~pes made by Oded Robin, who
also gave me many great suggestions for my research. I thank Laura Doughty for
everything including teaching me about the American culture. "Mechi-Mechi" time
with the group members was one of my most treasured memories of MIT. Dr. Gene
Dresselhaus is a true physicist, and I admire his courage and spirit to fight with
Microsoft.
Thanks to Prof. Jackie Ying, I have the chance to know another group of interesting people at the Nanostructured Materials Research Laboratory. There is always
laughter being with Justin McCue, who happened to have the same birthday as I, and
who helped me a lot in improving the anodic alumina film synthesis. I am indebted
4
to Dr. Mark Fokema, Dr. Martin Panchula, Neeraj Sangar, and Dr. Andrey Zarur
for assisting me with the equipment set-up and for the useful advice in experiments.
Chen-Chi Wang, Duane Myers, Ed Ahn, John Lettow, Suniti Moudgil, and Dr. Jinsuo
Xu have all been supportive in my research.
To Emily Fang I owe my wonderful and fondest memories in Boston. Without her
support and companionship, life would have never been the same.
This thesis is dedicated to my family, especially to my parents, Lloyd Lin and
Spring Ho, who have constantly given me their unreserved love and support throughout my life.
5
Contents
1
2
Introduction
20
1.1
B ackground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.2
Motivation for Studying Te-doped Bi Nanowires . . . . . . . . . . . .
22
1.3
T hesis O utline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Fabrication and Characterization of Te-Doped Bi Nanowires
24
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.2
Synthesis of Porous Anodic Alumina Templates
. . . . . . . . . . . .
27
2.2.1
Al Substrate Preparation . . . . . . . . . . . . . . . . . . . . .
27
2.2.2
Two-Step Anodization Process . . . . . . . . . . . . . . . . . .
29
2.2.3
Experimental Results . . . . . . . . . . . . . . . . . . . . . . .
34
2.3
2.4
3
Preparation of Te-doped Bi Nanowires
. . . . . . . . . . . . . . . . .
38
2.3.1
Preparation of Te-doped Bi Alloy . . . . . . . . . . . . . . . .
38
2.3.2
Vacuum Melting/Pressure Injection Process . . . . . . . . . .
39
2.3.3
Subsequent Processing . . . . . . . . . . . . . . . . . . . . . .
40
2.3.4
Preparation of Free-Standing Nanowires
. . . . . . . . . . . .
41
Characterization of Te-doped Bi Nanowires . . . . . . . . . . . . . . .
42
2.4.1
SEM Study of Te-doped Bi Nanowire Arrays . . . . . . . . . .
42
2.4.2
XRD Study of Te-doped Bi Nanowire Arrays . . . . . . . . . .
44
Theoretical Modeling of ID Bi Quantum Wires
48
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.2
Crystal and Band Structures of Bi......
49
6
. . ...
. . . . . . . . .
3.3
3.4
4
Electronic States in Bi Nanowires . . . . . . . . . . . . . . . . . . . .
55
3.3.1
Solution to the Schr6dinger Equation for Bi Nanowires
57
3.3.2
Numerical Solutions to the Differential Equation:
3.3.3
=
A .. .. ... .. .. .. .. .. .. .. .
(a X + aY2
Non-Parabolic Band Structure of Bi Nanowires . . . . . . . .
. . . .
67
Band Shifting and Semimetal-Semiconductor Transition in Bi Nanowires 70
Transport Properties of Te-Doped Bi Nanowire Systems
77
4.1
Semi-Classical Transport Model . . . . . . . . . . . . . . . . . . . . .
77
4.2
Carrier Density in Pure Bi Nanowires . . . . . . . . . . . . . . . . . .
80
4.3
Te Doping of Bi Nanowires . . . . . . . . . . . . . . . . . . . . . . . .
85
4.4
Thermoelectric Investigations of Te-doped Bi Nanowires at 77K . . .
88
4.5
Effect of the T-point Holes on the Thermoelectric Properties of Bi
N anow ires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6
4.7
5
60
Two-Point Resistance Measurements
95
. . . . . . . . . . . . . . . . . .
106
4.6.1
Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . .
106
4.6.2
Transport Measurements of Bi Nanowires . . . . . . . . . . . .
109
4.6.3
Experimental R(T) Results for Te-doped Bi Nanowires
. . . .
114
Seebeck Measurements . . . . . . . . . . . . . . . . . . . . . . . . . .
119
4.7.1
Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . .
120
4.7.2
Experimental Results and Discussion . . . . . . . . . . . . . .
122
Conclusions and Future Directions
126
5.1
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
126
5.2
Suggestions for Future Studies . . . . . . . . . . . . . . . . . . . . . .
127
7
List of Figures
2-1
Schematic of the fabrication process of Te-doped Bi nanowire arrays
and free-standing nanowires: (a) the aluminum film after mechanical
polishing and electrochemical polishing, (b) porous anodic alumina
template produced on an Al substrate, (c) pressure injection of molten
Te-doped Bi into the evacuated channels of the porous template, (d)
anodic alumina filled with Te-doped Bi detached from the Al substrate,
and (e) free-standing Te-doped Bi nanowires obtained by dissolution
of the anodic alumina template using a special etching solution.
2-2
. . .
28
Schematic of the experimental set-up for electrochemical polishing.
The experimental set-up for the subsequent anodization process is similar, except that the heater is replaced with a magnetic stirrer, and the
oil bath is changed to an ice or water bath. . . . . . . . . . . . . . . .
2-3
Schematic illustrating the structure of anodic alumina template: (a)
side view, (b) cross-sectional view. . . . . . . . . . . . . . . . . . . . .
2-4
29
31
Schematic of the two-step anodization process: (a) a mechanically and
electrochemically polished Al substrate, (b) porous anodic alumina
layer produced in the first anodization, (c) textured Al substrate after
the removal of anodic alumina layer, and (d) porous anodic alumina
template produced in the second anodization.
2-5
. . . . . . . . . . . . .
33
SEM images of the top surfaces of porous anodic alumina templates
after a first anodization in (a) 4 wt% H2 C 2 0
4
and (b) 20 wt% H 2 SO 4 .
The average pore diameters in (a) and (b) are 26 nm and 12 nm, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
35
2-6
SEM image of the bottom side of an anodic alumina template after the
barrier layer was partially etched away. The sample has been subjected
to a first anodization in 4 wt% H 2 C 2 0
2-7
4
at 45 V for 180 minutes.
. . .
SEM images of the top surfaces of porous anodic alumina templates
after the second anodization in (a) 4 wt% H 2 C 2 0
H 2 SO 4 .
4
and (b) 20 wt%
The average pore diameters in (a) and (b) are 44 nm and
18 nm , respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-8
36
37
A lower magnification SEM image of the top surface of the porous
anodic alumina template presented in Fig. 2-7(a) after the second anodization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-9
38
Schematic of the experimental set-up for pressure injection of Te-doped
Bi alloy into the pores of an anodic alumina template [19].
. . . . . .
39
2-10 SEM micrographs of the bottom surface of nanowire arrays after the
barrier layer has been removed.
was prepared in 4 wt% H 2 C 2 0
4
The alumina template used in (a)
and has an average pore diameter of
~ 40 nm. The alumina template in (b) was prepared in 20 wt% H2SO
4
and has an average pore diameter of ~ 17 nm. In sample (a), most
of the pores have been thoroughly filled with Bi-Te alloy (shown as
bright spots); while in sample (b), pore filling was not extended to the
bottom surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
2-11 SEM micrograph of the bottom surface of a nanowire array after the
barrier layer has been removed. The alumina template used was anodized in 4 wt% H 2 C 2 0 4 and has an average pore diameter ~ 40 nm.
Some of the nanowires spread out from the pore region in the form of
"m ushroom s". . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
2-12 Schematic of the set-up for XRD experiments, illustrating the orientation of the nanowire array relative to the X-ray beam. . . . . . . . . .
9
45
2-13 XRD patterns of Bi/anodic alumina nanocomposites with average Bi
wire diameters of: (a) 40 nm, (b) 52 nm, and (c) 95 nm. The XRD
peaks of all three samples appeared at the same positions as those of
the polycrystalline Bi standard. The Miller indices corresponding to
the lattice planes of bulk Bi are indicated above the individual peaks.
46
2-14 XRD patterns of 40-nm Bi nanowire arrays containing (a) no Te, (b)
0.075 at% Te, and (c) 0.15 at% Te. The XRD peaks of all three samples
appeared at the same positions as those of the polycrystalline Bi standard. The Miller indices corresponding to the lattice planes of bulk Bi
are indicated above the individual peaks.
3-1
. . . . . . . . . . . . . . .
47
The crystal structure of Bi, which can be derived by distorting two
interlaced fec lattices along the body diagonal, followed by a small
relative displacement of the two fcc lattices along the trigonal axis.
The bisectrix axis lies on the mirror plane, and is perpendicular to the
binary and trigonal axes. . . . . . . . . . . . . . . . . . . . . . . . . .
3-2
50
The Brillouin zone of Bi, showing the Fermi surfaces of the three electron pockets at the L points and one T-point hole pocket. The total
number of electrons is equal to that of holes. The mirror plane going
through the L(A) pocket and the T-point holes is indicated by dashed
lines. .. . ... . . ..
3-3
. . .
. . . . . . . . .. . . . . . . . . ..
. . . . . .
51
Schematic of the Bi band structure at the L point and the T point
near the Fermi energy level, showing the band overlap A0 of the Lpoint conduction band and the T-point valence band. The L-point
electrons are separated from the L-point holes by a small bandgap
E9 L.
3-4
At T = 0 K, Ao = -38 meV and EL = 13.6 meV [35, 36]. .....
51
Schematic of the grid points used to transform the differential equation
into a difference equation. The mesh in the circular wire cross-section
consists of M concentric circles and N sectors. In the figure illustrated,
M = 5 and N = 12..
. . . . . . . . . . . . . . . . . . . . . . . . . . .
10
62
3-5
The wavefunctions u(r, 0) of the lowest four eigenstates for a = 1,
/ = 3 in a circular wire with unity radius. The eigenvalues of the four
states are
AN
=
11.4759, 21.7205, 36.2447 and 36.2874. We note that
although the second and the fourth states have similar wavefunctions
except for nodes or oscillations in different directions, they have different energies due to the inequivalence of effective masses in the x and
y directions. The node numbers (n, m) of the four states are (0,0),
(1,0), (2,0) and (0,1), respectively. The energy (eigenvalue) increases
with node numbers n and m, but the energy increment for n may be
different from that for m. Since a < /3, the effective mass along the x
direction is larger than that in the y direction, and the energy increment will be smaller for the increase in the x-nodes. Therefore, the first
two excited states (second and third states) have nodes or oscillations
increased in the x direction, while to increase nodes or oscillations in
the y direction would require more energy; the eigenstate with the first
node in the y direction is the fourth state, which is at a higher energy.
3-6
66
The subband structure of Bi quantum wires oriented along the trigonal direction at 77 K, showing the energies of the first subbands for
the T-point hole pocket, the L-point electron pockets (A, B and C),
as well as the L-point hole pockets. The zero energy refers to the conduction band edge in bulk Bi. As the wire diameter d decreases, the
conduction subbands move up while the valence subbands move down.
At d, = 55.1 nm, the lowest conduction subband edges formed by the
L-point electrons cross the highest T-point valence subband edge, and
a semimetal-semiconductor transition occurs. . . . . . . . . . . . . . .
11
71
3-7
The subband structure of Bi quantum wires oriented along the binary
direction at 77K, showing the energies of the first subbands for the
T-point hole pocket, the L-point electron pockets (A, B, and C), as
well as the L-point hole pockets. The zero energy refers to the conduction band edge in bulk Bi. As the wire diameter d decreases, the
conduction subbands move up while the valence subbands move down.
At dc = 39.8nm, the lowest conduction subband edge formed by the
L(A) electrons crosses the highest T-point valence subband edge, and
a semimetal-semiconductor transition occurs. . . . . . . . . . . . . . .
3-8
72
The subband structure of Bi quantum wires oriented along the bisectrix direction at 77K, showing the energies of the first subbands for
the T-point hole pocket, the L-point electron pockets (A, B and C),
as well as the L-point hole pockets. The zero energy refers to the conduction band edge in bulk Bi. As the wire diameter d decreases, the
conduction subbands move up while the valence subbands move down.
At dc = 48.5 nm, the lowest conduction subband edges formed by the
L(B, C) electrons cross the highest T-point valence subband edge, and
a semimetal-semiconductor transition occurs. . . . . . . . . . . . . . .
3-9
73
The subband structure of Bi quantum wires oriented along the [0112]
direction at 77 K, showing the energies of the first subbands for the
T-point hole pocket, the L-point electron pockets (A, B and C), as
well as the L-point hole pockets. The zero energy refers to the conduction band edge in bulk Bi. As the wire diameter d decreases, the
conduction subbands move up while the valence subbands move down.
At dc = 49.0 nm, the lowest conduction subband edges formed by the
L(B, C) electrons cross the highest T-point valence subband edge, and
a semimetal-semiconductor transition occurs. . . . . . . . . . . . . . .
74
3-10 Calculated critical wire diameter dc for the semimetal-semiconductor
transition as a function of temperature for Bi nanowires oriented in
different directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
75
4-1
Calculated total carrier density (for electrons and holes) of bulk Bi as
a function of temperature. . . . . . . . . . . . . . . . . . . . . . . . .
4-2
81
Calculated density of states for 40-nm Bi nanowires oriented along the
[0112] growth direction at 77K (solid lines), compared to the density
of states of bulk Bi (dashed lines). The zero in energy represents the
band edge of the L-point conduction band in bulk Bi. . . . . . . . . .
4-3
82
Calculated total carrier density (for electrons and holes) of Bi nanowires
of different diameters oriented along the [0112] direction as a function
of temperature. The carrier density of 10-nm Bi nanowires has a temperature dependence similar to that of a narrow-gap semiconductor,
while 80-nm nanowires behave like a semimetal. The carrier density of
40-nm Bi nanowires has a semiconductor-like temperature dependence
at low temperatures (T < 180 K), and a semimetal-like temperature
dependence at high temperatures (T > 180 K). . . . . . . . . . . . . .
4-4
84
Calculated total carrier density (for electrons and holes) of 40-nm Bi
nanowires oriented along various directions as a function of temperature. The differences in carrier densities for different wire orientations
become negligible for T > 150 K.
4-5
. . . . . . . . . . . . . . . . . . . .
84
Calculated total carrier density (for electrons and holes) of 40-nm Bi
nanowires of different Te dopant concentrations oriented along the
[0112] direction as a function of temperature . . . . . . . . . . . . . .
4-6
87
Calculated ZlDT for Te-doped Bi nanowires of three different wire
diameters oriented along the trigonal axis at 77K as a function of
Te dopant concentration.
4-7
. . . . . . . . . . . . . . . . . . . . . . . .
92
Calculated ZlDT at 77K as a function of Te dopant concentrations
for 10-nm Te-doped Bi nanowires oriented along trigonal, binary,
bisectrix, [0112] and [1011] directions. . . . . . . . . . . . . . . . . . .
Nd
13
93
4-8
Calculated ZlDT as a function of the chemical potential at 77 K for (a)
40-nm, (b) 20-nm, (c) 10-nm and (d) 5-nm Bi nanowires oriented along
the trigonal direction. The zero in energy refers to the conduction band
edge in bulk Bi. EO)
h(T) and E0)
h(L) denote the highest subband edges of the
T-point holes and L-point holes, respectively, and the lowest subband
edge of the L-point electrons is labeled as E(.
The solid curves are
the ZlDT calculated with both the T-point holes and L-point holes
present, and the dashed curves show the Z1DT calculated assuming
there are no T-point holes.
For 5-nm Bi nanowires, the solid and
dashed curves coincide with each other, indicating that the T-point
holes have a negligible effect on the transport properties. . . . . . . .
4-9
97
Calculated ZlDT for p-type Bi nanowires of different diameters oriented along the trigonal axis at 771K as a function of acceptor dopant
concentration Na. The dashed curves represent the results obtained
assuming there are no T-point holes. The inset shows Z1DT calculated
for 5-nm nanowires. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
4-10 Calculated ZlDT as a function of the chemical potential at 77 K for (a)
40-nm, (b) 20-nm, (c) 10-nm and (d) 5-nm Bi nanowires oriented along
the binary direction. The zero in energy refers to the conduction band
edge in bulk Bi. E
and EO) denote the highest subband edges of the
T-point holes and L-point holes, respectively, and the lowest subband
edge of the L-point electrons is labeled as E
)
The solid curves show
the ZlDT calculated with both the T-point holes and L-point holes
present, and the dashed curves show the ZlDT calculated assuming
there are no T-point holes. . . . . . . . . . . . . . . . . . . . . . . . .
14
100
4-11 Calculated ZlDT as a function of the chemical potential at 771K for (a)
40-nm, (b) 20-nm, (c) 10-nm and (d) 5-nm Bi nanowires oriented along
the bisectrix direction. The zero in energy refers to the conduction
band edge in bulk Bi. E
) and E(
denote the highest subband
edges of the T-point holes and L-point holes, respectively, and the
lowest subband edge of the L-point electrons is labeled as E0)
The
solid curves show the Z1DT calculated with both the T-point holes and
L-point holes present, and the dashed curves show the ZlDT calculated
assuming there are no T-point holes.
. . . . . . . . . . . . . . . . . .
101
4-12 Calculated ZlDT for p-type Bi nanowires of different diameters oriented along the binary axis at 77K as a function of acceptor dopant
concentration Na. The dashed curves represent the results obtained
assuming there are no T-point holes. The inset shows ZlDT calculated
for 5-nm nanowires. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
102
4-13 Calculated ZlDT for p-type Bi nanowires of different diameters oriented along the bisectrix axis at 77 K as a function of acceptor dopant
concentration N.
The dashed curves represent the results obtained
assuming there are no T-point holes. The inset shows ZlDT calculated
for 5-nm nanowires. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103
4-14 Calculated Z1DT as a function of the chemical potential at 77 K for (a)
40-nm, (b) 20-nm, (c) 10-nm and (d) 5-nm Bi nanowires oriented along
the [1011] direction. The zero in energy refers to the conduction band
edge in bulk Bi. E()h(T) andE h(L)
(0) denote the highest subband edges of the
T-point holes and L-point holes, respectively, and the lowest subband
edge of the L-point electrons is labeled as EJ)
The solid curves show
the ZIDT calculated with both the T-point holes and L-point holes
present, and the dashed curves show the ZlDT calculated assuming
there are no T-point holes. . . . . . . . . . . . . . . . . . . . . . . . .
15
104
4-15 Calculated ZlDT as a function of the chemical potential at 77 K for (a)
40-nm, (b) 20-nm, (c) 10-nm and (d) 5-nm Bi nanowires oriented along
the [Oli2] direction. The zero in energy refers to the conduction band
edge in bulk Bi. EO)
h(L) denote the highest subband edges of the
h(T) and Jo)
T-point holes and L-point holes, respectively, and the lowest subband
edge of the L-point electrons is labeled as EO) The
The so lid curves show
e(L)
the ZlDT calculated with both the T-point holes and L-point holes
present, and the dashed curves show the ZlDT calculated assuming
there are no T-point holes. . . . . . . . . . . . . . . . . . . . . . . . .
105
4-16 Schematic of the experimental set-up for pseudo-four-point measurem ents. . . . . . . .. . .
. . . . . . . ....
. . . . . . . . . . . . . ..
107
4-17 Temperature dependence of resistance for Bi nanowire arrays of various
diameters prepared by the vapor deposition method, compared to bulk
B i [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
110
4-18 The calculated temperature dependence of the resistance for Bi nanowires
of 36 nm and 70 nm, using a semi-classical transport model. . . . . . .111
4-19 Temperature dependence of resistance for 40-nm pure or Te-doped
Bi nanowire arrays. The resistance of each sample is normalized to
R (270 K ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-20 Calculated temperature dependence of
[,-lcr
and
I- 1
115
,ed for 40-nm pure
and Te-doped Bi nanowires, respectively. The calculation is based on
the measured T-dependent resistance in Fig. 4-19 and the calculated
carrier concentration similar to Fig. 4-3. The dashed line is a curve
fitting of the mobility of pure Bi nanowires to a T- 2
dependence for
T > 100 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
118
4-21 Schematic of the experimental set-up used for the Seebeck coefficient
measurement of Bi nanowire arrays. . . . . . . . . . . . . . . . . . . .
16
121
4-22 Measured thermoelectric voltage AV as a function of the temperature gradient AT of a 40-nm 0.075 at% Te-doped Bi nanowire array at
300 K. The Seebeck coefficient of the Bi nanowire array is calculated
as S = -58.4 + 23.0 = -35.4pV/K. . . . . . . . . . . . . . . . . . . .
122
4-23 Measured temperature dependence of Seebeck coefficient for 40-nm Bi
nanowire arrays with 0 at% (),
0.075 at% (A) and 0.15 at%
(0) Te.
The cooling and heating curves are shown as solid and dashed lines,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
4-24 Thermoelectric power of 1 jim-thick BiO. 91 Sb0 .0 9 films with different Te
dopant concentrations as a function of temperature (Cho et al. [65]). .
17
124
List of Tables
2.1
The first anodization conditions for anodic alumina templates shown
in F ig. 2-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
The second anodization conditions for anodic alumina templates shown
in Fig. 2-7............
36
................................
. . . . . . . . . . . . .
54
. .
54
3.1
Band structure parameters of Bi at T = 0 K.
3.2
Temperature dependence of the band structure parameters of Bi.
3.3
Comparisons of the lowest four eigenvalues derived from analytic solutions and numerical calculations for a = #.
3.4
35
. . . . . . . . . . . . . .
65
Calculated effective mass components of each carrier pocket for determining the band structure of Bi nanowires at 77 K, based on the values
given in Table 3.1. The z' direction is chosen to be along the wire axis.
All mass values in this table are in units of the free electron mass, MO.
3.5
70
Calculated critical wire diameters d, for the semimetal-semiconductor
transition of Bi nanowires at 77 K and 300 K for various crystallographic orientations.
3.6
. . . . . . . . . . . . . . . . . . . . . . . . . . .
75
Calculated critical wire diameters d, and critical wire widths a, for Bi
nanowires oriented along various directions at 77 K, using the cyclotron
effective mass approximation and the square wire approximation, respectively.
4.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
The values of mobility tensor elements for electron and hole pockets of
Bi at 77 K [55]. The values are given in cm 2V--s-..
18
. . . . . . . . . .
89
4.2
Calculated mobility along the wire for each carrier pocket in Bi nanowires
of various wire orientations at 77 K. The values are given in m 2 V--1.
4.3
90
The sound velocities v of Bi nanowires oriented along various directions. The values at 77 K are interpolated from the experimentally
measured results at 1.6 K [57] and 300 K [58] . . . . . . . . . . . . . .
4.4
Calculated phonon mean free paths fp at 77K for Bi crystal orientations
along the three principal axes and the [0112] and [1011] directions. . .
4.5
91
91
The optimal carrier concentrations Nd(0 pt) (in 1018 cm-3) and the cor-
responding ZlDT at 77K for Te-doped Bi nanowires of various wire
diameters and orientations.
4.6
. . . . . . . . . . . . . . . . . . . . . . .
94
The optimum acceptor concentrations Na(opt) (in 1018 cm-3) and the
corresponding ZlDT at 77 K for p-type Bi nanowires of various wire
diameters and orientations . . . . . . . . . . . . . . . . . . . . . . . .
19
95
Chapter 1
Introduction
1.1
Background
In the last decade, rapid and significant advances in fabrication techniques have been
made to produce high-quality nanostructured materials, and these developments have
opened up new areas for both applications and research. One of the major incentives
is to be able to fabricate smaller and smaller devices for improved cost efficiency and
functionality. The miniaturization of electronic and mechanical components not only
decreases the physical size of the devices, but also offers novel materials properties.
As the length scale of a material shrinks to a size comparable to the de Broglie
wavelength of electrons, quantum mechanics are used to describe the system. Many
unique properties unlike those of the bulk material are observed. The emergence of
quasi-low-dimensional systems, for example, is one of the consequences of quantum
effects. These systems are formed when electrons are confined in one or more directions.
The electron energy associated with motions in the confined direction(s) becomes
quantized, and forms a series of discrete energy levels that electrons can occupy.
When the thermal excitation energy is much smaller than the separation of the energy
levels, the shift between these discrete energy levels becomes very unlikely, and the
electrons essentially lose their degrees of freedom in the confined directions. The
systems confined in one, two and three dimensions can be treated as 2D (quantum
well), ID (quantum wire) and OD (quantum dot) electron gas, respectively.
20
Because of the unique electronic states in these low-dimensional systems, the
confined electrons exhibit a very different behavior from that of bulk materials. Therefore, low-dimensional materials provide unique opportunities to examine new physical
phenomena, such as the quantum Hall effect, the fractional quantum Hall effect, and
the quantum-confined Stark effect. The study of low-dimensional systems is of interest
for potential applications in novel electronic, optical, magnetic, and thermoelectric
devices.
Carbon nanotubes, for instance, are ID materials that have been widely
explored for nanotechnology applications since their discovery in 1991 [1]. A broad
range of potential nanodevices has been proposed based on the application of carbon
nanotubes, such as field emission displays
[2],
data storage, microscope probe tips [3],
hydrogen storage, and sensors.
Another promising application of low-dimensional systems involves advanced thermoelectric materials.
The search for good thermoelectric materials used to be an
active field during the 1957-1965 period, after Joffe's suggestion of using thermoelectrics as solid-state refrigerators [4].
However, the best thermoelectric material
found in that period had only moderate performance efficiency, and there has been
little progress in the thermoelectrics field since then. Recently, Hicks and Dresselhaus
predicted a dramatic enhancement in thermoelectric performance with low-dimensional
systems [5, 6].
The proposed improvement in the thermoelectric efficiency is asso-
ciated with the enhanced density of states at the band edge in the low-dimensional
systems, the possibility to manipulate multiple anisotropic carrier pockets to optimize
their thermoelectric performance, and opportunities to increase the phonon scattering at the quantum-confined interfaces.
This promising prediction, along with the
increasing demand for CFC-free, efficient and small power generators or refrigerators,
has rendered the study of low-dimensional thermoelectric materials a rapidly growing
field over the past few years.
21
1.2
Motivation for Studying Te-doped Bi Nanowires
The study of bismuth nanowire systems has attracted a great deal of research interest
because these nanowires provide us with a unique system to examine ID electronic
transport properties, as well as a promising candidate for thermoelectric applications.
Bismuth, a group V element, has several exceptional properties attractive for studies
as a low-dimensional system, and for thermoelectric applications. It has one of the
smallest electron effective masses
(~
0.001mo) [7] of all known materials and a very
long carrier mean free path. In addition, the highly anisotropic Fermi surface of Bi
[8]
provides another parameter that can be used to manipulate the electronic states by
varying the crystal growth orientation along the wire axis.
Since Bi is a semimetal with equal numbers of electrons and holes, the transport properties are determined by both types of carriers. However, in most applications, such as thermoelectrics, it is necessary to control the Fermi energy so that
the transport phenomena are dominated by a single type of carriers, i.e. electrons or
holes. It has been experimentally shown that group VI elements, such as Te, act as
electron donors in Bi; and group IV elements, such as Pb and Sn, act as electron
acceptors in Bi. Therefore, n-type or p-type Bi can be synthesized by introducing Te
or Pb (or Sn) dopants into Bi, respectively. The study of n-type Bi is especially desirable due to the small electron effective masses and highly anisotropic Fermi surfaces.
In fact, Te-doped Bi has been investigated extensively as a possible candidate for
thermoelectric materials. However, due to the presence of holes in bulk Bi, a large
Te dopant concentration must be introduced to eliminate the effect of hole carriers,
and the resulting high ionized impurity scattering has a rather adverse effect on the
electron mobility and thermoelectric performance. As will be discussed in Chapter
3, the unique semimetal-semiconductor transition that occurs in Bi nanowires offers
one possible solution to this doping problem. With the disappearance of the overlap
between the valence and conduction bands, the electron densities can be altered
significantly without reducing the mobility by very much. Therefore, Te-doped Bi
nanowire systems are expected to provide a promising ID system with highly mobile
22
carriers, and are predicted to exhibit superior thermoelectric performance.
1.3
Thesis Outline
In Chapter 2, a detailed description of the fabrication process of Te-doped Bi nanowire
arrays will be presented, and the characterization of the nanowires derived will be
discussed.
Chapter 3 consists of a theoretical modeling of the electronic band structure of Bi
nanowires. After a brief discussion of the approximation methods used in previous
calculations, an improved model, considering more realistic wire boundary conditions,
anisotropic carrier pockets and non-parabolic dispersion relations of the electrons
will be presented. A powerful numerical method, which can be generalized to other
problems, is designed to solve the band structure and to derive the wavefunctions in
Bi nanowires. The band shifting and the semimetal-semiconductor transition in Bi
nanowires are discussed for various wire orientations.
Chapter 4 is devoted to the study of thermoelectric-related transport properties of Bi nanowire systems.
The semi-classical model based on the Boltzmann
transport equation for a ID system will first be developed for a general nanowire
system, followed by a detailed investigation of thermoelectric properties of Te-doped
Bi nanowires. The effect of the T-point holes on the thermoelectric properties of Bi
nanowires is also discussed. Transport experiments, including resistance and Seebeck
coefficient measurements are conducted as a function of temperature for pure and
Te-doped Bi nanowires.
Lastly, conclusions of this thesis are presented along with suggestions for future
research in Chapter 5.
23
Chapter 2
Fabrication and Characterization
of Te-Doped Bi Nanowires
2.1
Introduction
Advances in the synthesis of low-dimensional systems have attracted great interest in
the research of these novel materials. The first high-quality 2D system was produced
by molecular beam epitaxy (MBE) [9]. The crystallinity and the well-defined boundaries between materials of different bandgaps produced by the MBE technique were
essential for studying the effect of low dimensionality.
Similarly, to investigate the transport properties of ID nanowire systems, two
conditions must be satisfied. First, the nanowires have to be dense and continuous
so that electrical and thermal currents can be conducted without much scattering
from lattice dislocations. In addition, the host material should have a large bandgap
compared to the nanowires so that the electronic wavefunctions are well-confined
within the nanowires.
Since the present project was in part motivated by thermoelectric applications of
low-dimensional systems, several additional criteria are considered. First, in order to
carry a high electrical current density in practical applications, it is favorable to have
a large number of parallel nanowires connected. Secondly, in thermoelectric applications, it is desirable to minimize the thermal conductivity and maximize the electrical
24
conductivity. Therefore, the thermal conductivity of the host material should be small
so that heat flow through the host material is negligible, and a high crystallinity in
the nanowires is required to achieve a high electrical conductivity.
Template-mediated synthesis [10, 11, 12] is a rather promising approach for producing nanowire systems with the above desired characteristics. There are several
porous materials that may serve as the host template for Te-doped Bi nanowire arrays.
One such possible template is porous mica prepared by carbon ion bombardment [13].
The advantage of this fabrication technique is that pore diameters as small as 5 nm can
be easily achieved. However, the packing density of pores in the ion-bombarded mica
is only 1013 cm- 2 and the pores are randomly distributed. It is also difficult to systematically alter the pore size of the nanochannels with this technique. Another possible
porous template is the mesoporous molecular sieve termed MCM-41 [10], which possesses very small channel diameters that can be varied systematically between 20 A
and 100 A [14]. The well-defined pores of MCM-41 are hexagonally-packed with a
high density. However, MCM-41 is usually produced in the form of powder with a
particle size of
-
1 pm. As a result, it is very difficult to handle in the pore-filling
process and for device fabrication.
Another promising host material is porous anodic alumina. It is well known that
an alumina film with regular hexagonally-packed pores can be formed on aluminum
substrate through electrochemical oxidation or anodization [15]. The pore diameter
of anodic alumina ranges from several to hundreds of nanometers depending on the
processing conditions, and the pore diameters can be adjusted in subsequent treatment. Channel lengths (or film thicknesses) as long as 100 pm have been reported in
the literature, and there is no theoretical limit on the achievable cross-sectional area
for the films. Moreover, pore regularity and pore size distribution can be controlled by
producing a regular pattern of concave indentations on the aluminum surface before
the anodization process [16]. The unique pore structure of anodic alumina films has
made these materials promising for hosting the synthesis of various nanowire systems
[11, 17, 18], and this fabrication process offers a much less expensive alternative to
the traditional lithographic approaches [12]. The wide bandgap of anodic alumina
25
(-
3.2 eV) also makes it an excellent host material for quantum wires. Furthermore,
due to its amorphous nature, anodic alumina has a low thermal conductivity, which
is attractive for thermoelectric applications.
Recently, fabrication technique for pure Bi nanowires in porous anodic alumina
templates has been developed using a vacuum melting and pressure injection process
[19, 20, 21].
is
-
The smallest pore that has been filled with this injection technique
13 nm with a channel length of 50 pum. Due to the high thermal stability of
the alumina template (up to 800 C), this approach can be readily extended to the
nanowire fabrication of other materials with a low melting point. It is also straightforward to incorporate dopants into Bi by this technique. In addition, nanowires can
be synthesized in nanochannel templates by electrochemical deposition and vaporphase deposition. These two techniques may offer the possibililty of filling templates
of smaller pores with Bi without the ultrahigh pressure requirement associated with
the pressure injection process. It has been reported that Bi nanowires as small as
5nm and 7nm have been produced by electrochemical depostion [13] and vaporphase deposition [22], respectively. However, electrochemical deposition of Bi is not
well suited for working with anodic alumina templates since the latter are not stable
in low pH, while most Bi salts are water-insoluble and only Bi(N0 3 ) 3 has a low solubility at low pH. Futhermore, Bi nanowires produced by this method are more likely
polycrystalline. It is also more difficult to control the doping level and to produce
a uniformly doped nanowire by the electrochemical deposition process because the
deposition rates for different chemical precursors are usually different. The templates
filled by vapor-phase deposition are typically curved due to the rapid cooling rate in
such process. More caution is required to produce doped nanowires by this approach
since the concentration of the dopant in the vapor phase is different from that in the
liquid phase, and the doping is very sensitive to the vapor pressure.
Considering the advantages and drawbacks of the various template-filling methods,
we decided to use the vacuum melting and pressure injection approach to synthesize
Te-doped Bi nanowires. In the following sections, detailed descriptions of the fabrication of anodic alumina templates and Te-doped Bi nanowire arrays will be presented.
26
2.2
2.2.1
Synthesis of Porous Anodic Alumina Templates
Al Substrate Preparation
In this section, we describe the detailed procedures for producing porous anodic
alumina templates (see Figs. 2-1(a)-(b)). We found that to produce anodic alumina
templates with regular, hexagonally-packed pores with uniform sizes, the substrate
has to be first polished mechanically and electrochemically to yield a highly smooth
surface [19].
High-purity aluminum film (99.997%, Alfa AESAR) with a thickness of 0.25 mm
was cut into rectangular pieces of the desired dimensions (typically 1 cm x 2.5 cm).
The aluminum foil was then placed between two glass slides and flattened under a
pressure of 1.0 x 106 psi. Next, the film was mechanically polished in three stages. The
Al film was first polished with a 600-grit sandpaper, followed by a special polishing
cloth (TEXMET 1000, Buehler) with 3-um diamond paste. In the final polishing
stage, Mastermate (Buehler), which is an aqueous dispersion of 20-nm SiO 2 particles,
was used as the polishing solution with Microcloth (Buehler).
After mechanical
polishing, the Al film has a dull mirror finish. It was then washed with de-ionized
water and acetone to remove residues of the polishing agent. Since aluminum oxidizes
readily when exposed to air, the polished aluminum film was annealed at 350'C for
1 hour in air to produce a uniform alumina layer on the polished substrate.
The experimental set-up for electrochemical polishing is shown in Fig. 2-2. The
aluminum film is placed at the anode and a platinum film of comparable size is placed
at the cathode as the counter electrode. The electrolyte used for this polishing process
was 95 vol% H 3 PO 4 (85%, Alfa AESAR) and 5 vol% H2 SO
4
(97%, Alfa AESAR) plus
20 g/l Cr0 3 (Alfa AESAR) [23]. The beaker containing electrolyte was placed in a
silicone oil bath and heated up to
-
85'C. The heater was connected to a temperature
controller so that the temperature of the electrolyte is maintained at 85'C throughout
27
(a)
Aluminum Film
Anodization
InIect'on
(b)
Pores
Al2Q
.
2
Barrier Layer
Al
PressureInjection
(C)
Te-doped Bi
Al and Barrier
Layer Removal
(d)
Te-doped Bi Nanowire Array
Template
Dissolution
(e)
iFree-standing
Wires
Figure 2-1: Schematic of the fabrication process of Te-doped Bi nanowire arrays
and free-standing nanowires: (a) the aluminum film after mechanical polishing and
electrochemical polishing, (b) porous anodic alumina template produced on an Al
substrate, (c) pressure injection of molten Te-doped Bi into the evacuated channels
of the porous template, (d) anodic alumina filled with Te-doped Bi detached from the
Al substrate, and (e) free-standing Te-doped Bi nanowires obtained by dissolution of
the anodic alumina template using a special etching solution.
28
V
Elec trolyte
A
A --
Thermocouple
Oil Bath
.
.
.
..
.Pt
Film
Al Film
... . . . .
....
Heater
Te pr.r
Controller
Figure 2-2: Schematic of the experimental set-up for electrochemical polishing. The
experimental set-up for the subsequent anodization process is similar, except that the
heater is replaced with a magnetic stirrer, and the oil bath is changed to an ice or
water bath.
the entire electrochemical polishing process. The optimum voltage used for electrochemical polishing was found to be 20 V [23], and the current was limited to < 2 A.
A larger initial current was observed to result in a non-uniform polished surface. The
current decayed gradually during the polishing process, which was stopped once the
current approached a steady value. The required polishing time was usually about
15-20 sec.
The duration of this intense reaction was found to be very critical to
the quality of the polished surface. If the Al substrate was over-polished, mm-sized
defects would appear on the substrate surface. When properly timed, the electrochemical polishing would yield a shiny mirror finish on the substrate surface.
2.2.2
Two-Step Anodization Process
The self-organized formation of porous structures in anodic alumina has been studied
for more than 40 year [24, 25]. The growth of the pores was found to be dependent
on parameters such as temperature, voltage, electrolyte solution, and anodization
duration. In order to achieve a greater regularity in the pore ordering and to improve
29
the pore size distribution, a two-step anodization process [26] was adopted for the
preparation of anodic alumina templates.
The experimental set-up for the anodization process was similar to that for the
electrochemical polishing shown in Fig. 2-2. The hot oil bath in Fig. 2-2 was replaced
by an ice bath or a water bath, since the temperature for the anodization process
is usually kept low. Before the anodization process, the polished substrate was first
dipped in a solution of 3.5 wt% H3 PO 4 (85%, Alfa AESAR) with 45 g/l Cr0
3
(Alfa
AESAR) for about 50 minutes to remove the oxide layer [23]. The typical electrolytes
used were 4 wt% oxalic acid (H 2 C 2 0 4 , Alfa AESAR) and 20 wt% H2 SO 4 , with anodization voltages of 30-65 V and 10-25 V, respectively. Since the anodization process is an
exothermic process, the local temperature on the electrode surface would rise during
reaction [15]. Therefore, the electrolyte solution was constantly but gently agitated
with a magnetic stir bar to disperse the heat generated, and the heater in Fig. 2-2
was replaced by a magnetic stirrer. To minimize the unwanted anodization reaction
occurring on the back side of the aluminum substrate, a piece of scotch tape was
attached to that surface.
Figures 2-3(a) and (b) represent the schematics of the side view and cross-sectional
view of the anodic alumina template, respectively [23]. The geometric structure of
the porous anodic alumina can be characterized by the cell size D, pore diameter
d and pore length (or film thickness) L. The cell size or interpore distance D is
found to be dependent on the anodization voltage V, and D is described by the
empirical relation: D (nm) = -1.7 + 2.81xV (volts) [27].
The pore growth rate
(dL/dt) is governed by the current I. An empirical relation has been established:
dL/dt (ptm/min) ~_ 0.03 x I/A (mA/cm 2 ) [23], where I is the total current and A
is the reaction area of the substrate. The current I is found to be a function of temperature T, electrolyte type, anodization voltage V and surface area A. The optimal
current density I/A is usually on the order of 10-20 mA/cm 2 . The pore diameter d is
also found to depend on the temperature, voltage, electrolyte type and anodization
time, and it can be enlarged by subsequent etching. It should be noted that a layer of
insulating anodic alumina, called barrier layer, is grown between the pore channels and
30
Pores
Cell Size D
Pore
-
A NN~z
Length......
A120_1
Barrier Layer
Al
(a)
(b)
Figure 2-3: Schematic illustrating the structure of anodic alumina template: (a) side
view, (b) cross-sectional view.
the Al substrate. The thickness of this barrier layer
dbarrie,
the anodization voltage V with an empirical relation:
is mainly determined by
dbarrier
(nm) ~- V (volts) [23].
At the initial stage of anodization, the pores grow randomly over the Al substrate.
The self-ordering of the pores proceeds with time after the process reaches a steady
state [16]. Therefore, ordered structure is not observed on the top surface of the anodic
alumina layer. However, it was found that locally ordered regions with a domain size
of
-
1 tm can be obtained on the bottom side of the anodic alumina template. The
difference in the structures of the top side and the bottom side implied that most of
the pores were not parallel to one another and the channels might combine or split
within the porous film.
It has been suggested that regularly ordered pores can be obtained also on the top
surface if the pore growth can be initiated with a textured surface that has the right
geometric structure corresponding to the optimum anodization conditions [16, 26, 28].
One way to produce such a textured pattern on the Al substrate is by using the
molding technique [16], whereby an array of hexagonally-ordered convex features is
generated on a SiC substrate by electron beam (EB) lithography. The patterned SiC
then serves as the master mold and is pressed onto the polished Al substrate, thereby
producing a regular array of concave features that would serve as the initiation points
for the pore growth. By this technique, an array of defect-free hexagonally-packed
31
pores has been synthesized [16].
However, due to the limitation of the processing
area and the writing speed of EB lithography, the textured area is usually limited
to several mm 2 , which is too small for our purpose.
Another promising method
to improve the pore structure regularity in anodic alumina is the so-called two-step
anodization [26]. The principle behind this approach is similar to the molding process
discussed above, whereby a textured pattern is used to facilitate the growth of ordered
pores. In the two-step anodization process, the anodic alumina layer was completely
removed after the first anodization to produce an Al substrate with a self-textured
surface, which resulted from the curvature of the interface between the barrier layer
and the underlying Al substrate (see Fig. 2-4). The textured Al substrate was then
re-anodized under the same conditions as the first anodization step so that ordered
pores could grow with the aid of the textured surface. Although it is not possible to
produce a large array of defect-free ordered pores in the two-step anodization process
as in the SiC molding process, the regularity of the pore structure and the pore
size distribution are dramatically improved by the second anodization. The two-step
anodization method is also capable of producing a large area of ordered pores more
economically and easily. Therefore, we decided to use this approach to prepare the
anodic alumina templates for pressure injection of Bi nanowires.
In our process, the anodic alumina layer from the first anodization was dissolved
at room temperature in 3.5 wt% H 3 PO 4 with 45 g/l Cr0 3 , which is exactly the same
acid solution used for removing the aluminum oxide surface layer before the first
anodization process. The etching rate of the anodic alumina is not constant and the
etching process usually proceeds overnight to ensure that the anodic alumina layer
is removed completely. The porous anodic alumina film from the first anodization
process appears as a transparent surface layer; the shiny metallic surface of the Al
substrate is restored after removal of the anodic alumina layer.
The pore diameter d of the anodic alumina from the second anodization is usually
slightly larger than that from the first anodization.
The pore size of the anodic
alumina can be further enlarged by dipping the template in a second acid solution.
For templates anodized in 4 wt% H2 C 2 04, 5 wt% H3 PO 4 was used for the pore
32
(a)
Polished Al Film
FirstAnodization
Template Surface
(b)
Porous Anodic
.. .. ..
Y'Y'Y
..
'YY
Removal ofRemovl
Anodic Alumina
Layer
of'Al
Alumina
-Barrier Layer
Substrate
(c)
Second Anodization
(d)
I
I
l
Figure 2-4: Schematic of the two-step anodization process: (a) a mechanically and
electrochemically polished Al substrate, (b) porous anodic alumina layer produced in
the first anodization, (c) textured Al substrate after the removal of anodic alumina
layer, and (d) porous anodic alumina template produced in the second anodization.
33
enlargement; while for templates prepared in 20 wt% H 2 SO 4 , another 20 wt% H 2 SO
4
solution was used as the pore etching agent.
The underlying Al substrate can be etched away by a solution of 0.2 M HgCl 2
(Alfa AESAR) at room temperature to produce a free-standing porous alumina thin
film, which has open pores on the top side and a barrier layer on the bottom side. The
barrier layer can also be etched away with the acid solution used for pore enlargement
(as described in the last paragraph). The time required to etch through the barrier
layer is determined by the empirical relation: t (min) ~
1.5 x V (volts) for templates
anodized in H 2 C 2 0 4 , and t (min) ~_ 3.0 x V (volt) for templates anodized in H 2 SO 4.
During the etching process, the top side of the template is protected by mounting on
a glass slide using paraffin (Crystal Bonder, Buehler). After the etching process, the
template could be recovered by using acetone to dissolve the paraffin.
2.2.3
Experimental Results
The surface structure of the porous anodic alumina was studied with a JOEL 6320
high-resolution scanning electron microscope (SEM). Figure 2-5 shows the SEM images
of the top surfaces of anodic alumina templates after a first anodization in two
different electrolytes (see Table 2.1). As shown in Fig. 2-5, the pores are randomly
distributed over the top surfaces of both templates. Figure 2-6 shows the SEM image
of the bottom side of an anodic alumina template after a first anodization.
The
Al substrate and part of the barrier layer have been etched away. A more regular
hexagonally-packed pore structure and a narrow pore size distribution are noted at
the bottom side of the template.
Figure 2-7 shows the SEM images of the top surfaces of anodic alumina templates
after the second anodization process.
The processing conditions for the second
anodization process for templates (a) and (b) shown in Fig. 2-7 are listed in Table 2.2,
and the conditions used in the first anodization of these templates are identical to
those listed in Table 2.1, respectively.
It was found that the current in the first
anodization process usually varied with time and depended on the surface condition
34
Figure 2-5: SEM images of the top surfaces of porous anodic alumina templates after
a first anodization in (a) 4 wt% H 2 C 2 0 4 and (b) 20 wt% H 2 SO 4 . The average pore
diameters in (a) and (b) are 26nm and 12nm, respectively.
Table 2.1: The first anodization conditions for anodic alumina templates shown in
Fig. 2-5.
Sample
Electrolyte
Fig. 2-5(a)
Fig. 2-5(b)
4 wt% H 2 C 2 0 4
20wt% H2 SO 4
Voltage
(V)
45.0
20.0
35
Temp.
('C)
19
0-11
Current
(mA/cm 2 )
16-17
18-37
Duration
(minutes)
90
85
Barrier
Layer
Figure 2-6: SEM image of the bottom side of an anodic alumina template after the
barrier layer was partially etched away. The sample has been subjected to a first
anodization in 4 wt% H2 C 2 0 4 at 45 V for 180 minutes.
Table 2.2: The second anodization conditions for anodic alumina templates shown in
Fig. 2-7.
Sample
Fig. 2-7(a)
Fig. 2-7(b)
Electrolyte
4wt% H 2 C 2 0 4
20 wt% H 2 SO 4
Voltage
(V)
Temp.
(0 C)
45.0
20.0
19
0-13
(mA/cm )
Duration
(minutes)
Pore Enlargement
(minutes)
11
22-32
250
120
0
20
Current
2
of the Al substrate. However, the current was found to be fairly stable throughout
the second anodization process and was reproducible with different samples.
Figure 2-7 shows that the pore packing order and the pore size distribution are
dramatically improved by using the two-step anodization technique. We note that
the pore size after the second anodization process was always larger than that after
the first anodization process.
Figure 2-8 shows a low-magnification SEM image of
the top surface of the anodic alumina template presented in Fig. 2-7(a).
The pore
arrangement is represented by ordered domains of several micrometers for templates
anodized in H 2 C 2 0 4 . The templates anodized in H 2 C 2 0
4
usually have a more ordered
pore packing compared to those anodized in H 2 SO 4 ; a longer first anodization period
is necessary to produce a comparable pore regularity for templates anodized in H 2 SO 4 .
36
Figure 2-7: SEM images of the top surfaces of porous anodic alumina templates after
the second anodization in (a) 4 wt% H 2 C 2 0
4
and (b) 20 wt% H 2 SO 4 . The average
pore diameters in (a) and (b) are 44 nm and 18 nm, respectively.
37
Figure 2-8: A lower magnification SEM image of the top surface of the porous anodic
alumina template presented in Fig. 2-7(a) after the second anodization.
2.3
2.3.1
Preparation of Te-doped Bi Nanowires
Preparation of Te-doped Bi Alloy
To produce Te-doped Bi nanowires, we used a technique similar to that developed
for synthesizing pure Bi nanowires, replacing the pure Bi metal by the Te-doped Bi
alloy in the pressure injection process.
The Te-doped Bi alloy was prepared by mixing the desired amount of high-purity
Bi (99.9999%, Alfa AESAR) and Te (99.999%, Alfa AESAR) in a quartz tube, which
was then connected to a vacuum pump and evacuated to
-
10-6 torr. To retain a
high vacuum in the quartz tube, the open end of the quartz tube was fused and
sealed while the vacuum pump was still connected. The Bi-Te mixture was melted
and maintained at 600'C for 24 hours to allow the system to reach equilibrium. To
prevent possible Te segregation from Bi, the alloy melt was then quenched to room
temperature with cold water. This vacuum sealing-melting-quenching technique was
found to be effective at producing oxygen-free Bi-Te alloy.
38
Thermocouple
Pressure Gauge
High-Pressure Ar
or Vacuum Pump
Screw
Pressure
Reactor
Glass
Beaker
Heater
Te-doped
Bismuth
Template
Figure 2-9: Schematic of the experimental set-up for pressure injection of Te-doped
Bi alloy into the pores of an anodic alumina template [19].
2.3.2
Vacuum Melting/Pressure Injection Process
The anodic alumina template was washed intensively with de-ionized water followed
by acetone, and then dried in air. With the Al substrate still attached to its bottom
side, the anodic alumina template was placed in a glass beaker with the as-prepared
Bi-Te alloy. Since Bi has a higher mass density than the alumina template, two pure
Al wires (99.999%, Alfa AESAR) with a diameter of 1.0 mm and a pure Cu wire
(99.999%, Alfa AESAR) were used to help fix the template to the bottom of the glass
beaker while the Bi-Te alloy was melted. The Al wires were used here because Al
has essentially zero solubility in liquid Bi at the temperatures used in the subsequent
pressure injection process (T < 325'C). Cu wire was used because Cu has a small
solubility (-
0.2 at%) in liquid Bi at 325 C, and the introduction of Cu impurity in
liquid Bi could reduce the surface tension of Bi, thus facilitating the injection of Bi
into the nanochannels of the anodic alumina template [23]. Since Cu has essentially
zero solubility in a solid Bi crystal, Cu atoms would be pushed out of the Bi-Te lattice
when the alloy crystallized. Consequently, the resulting Te-doped Bi nanowires would
be free of Cu impurities, and their electrical properties should not be affected by this
processing.
Figure 2-9 shows a schematic of the experimental set-up for pressure injection. The
39
chamber employed was a Parr high-pressure reactor that could sustain a pressure of up
to
-
5500 psi. It was attached to a vacuum pump that could achieve 102 mbar, and
was heated to 240 C for 8-10 hours to evacuate the channels of the alumina template.
After degassing, the temperature was slowly ramped (-
1 C/min) to 325 0 C to melt
the Bi-Te alloy, which has a melting point of 271'C. The reactor was maintained at
this temperature for two hours for the system to reach equilibrium. Before pressure
injection, the vacuum pump was removed from the chamber, and a high-pressure Ar
gas cylinder (6000 psi, BOC Gases) was connected to the inlet valve of the chamber
instead. Ar gas was then introduced into the chamber to increase the pressure to
4500 psi for the pressure injection of liquid alloy into the nanochannels of the anodic
alumina template. After 5 hours, the chamber was slowly cooled down to room temperature over a 12-hour period, and the Bi-Te alloy was allowed to crystallize within
the nanochannels. The slow cooling process was found to be critical towards producing single-crystalline nanowires. After the reactor was cooled to room temperature,
the high-pressure Ar gas was slowly released from the chamber.
2.3.3
Subsequent Processing
After pressure injection, the alumina template was buried in a large piece of BiTe alloy. A method has been developed to recover the Bi-Te-filled alumina template
without breaking it into small pieces. First, the majority of the alloy block was shaved
off slowly with a diamond wire saw so that the resulting piece had a thickness of only
1 mm. This was then mounted on a glass slide with the alumina template protected
by paraffin. The excess alloy was removed by mechanical polishing using a 2400-grit
sandpaper with water as the coolant. After the polishing process, the paraffin can be
dissolved with acetone. The template recovered in this fashion usually had a layer
of Bi-Te alloy with a thickness of several micrometers. This thin layer of alloy was
found to be useful in making electrical contacts for the nanowire array.
In cases where the Al substrate remained attached to the anodic alumina template
after the sample recovery, the Al substrate was etched away with a 0.2 M HgCl 2
40
solution. The alumina barrier layer that capped the ends of the nanochannels on
the bottom side of the anodic alumina template could be etched away using 5 wt%
H3 PO 4 or 20 wt% H 2 SO 4 for the template anodized in oxalic acid and sulfuric acid,
respectively. The etching time depended on the thickness of the barrier layer. It was
found that the time needed to dissolve the barrier layer was about twice as long for
the Bi-Te-filled templates than for the as-prepared unfilled templates.
2.3.4
Preparation of Free-Standing Nanowires
The free-standing nanowires can be obtained by dissolving the anodic alumina template in an acid solution of 3.5 vol% H3 PO 4 plus 45 g/l Cr0
3
at 20 C. After the
template was completely dissolved in the acid solution, de-ionized water was added
to dilute the etching solution. After repeated dilution with de-ionized water, ethanol
was added to the solution. Since the free-standing nanowires tended to agglomerate,
the solution was sonicated to disperse the nanowires evenly in the ethanolic solvent.
Te-doped Bi nanowires were found to be stable when embedded in the anodic
alumina matrix. In contrast, the free-standing nanowires were oxidized easily when
exposed to air. Even when dispersed in ethanol, the free-standing nanowires can
become slowly oxidized. The oxide layer on the Bi-Te nanowire can be dissolved in
a 1:10 HF-ethanol solution. Recently, the technique used for studying the transport
properties of a single carbon nanotube using nanocontacts with electron beam (EB)
lithography has been extended to the investigation of a single Bi nanowire [29]. The
transport properties and electronic density of states of the quasi-iD system can be
revealed by directly examining the electronic properties of a single nanowire.
41
2.4
2.4.1
Characterization of Te-doped Bi Nanowires
SEM Study of Te-doped Bi Nanowire Arrays
To ensure that the Bi-Te alloy was injected as nanowires from the top surface all the
way to the bottom side of the anodic alumina template, the bottom surface of the
Te-doped Bi/A120 3 nanocomposite was examined by SEM after removal of the barrier
layer. Shown in Figs. 2-10(a) and (b) are two SEM images of nanowire arrays with
average wire diameters of 40 nm and 17 nm, respectively. We note that for sample
(a), the template was prepared in 4 wt% H2 C2 0 4 and most of the channels were filled
with Bi-Te alloy. However, for sample (b), whose alumina template was anodized in
20 wt% H2 SO 4 and has a smaller channel diameter, we did not observe channel filling
of Bi-Te alloy to the bottom surface. There are two major reasons for the dramatic
difference in the nanowire filling for the two samples. First, since liquid Bi has a high
surface tension
(-
~ 375 dyne/cm), a high pressure was required to force the liquid
Bi into the nanochannels of the anodic alumina template. From a simple estimation
of fluid dynamics, the pressure required to inject liquid into a cylindrical channel is
given by
pa >_ 471, cos 0
d
(2.1)
where Pa is the external pressure, d is the channel diameter, 0 is the contact angle,
and -yl, is the liquid-vapor surface tension [30]. Without the use of any wetting agent,
the contact angle 0 ~ 1800 for a ceramic-liquid metal interface since the metallic
bonding energy is much stronger than the metal-ceramic van der Waals interaction
energy. Using Eq. (2.1), the smallest channel diameter that can be filled with Bi in
our set-up (~ 5000 psi) is about 43 nm. A higher pressure of > 10000 psi is required
to inject liquid Bi into channels with of d < 20 nm. In addition, with smaller channel
diameters, it is more probable that blockage by impurities would prevent the pores
from being completely filled.
Figure 2-11 shows another SEM image of the bottom surface of Te-doped Bi
nanowire array with an average wire diameter of 40 nm. We note that more than
42
100 nm
Figure 2-10: SEM micrographs of the bottom surface of nanowire arrays after the
barrier layer has been removed. The alumina template used in (a) was prepared in
4 wt% H2 C2 0 4 and has an average pore diameter of ~ 40 nm. The alumina template
in (b) was prepared in 20 wt% H2 SO 4 and has an average pore diameter of - 17 nm.
In sample (a), most of the pores have been thoroughly filled with Bi-Te alloy (shown
as bright spots); while in sample (b), pore filling was not extended to the bottom
surface.
43
Figure 2-11:
barrier layer
H2 C2 0 4 and
out from the
SEM micrograph of the bottom surface of a nanowire array after the
has been removed. The alumina template used was anodized in 4 wt%
has an average pore diameter - 40 nm. Some of the nanowires spread
pore region in the form of "mushrooms".
80% of the pores were filled all the way to the bottom side of the template and the
majority of the Bi wires have the same diameter and length as the pores. Some of the
Bi nanowires have larger size terminations, which we called "mushrooms", spreading
out from the pore region onto the bottom side of the template. The formation of these
features could be attributed to the 3.2% volume expansion of Bi during the liquidsolid phase transition. The alloy inside the nanochannels might have broken through
the amorphous alumina barrier layer during solidification, forming mushroom-like
features. These features could be very useful in making nano-electrical contacts with
nanowires in the array since they provide a larger contact area for each nanowire.
2.4.2
XRD Study of Te-doped Bi Nanowire Arrays
The crystal structure of the Te-doped Bi nanowires was investigated by X-ray diffraction (XRD) with the reflection plane perpendicular to the wire axis.
The XRD
studies were performed on a 0 - 0 diffractometer (Siemens D5000, 45 kV, 40 mA, Cu44
0
0
Figure 2-12: Schematic of the set-up for XRD experiments, illustrating the orientation
of the nanowire array relative to the X-ray beam.
Ka) using an experimental set-up depicted in Fig. 2-12. Before XRD experiments
were performed, the nanowire arrays were thermally annealed in vacuum at 2000 C
for 8-10 hours to release the high stress built up within the nanowires.
Figure 2-13 shows the XRD patterns of pure Bi nanowire arrays of various diameters. All the diffraction peak positions observed for these samples were similar to
those of polycrystalline bulk Bi, indicating that the crystal structure of bulk Bi was
preserved in these nanowires. However, differences were noted in the relative XRD
peak intensities for nanowires of different diameters, which implied that there were
preferential crystal orientations in the nanowire arrays, depending on the wire diameters. By normalizing the areas of the XRD peaks in Fig. 2-13 to the peak intensities
of a polycrystalline bulk Bi standard, we found that for the 40-nm Bi nanowire array,
more than 80% of the wire axes were oriented along a direction perpendicular to the
(012) lattice plane; whereas for the 95-nm Bi nanowire array, the dominant crystal
orientation is perpendicular to the (202) lattice plane.
These preferential crystal
orientations of Bi nanowire arrays were consistent with those observed by Zhang and
co-workers [20, 23, 31]. They reported that for the 65-nm and 109-nm Bi nanowire
arrays prepared by the pressure injection process, more than 90% of the wires were
oriented perpendicular to the (202) lattice plane, while for the 23-nm Bi nanowire
array, more than 99% of the nanowires were oriented perpendicular to the (012) lattice
plane. Collectively, these observations suggested that for Bi nanowires with diameters
> 60 nm, most of the wires were oriented perpendicular to the (202) lattice plane;
45
(202)
(c)
(012)
(110)
(024)
(b)
(a)
20
30
40
20 ()
50
60
Figure 2-13: XRD patterns of Bi/anodic alumina nanocomposites with average Bi
wire diameters of: (a) 40 nm, (b) 52 nm, and (c) 95nm. The XRD peaks of all three
samples appeared at the same positions as those of the polycrystalline Bi standard.
The Miller indices corresponding to the lattice planes of bulk Bi are indicated above
the individual peaks.
whereas for nanowires with smaller diameters (< 50 nm), the preferential growth
direction was perpendicular to the (012) plane. For Bi nanowires with intermediate
diameters (e.g. 52 nm in Fig. 2-13(b)), three major crystal orientations were observed,
which were perpendicular to (012), (110) and (202) lattice planes. The existence of
more than one dominant crystal orientations in the 52-nm Bi nanowire array might be
attributed to the transitional behavior as the preferential crystal orientation shifted
in direction perpendicular to the (202) and (012) planes.
Figure 2-14 shows the XRD patterns of the Bi nanowire arrays with different Te
dopant concentrations. The wire diameters of the three samples in Fig. 2-14 were
all about 40 nm. We note that the XRD peak positions as well as the relative peak
intensities of the Te-doped Bi nanowire arrays were similar to those of the undoped
Bi nanowire array, indicating that the small amount of Te dopants did not affect
the crystal structure of Bi in the former, and the dominant crystal orientation along
the wire axes was perpendicular to the (012) lattice plane for these 40-nm pure and
46
(0 2)
(024)
(202)
(c)
CO
a
(b)
c
U,
C
(a)
20
30
40
50
60
20 (()
Figure 2-14: XRD patterns of 40-nm Bi nanowire arrays containing (a) no Te, (b)
0.075 at% Te, and (c) 0.15 at% Te. The XRD peaks of all three samples appeared
at the same positions as those of the polycrystalline Bi standard. The Miller indices
corresponding to the lattice planes of bulk Bi are indicated above the individual peaks.
Te-doped Bi nanowire arrays.
XRD study of the pure and Te-doped Bi nanowires indicated that the rhombohedral crystal structure of Bi was preserved. This result has important implications
for transport-related experiments, since it suggests that the unique band structure and
electronic properties of bulk Bi are retained by the pure and Te-doped Bi nanowires.
Since the carrier pockets of Bi are highly anisotropic, the preferred crystal orientation
provides valuable information on the electronic states in the Te-doped Bi nanowire
systems. The directions perpendicular to the (202) and (012) lattice planes in real
space are [1011] and [0112] (using the space group R-3m and the hexagonal structural
description for Bi), respectively. As will be discussed in Sec. 3.2, the crystal structure
of Bi is characterized by three orthogonal principal axes that are defined as the
binary, bisectrix, and trigonal axes. The two preferential directions for Bi nanowires
of different wire diameters both lie in the trigonal-bisectrix plane, with the
[1011]
and
[0112] directions making an angle of 71.870 and 56.50 to the trigonal axis, respectively.
47
Chapter 3
Theoretical Modeling of ID Bi
Quantum Wires
3.1
Introduction
The study of low-dimensional systems has attracted much research interest since
the advent of the molecular beam epitaxy (MBE) technique, which made possible the production of high-quality 2D systems, i.e. quantum well superlattices
[91.
Due to quantum confinement in one of the dimensions, these 2D systems exhibited very different properties from their bulk counterparts. The quantum effects are
expected to be even more pronounced in quantum wire (ID) or quantum dot (OD)
systems. The study of low-dimensional systems has enabled us to understand many
interesting physical properties and uncover some important phenomena, such as the
quantum Hall effect.
In addition, in some practical applications, such as thermo-
electric power generation and refrigeration, low-dimensional systems are predicted to
provide superior performance to the conventional bulk materials
[5].
Recently, there is increased interest in examining ID systems due to rapid advances
in the fabrication of quantum wires [17, 18, 19, 21].
Bi nanowires are especially
intriguing because Bi has very unique electronic properties that make it a promising
material for transport study of ID systems, as well as a potential material for novel
thermoelectric applications. However, due to the complicated band structure of Bi,
48
the theoretical investigation of ID Bi systems has been very challenging. Progress
in understanding the electronic states and transport properties of Bi quantum wires
has been made recently [31, 32, 33]. However, due to the special geometric configurations (circular wire cross-section and high length/diameter aspect ratio) and the
highly anisotropic carrier pockets in Bi, several approximations were made in the
previous calculations to derive the electronic states and transport-related properties
of Bi nanowires.
In this chapter, a solution to the complicated band structure of Bi quantum wire
systems is presented, which explicitly takes into account the circular wire boundary
conditions for electron wavefunctions and the anisotropic carrier effective masses in
Bi. In my calculations, the non-parabolic features of the L-point conduction band and
the temperature dependence of various band parameters are also included to provide
a more realistic model of Bi nanowires.
3.2
Crystal and Band Structures of Bi
Bi is a semimetal with two atoms in a primitive unit cell. The crystal structure of Bi
can be visualized by taking two interpenetrating face-centered cubic (fcc) lattices, and
stretching the cubic lattices along the diagonal axis with a small relative displacement
between the two slightly distorted fcc lattices along their body diagonal, as shown in
Fig. 3-1. The two angles shown in Fig. 3-1 are a' = 87.57 and a = 57.23', whereas
&' = 90' and a = 60 for an ideal undistorted fcc structure
[34].
The basis vectors
for the Bi structure are given in Fig. 3-1 as tj (i = 1, 2, 3). The crystal structure of
Bi is also described as rhombohedral, which can be conveniently presented in terms
of a hexagonal unit cell with ao = 4.5460
A
and co = 11.862
A
[34]. In the Bi crystal
structure, there are three principal symmetry axes: trigonal, binary, and bisectrix, as
indicated in Fig. 3-1. Due to the deformation of the Bi crystal structure with respect
to the fcc lattice, the trigonal axis is along the body diagonal and has a three-fold
symmetry. There are three two-fold binary axes perpendicular to the three mirror
49
trigonal (z)
mirrorplane
bisectrix (y)
binary (x)
Figure 3-1: The crystal structure of Bi, which can be derived by distorting two interlaced fcc lattices along the body diagonal, followed by a small relative displacement
of the two fec lattices along the trigonal axis. The bisectrix axis lies on the mirror
plane, and is perpendicular to the binary and trigonal axes.
planes spanned by the trigonal axis and one of the basis vectors t, (see Fig 3-1). The
three bisectrix axes lie in the mirror planes, and are perpendicular to the trigonal
and binary axes. It is customary to take the x, y and z directions as parallel to
the binary, bisectrix and trigonal axes, respectively, thereby forming an orthogonal
coordinate system.
The Brillouin zone of Bi is similar to that of a fcc lattice with a little compression
along one of the three-fold axis, as shown in Fig. 3-2. Due to the small distortion
along the body diagonal, the T point in the Brillouin zone of Bi is not equivalent
to the three L points. The principal symmetry axes corresponding to the trigonal,
binary and bisectrix directions are also shown in Fig. 3-2.
In Bi, there are three electron pockets at the L points of the Brillouin zone and
one hole pocket at the T point (see Fig. 3-2). Figure 3-3 shows the schematic diagram
of the band structure at the L point and T point near the Fermi energy level. The
deformation from the cubic lattice is responsible for the different band structure at
the L point and the T point in Bi. The small band overlap AO (which is 38 meV at
0 K [36]) between the L-point conduction band extremum and the T-point valence
50
trigonal (z)
[0001]
hole pocket
QI
LC)
electron pocket (A)
LA
bisectrix (y)
[1010]
binlary (x)
L (B)
[1210]
Figure 3-2: The Brillouin zone of Bi, showing the Fermi surfaces of the three electron
pockets at the L points and one T-point hole pocket. The total number of electrons
is equal to that of holes. The mirror plane going through the L(A) pocket and the
T-point holes is indicated by dashed lines.
,-AO1
EgL
T
L
Figure 3-3: Schematic of the Bi band structure at the L point and the T point near the
Fermi energy level, showing the band overlap AO of the L-point conduction band and
the T-point valence band. The L-point electrons are separated from the L-point holes
by a small bandgap EgL. At T = 0 K, AO = -38 meV and EL = 13.6 meV [35, 36].
51
band extremum causes electrons to be transferred from the top of the T-point valence
band to the bottom of the L-point conduction band, which is lower in energy, thereby
forming an equal number of holes and electrons in Bi. The hole ellipsoid at the T
point can be characterized by the effective mass tensor,
0
0
0
Mh2
0
0
0
Mh3
/MhI
Mh
where min
= mh2
due to symmetry, and nh3 >
)
mhl,
(3.1)
indicating a large anisotropy in
the hole Fermi surface. The band structure near the T point can be approximated
by a parabolic dispersion relation,
h2
ET(k) = ETo -k
2
-ah - k,
where ETO is the band edge of the T-point valence band extremum, and a
(3.2)
1
= Mi-
is the inverse mass tensor of the T-point holes. The three L-point electron pockets
are even more anisotropic and the effective mass tensor has the form
Me1
Me,A
("0
m
K0
0
0
e2
me 4
ne4
ine3
(3.3)
for the electron pocket with its longest axis oriented near the bisectrix direction
(labeled by A in Fig. 3-2). The presence of a positive off-diagonal element
in
4
[37]
indicates that the longest axis of the electron pocket is tilted towards the trigonal (z)
direction with respect to the x-y plane by 0 ~ 60, which is determined by
=
2 tea2
(tan 2me4
-
me3 /
.
(3.4)
The dispersion relations for the L-point conduction band, however, are more complicated. Due to the strong coupling between the L-point electrons and holes, which
52
are separated by a narrow direct bandgap (~
13.6 meV at T = 0 K [35]), the parabolic
band structure used to describe the T-point holes is not appropriate for the L-point
bands. The band structure at the L point is best described phenomenologically by
the two-band Lax model, which is based on k - p perturbation theory [38, 39, 40].
Taking the energy at the band edge of the L-point conduction band to be zero, the
Lax model gives non-parabolic dispersion relations
E
~E12h 2
+ 22k-a-k
k
~
EE(k) = -
2
I
\
(3.5)
EgL
for the L-point electrons (- sign) and L-point holes (+ sign), where EgL is the direct
L-point bandgap and ae is the inverse mass tensor at the conduction (or valence)
band extremum and has the form
ace = me
for the electron pocket A (see Fig. 3-2).
a0
0
0
a2
a4
0
a4
a3
(3.6)
The inverse effective mass tensors for the
other two electron ellipsoids can be obtained by rotations of ±120 about the trigonal
(z) axis. It can be shown that for energies very close to the L-point conduction band
edge, the non-parabolic band given by Eq. (3.5) becomes parabolic,
EL(k)
ii2
2
kae
k.
However, this parabolic approximation is valid only for E <
(3.7)
EgL, and the full ex-
pression of Eq. (3.5) must be used for E > EgL.
The values for important band structure parameters of Bi are listed in Table 3.1
for T = 0 K. It should be noted that these band structure parameters are strongly
temperature-dependent for temperatures above 80K [41, 42] (see Table 3.2).
Using
the electron effective mass tensor components given in Table 3.1 for the L(A) pocket
at T = 0 K, the electron effective mass tensors for the L(B) and L(C) pockets at the
53
Table 3.1: Band structure parameters of Bi at T = 0 K.
Parameters
Band Overlap
Bandgap at the L point
Notation I
A0
EgL
Value
-38 meV
13.6 meV
Electron Effective
mel
0.00119 mO
Mass Tensor Elements
at the Band Edge for
me2
me3
0.263 mo
0.00516 mo
the L(A) pocket
me 4
0.0274 mo
T-point Hole Effective
Mass Tensor Elements
at the Band Edge
mhl
mh2
mU3
0.059 mo
0.059 mo
0.634 mo
Reference
[36]
[35]
[7]
[36]
Note: mo is the free electron mass.
Table 3.2: Temperature dependence of the band structure parameters of Bi.
Parameters
Temperature Dependence
-38 (meV)
Band Overlap [41]
Ao =
(T < 80K)
-38 - 0.044(T - 80)
+4.58 x 10-4(T - 80)2
-7.39 x 10 6 (T - 80)3 (meV)
Direct Bandgap [42]
EgL = 13.6 + 2.1
L-point Electron
(me(T))
Effective Mass
-
(T > 80K)
10-3T + 2.5 x 10- 4 T 2 (meV)
(me (0)) i
1 - 2.94 x 10- 3 T + 5.56 x 10- 7T 2
Components [7, 42]
Note: me(0) = effective mass tensor at 0K.
54
x
(3.8)
(3.9)
(3.10)
band edge are calculated as :
me(B)
me(c)
=
=
0.1975
0.1134
-0.0237
0.1134
0.0666
-0.0137
-0.0237
-0.0137
0.0052
0.1975
-0.1134
0.0237
-0.1134
0.0666
-0.0137
0.0237
-0.0137
0.0052
jm
0
(3.11)
mo
(3.12)
)
in Cartesian coordinates.
3.3
Electronic States in Bi Nanowires
Due to the small electron effective mass components in Bi, the quantum confinement
effects are more prominent in Bi nanowires than in other materials at a given wire
diameter. Since the electrons in the nanowires are restricted in directions normal to
the wire axis, the confinement causes the energies associated with the in-plane motion
to be quantized, and the lowest energy level is approximated by
AE ~
7r2 h2 2
(3.13)
M*d2
where m* is the in-plane effective mass of the electrons, and d is the wire diameter.
Since motion is only allowed along the wire axis, the electrons would behave like a
ID system with a dispersion relation of
Enm(ki) = Enm +
2mk*1,
(3.14)
where Enm represents the quantized energy level labeled by two quantum numbers
(n, m), k, is the wavenumber of the electron wavefunctions traveling along the wire
axis, and m* is the effective mass for electrons moving along the wire. We note that
for materials such as Bi with highly anisotropic carriers, the mass m* that determines
55
the subband energies Enm can be very different from the mass m* that characterizes
the motion along the wire.
In the nanowire system, the quantized subband energy Enm and the transport
effective mass m* along the wire axis are the two most important parameters that
determine almost every electronic property of this unique ID system.
Due to the
highly anisotropic electron and hole pockets, calculation of the band structure of
Bi nanowires was found to be very challenging. In the first theoretical modeling of
Bi nanowires, the quantized energy levels were evaluated by Zhang et al. using a
cyclotron mass approximation [31, 33]. In this case, the in-plane mass perpendicular
to the wire axis was approximated by the cyclotron effective mass given by
MP*,M*
=
detM,
"-
1
/2
,
(3.15)
(I- Me - 1
where 1 is the unit vector along the wire axis, and the transport effective mass is
approximated by
rnz*
. Me - 1
(3.16)
for a parabolic electron pocket. In this early model [31], parabolic dispersion relations were assumed for both electrons and holes.
With these approximations, the
quantized energy levels could be readily derived by solving the 2D Schr6dinger equation with a circular boundary condition [31]. However, the cyclotron effective mass
approximation is an oversimplification for the in-plane effective mass, and is not
valid for electrons with a highly anisotropic in-plane effective mass tensor. Instead of
mim2 for the average in-plane effective mass m*, where m, and m 2 are the
two principal mass components in the plane normal to the wire axis, the quantized
energy states Enm are to a much closer approximation determined by m* given by
1
1 1
m*
2 m,
- ~ -
+
1
M2
(3.17)
The discrepancies between m* and m* can be very significant if mi and m 2 are very
different, as they are for bismuth.
56
Subsequently, an improved model to describe the electronic states in Bi nanowires
was developed by Sun et al.
[32,
43], based on a square wire approximation for wires
oriented along the three principal axes. With the edges of the square wire set to be
parallel to the principal axes of the in-plane mass components for each carrier pocket,
the square cross-section boundary condition greatly simplified the eigenvalue problem
for solving the Schr6dinger equation, and allowed us to obtain analytical solutions for
the quantized energy levels. In this case, the transport effective mass along the wire
axis was derived as
m*=(l M-1 .
(3.18)
for a parabolic carrier pocket. Although the square wire approximation provided a
better solution for the in-plane effective masses and the quantized energy states Enm
than the previous cyclotron effective mass approximation, further improvements are
needed to describe the actual circularwires used in the experiments, and their proper
symmetries. In a wire with circular cross-section, the physical quantities should be
invariant in rotations about the wire axis. However, in the square wire approximation,
the quantized subband energies were highly dependent on the rotations of the square
cross-section about the wire axis, and therefore, the subband energies were not welldefined in this approximation. In addition, the different expressions of Eqs. (3.16)
and (3.18) for the ID transport effective mass also gave rise to very different values
for wires oriented along directions other than the principal axes of ellipsoidal pockets.
Thus, further investigation and clarification of the transport effective mass m* along
the wire axis are needed.
3.3.1
Solution to the Schr6dinger Equation for Bi Nanowires
In this section, solutions to the quantized energy levels and the transport effective
mass along the wire axis will be derived for the 1D Bi quantum wire system, taking
into consideration the anisotropic effective masses and the circular wire cross-section.
As an illustration, we first focus on the T-point holes that have a simple parabolic
57
dispersion relation; solutions to the non-parabolic L-point carriers will be dealt with
in Sec. 3.3.3.
For an infinitely long wire with a circular cross-section of diameter d, the z' axis is
taken to be parallel to the wire axis with the x' and y' axes lying in the cross-sectional
plane of the wire. Since the wire axis is allowed to be oriented along an arbitrary
direction with respect to the crystallographic directions, the effective mass tensor of
an anisotropic carrier pocket in the wire coordinates (X', y', z') has the general form
12
M 13
M 21
M 22
M23
Tn
in 3 2
in 3 3
n 1 1 iM
=
M
31
,
(3.19)
where mij =n.... The dispersion relation of the carrier pocket is then written as
E(k) =
/i2
2
k - a - k,
(3.20)
where
a -M
-
all
OZ12
a 21
a2 2 a2 3
a
a
31
32
Ce13
a
(3.21)
33
is the inverse mass tensor. From the effective mass theorem [44], the envelope wavefunction of electrons J(r) is described by the Schr6dinger equation,
h
-
2
2
V
VI(r)
VTa
ET(r),
(3.22)
which is obtained by replacing k with (-iV) in Eq. (3.20). For Te-doped Bi nanowires
embedded in the anodic alumina matrix, electrons are well confined within the wires
due to the very large bandgap of alumina. Thus, to a good approximation, the
electron wavefunction I(r) can be assumed to vanish at the wire boundary.
We first utilize the symmetry properties of the circular wire to simplify the differential equation, Eq. (3.22). Since the wire has cylindrical symmetry, all the physical
58
quantities are invariant under rotations about the wire axis z'. Therefore, with an
appropriate rotation about the z' axis, we can always make the matrix elements
a 12 =a
21
= 0.
Physically, the effect of this particular rotation is equivalent to
choosing a proper set of x' and y' axes so that the ellipse obtained by projecting the
constant energy ellipsoid onto the x'-y' plane will have major and minor axes parallel
to the x' or y' axes. With a 12 = a 21 = 0, the Schr6dinger equation can be rewritten
as
1929
qf
all-
+ a 22
qj29
229
+
OX12 Oy 0
2a
qj29
13
'
09x'a9z'
2a 2 3 _' y
qf2
Y0z
+
33
(2E)2E\
2q
OJ'
f.
(3.23)
Since the electrons are unbounded in the z' direction, the wavefunction can be written
as a product of a traveling wave in the z' direction and a bound-state wavefunction
in the x'-y' plane:
T (r) = u(x, y) exp(idz') exp(ftly') -exp(Zikz/ z')
where
(3.24)
and q are some constants to be determined, and kz, is the wavenumber of
the traveling wave in the z' direction. By substituting Eq. (3.24) into Eq. (3.23), we
obtain
a
1
, + 2i(a 22 + a23kz )
all 2 + a 22 -yr2+ 2i(ant + a13 kz,)
OX2 E
9
X
2E U + (a(2
+ a22T 2 + a 33 k2 , + 2a13kz, + 2a23Tykz,)u.
We note that by letting (
-(a1 3/ai)kz, and
= -(a
23 /aII)kz,,
,=
y
(3.25)
the coupling terms
between x', y' and kz, are eliminated, and Eq. (3.25) results in a simple second-order
differential equation in x' and y' only:
/i2
_-
2
12
192
(ell-+
a22
19X'2
OY/2
where
a2
M33 -- (O33
=
_
a
E-
h2k
,
1)
2M33
(3.26)
U,
'
Oz2
23
13
22
a 11
59
)_
= Z' M - z'.
(3.27)
Eq. (3.26) is reminiscent of a 2D Schr6dinger equation with in-plane effective masses
Mi ,
m,
all 1
a-
(X^- M
(^
X
(3.28)
M-1 -9y')1
in the x' and y' directions, respectively. Since u(x', y') must satisfy the boundary
condition u(x', y') = 0 when X' 2 +y'2 = (d/2) 2 according to Eq. (3.24), the eigenvalues
of u(x', y') in Eq. (3.26) are quantized and the energy of the electrons can be written
as
Enm(kz,) = Enm +
(3.29)
Z
2M33
where Enm is the eigenvalue of Eq. (3.26) corresponding to the band edge eigenstate
at kz, = 0 labeled by the quantum numbers (n, m). In Eq. (3.29), we see that the
electron states split into many subbands with band edges at E = Enm, and each
subband behaves like a ID electron system in the z' direction with the transport
effective mass mi,
= M 3 3 . It should be noted that only cylindrical symmetry of the
wire was actually assumed in the derivation of Eqs. (3.26)-(3.28), and these results
are also applicable to circular wires with a finite potential height. We also note that
the effective masses m, and my, that determine the bound-state energies have different expressions from the transport effective mass m2, that characterizes the ID
dispersion relation, as indicated by Eqs. (3.27) and (3.28).
Compared to the exact
expression of the transport effective mass mz, in Eq. (3.27), the transport effective
mass is better approximated using the cyclotron effective mass (Eq. (3.16)) vs. the
square wire simplification (Eq. (3.18)).
3.3.2
Numerical Solutions to the Differential Equation:
-A
(c<2 +o/32 )U
In Sec. 3.3.1, the problem of anisotropic carriers in an infinitely long wire with
a circular cross-section has been greatly reduced to an equivalent 2D differential
60
equation,
(Ce &
where a = al
and
+ 3y2)U = -Au
2
(3.30)
a22, and
2h2k2,
A = 2(E P
k)
2m
(3.31)
33
determines the quantized energies of the bound states. Also, the dummy variables x'
and y' in Eq. (3.26) are here replaced by x and y for simplicity. For a simple case
where
=
/3, the wavefunction of Eq. (3.30) has the analytic solution,
Umn(T)
(3.32)
~ Jm(Xmnr)Cimo,
where Jn is the mth Bessel function, and
Xmn
is determined by the n th root of
Jm(Xd/2) = 0 in order to satisfy the boundary condition. The eigenvalue A corresponding to the wavefunction in Eq. (3.32) is given by
Amn = axmn.
(3.33)
We note that the case of a =3 only applies to the T-point hole pocket in trigonal
Bi nanowires.
For the L-point electron pockets, however, a 4
/ holds for all wire
orientations, even for wires oriented along the principal axes of the electron ellipsoid.
For the general case where a :
3, there are no analytic solutions and the eigen-
values can only be derived through numerical methods.
In the following, numer-
ical solutions to this particular eigenvalue problem are developed by transforming
Eq. (3.30) into a corresponding difference equation, and the resulting Hamiltonian
matrix can be solved numerically with high accuracy through the aid of computers.
First, since the boundary condition for u(r) is determined by u(r = d/2) = 0, it
is useful to express Eq. (3.30) in terms of cylindrical coordinates:
a2
'9X2
=
cos
2
2U
2
19r2
+2
cos 0 sin 0 au
cos 0 sinO
- -2
r2
61
00
r
a92U
aa
+
Oroo
r=d/2
,fl)
r =d1
Figure 3-4: Schematic of the grid points used to transform the differential equation
into a difference equation. The mesh in the circular wire cross-section consists of M
concentric circles and N sectors. In the figure illustrated, M = 5 and N = 12.
sin 2 0 &2 U
sin2 0 Ou
r
02U
sin22o
Oy2
2
u
Or 2
cos 2 0 OU
r
r2
Or
Or
-2
(3.34)
02'
cos0sin0&u
72
190
+2
cos0sin0 9 2 71,
r
rO
cos 2 0 0 2 U
r2
+
(3.35)
002'
where 0 is the polar angle to the x axis and r is the perpendicular distance to the
z axis in cylindrical coordinates. To transform Eq. (3.30) into a difference equation,
we create a mesh consisting of M concentric circles and N sectors within the wire
cross-section, as shown in Fig. 3-4. The polar coordinates of the M x N grid points
in this circular mesh are
(rm, 0n) = (m6r, n6O),
m = 1, 2, .
1M
n =0,1,
(N - 1)
(3.36)
where 6r = d/2M and 60 = 27/N are the distance between adjacent concentric circles
and the angle spanned by each sector, respectively. With these assigned grid points,
62
the derivatives at (r, 0) = (rm, O)
in Eqs. (3.34) and (3.35) can be approximated by
(Um+1,n - Um-1,n)
an
an
02U
-
2
1I (Um,n+l 6 0 (U~~
()2
(3.37)
Um,n-1)
(3.38)
mni
(Um+i,n + Um-,n
-
(3.39)
2nm,n)
02U
1
(3.40)
(Um,n+l + Um,n-I -2Um,n)
2
arao
4(30)(3r) (Um+,n+i + Um-,n-1
-
Um+,n-1
Um-,n+1)
(3.41)
where Um,n = U(rm, On) is the value of the wavefunction u(r, 0) taken at the grid point
(rm,
On).
After substituting the derivatives in Eqs. (3.34) and (3.35) by their finite
difference counterparts, we obtain a difference equation for um,n of the form
Aum,n
Amnum+i,n + BmnUm-i,n + CmnUm,n~l +
Smn(Um+i,n+ i
Um-l,n-1
-
Um+1,n-1
where Amn, Bmn, - - - , Fmn are all functions of (m, n).
-
DmnUm,nr1
Un-1,n+1) +
+
.mnUm,n,
(3.42)
For each grid point (rm, On)
on the mesh, there is a corresponding difference equation, and therefore, we should
obtain M x N equations for the M x N points.
However, extra caution should be taken when applying Eq. (3.42) to grid points
at the outermost and innermost circles. For the grid points at the outermost circle,
the boundary condition requires that U,n
=- 0 for all n, and therefore, only the
difference relations of the other (M - 1) x N grid points need to be considered. For
the grid points at the innermost circle, the situation is more complicated. Although
the origin is excluded from the grid points to avoid infinite values for the coefficients
Amn, Bmn,
of Eq.
. ., etc. at r = 0, the value of Uo,n at the origin is needed for the evaluation
(3.42) for grid points at the first concentric circle (m = 1). Therefore, a method
is developed to estimate the value of u(0) from its neighboring points using the original
differential equation Eq. (3.30) in Cartesian coordinates. The difference equation of
63
Eq. (3.30) in Cartesian coordinates can be expressed as
&2
-Au(0)
= a
+ 13 12
0Y2r=0
z[u(r, 0) + u(6r, w) - 2u(0)] /#[u(6r,7r/2) + u(6r, 37r/2) - 2u(0)]
19X2 r=0
+±
(6r)2
(6r)2
(3.43)
at r = 0, and then u(0) can be solved as a linear combination of wavefunction values
at adjacent points:
a(ui,0 + Ui,O,=r) + #(Ui,O=-/2 +
(6r) 2 A + 2(a + 3)
a(Ui, 0 + uI,o,=7-) + t3 (ui,O,=r/2 +
2(a + 0)
u(0, 0)
Ui,O,=37r/2)
UI,On,=3n/2)
(3.44)
when
(6r)2 <
Thus, we are able to express the (M
2(a +f3)
a
(3.45)
1) x N difference equations from Eq. (3.42)
in a matrix form
/
H(i,o)(1,0)
/
H(1,0)(M-1,N-1)
/
U'1,0
Ul',1
+7
'1,1
U1,N-1
U,N-1
U2 ,0
112,0
\ H(M-1,N-1)(1,o)
...
H(M
1,N -1)(M-1,N -1)
j
'UM-1,N-1
U',0
)
71 'M-1,N-1
(3.46)
where 7- is the Hamiltonian matrix established from the coefficients in Eqs. (3.42) and
(3.44). Finally, the eigenvalues of 7- can be evaluated numerically using software such
as MATLAB or MATHEMATICA, and the number of eigenvalues obtained through
this numerical method is (M - 1) x N. However, it should be noted that higher
64
Table 3.3: Comparisons of the lowest four eigenvalues derived from analytic solutions
and numerical calculations for o = 0.
Analytic Solution
Numerical Solution
AA(d2/4a)
AN(d 2/4a)
5.783063
14.681925
26.374387
30.471504
5.7837
14.6688
26.2624
30.4678
Degeneracy
1
2
2
1
eigenvalues usually have larger discrepancies and should be discarded. The criterion
for dependable eigenvalues is given by
A <
a+3
(6r) 2 + (do)2
(3.47)
Table 3.3 compares the lowest four eigenvalues AA and their degeneracies derived
from the analytic solutions and AN from the numerical solutions for the special case
of oz =
. In this comparison, the number of concentric circles and sectors chosen
for numerical calculations are M = 64 and N = 40, respectively. We note that the
numerical solutions AN are in excellent agreement with the analytic solutions AA,
and the relative errors are smaller than 0.5% for all values listed in Table 3.3. By
increasing the number of concentric circles M and sectors N simultaneously, an even
higher accuracy can be obtained.
For a :4 13, there are no analytical solutions and the numerical technique that
we developed has provided a powerful approach to calculate the desired eigenvalues
accurately and efficiently. In addition, wavefunctions u(r) for the eigenstates can be
readily derived from solving the eigenvectors of the Hamiltonian matrix 'h. Figure 3-5
shows the calculated wavefunctions u(r, 0) of the lowest four eigenstates for o = 1 and
/
= 3 in a circular wire of unity radius. The eigenvalues for the lowest four states in
Fig. 3-5 are AN = 11.4759, 21.7205, 36.2447 and 36.2874, and the degeneracies noted
in Table 3.3 are lifted when oz #
/.
We note that although the second and the fourth
states have similar wavefunctions except for nodes or oscillations in different directions (see Fig. 3-5), their eigenvalues are different due to the inequivalence of effective
65
1st
0.03
-
0.02
-
state
2nd state
0.02 0.00 -
S0.01 -
-0.02
0.01
y
-1
y
-1
-1
-1
4th state
3rd state
0.05 0.020.00 -
0.00-0.02-
-0.05
1
1
-1
y
-0-
Figure 3-5: The wavefunctions u(r, 0) of the lowest four eigenstates for a = 1, f = 3
in a circular wire with unity radius. The eigenvalues of the four states are AN =
11.4759, 21.7205, 36.2447 and 36.2874. We note that although the second and the
fourth states have similar wavefunctions except for nodes or oscillations in different
directions, they have different energies due to the inequivalence of effective masses in
the x and y directions. The node numbers (n, m) of the four states are (0,0), (1,0),
(2,0) and (0,1), respectively. The energy (eigenvalue) increases with node numbers n
and m, but the energy increment for n may be different from that for m. Since a < #,
the effective mass along the x direction is larger than that in the y direction, and the
energy increment will be smaller for the increase in the x-nodes. Therefore, the first
two excited states (second and third states) have nodes or oscillations increased in
the x direction, while to increase nodes or oscillations in the y direction would require
more energy; the eigenstate with the first node in the y direction is the fourth state,
which is at a higher energy.
66
masses in the x and y directions. In Fig. 3-5, we see that the eigenstates can be
characterized by a pair of numbers (n, m) that represent the number of nodes in the
x and y directions, respectively. The pair of node numbers (n, m) provide us with an
unambiguous label to specify each eigenstate, as we have implicitly used in Eq. (3.29).
3.3.3
Non-Parabolic Band Structure of Bi Nanowires
In the previous sections, we have developed solutions for the T-point holes with
parabolic dispersion relations. At the L points, however, the bands are highly nonparabolic due to the strong coupling between the L-point electrons and holes
[8],
and this non-parabolic band structure must be taken into account in determining
the subband energy levels and the ID dispersion relations for the L-point carriers.
The dispersion relation for the L-point bands is described by Eq. (3.5), which can be
rewritten as
h2
2k-a-k
2
= E(k)
I + E(k)
(3.48)
EgL
for the conduction band at the L-point, where a is the inverse effective mass tensor
at the band edge and EgL is the direct bandgap at the L point. By the effective
mass theorem, we obtain a differential equation for the envelope function TJ(r) of the
L-point electrons
2
-V - a - Vqf(r) = E 1
E+
2
EgL)
(r).
(.9
We note that Eq. (3.49) would have the same form as the Schr6dinger equation
(Eq. (3.22)) for parabolic carrier pockets if we define a primitive energy
E (1 +
,
Eg L
67
(3.50)
and the same procedures described in Sec. 3.3.1 and Sec. 3.3.2 can be applied to solve
for the primitive energy E:
Enm(kz) = Enm +
where
Enm
h2 k2
~2
2mfi
(3.51)
is the primitive subband energy and rfnz is the primitive effective mass
along the wire axis. The real quantized subband energy levels and the ID dispersion
relations for each subband can be derived readily from solving E in Eq. (3.50) as
Enm(kz)
EgL
EgL
2
2
EgL
2
+
1+4En,(kz)
\
Eg
[nm
EgL
+
+
2 \
2
h2k2
Z
EgL mzEgL
(3.52)
The band edge energy of each subband is given by
Enm = Enm(kz = 0)
Eg L
=-
2
(Ynm
+ EgL
m
)
2
1+
____
EL
(.53
2
(3.53)
where
(3.54)
4?nm
7nm=
We note that the ID dispersion relation Enm(kz) in Eq. (3.52) for each subband is
more complicated than Eq. (3.29) and also non-parabolic. However, for energies near
the subband edge, it is a good approximation to treat the full dispersion relation in
Eq. (3.52) near the subband edge with a Taylor expansion as
Enm(kz)
EgL
EL
+
=
___m_
n
+
2 2
E
1+
(
2h 2 k2
Tn
m(Eg L +4Enm)
2
h k2
+
(3.55)
mEnm
2
2m*nm
68
where
1+
mzsnm =
n
nmmrnz
EgL
(3.56)
is the transport effective mass along the wire axis for the corresponding subband. It
should be noted that the Taylor expansion in Eq. (3.55) is valid only when
h 2 k2
E*Lnm
2m*,nm
4
()
where
E*Lnm
EL
+ 2Enm = 7nmEgL
(3.58)
is the effective bandgap between the two subbands in the valence band and the conduction band, which have the same quantized state labeled by (n, m). As indicated in
Eq. (3.56), the transport effective masses m*,nm are different for every subband, and
increase with the subband edge energy.
It should be noted that in deriving the non-parabolic dispersion relation in Eq. (3.5),
the far band contribution (h 2 k 2 /2mo) is usually neglected when applying k - p perturbation theory to bulk bismuth, due to the very small effective masses at the 3D
band edge for strongly coupled bands. In Bi nanowires, since the 3D band splits into
many subbands, the far band contributions are not negligible for higher subbands
where the effective masses at the subband edge are not small compared to the free
electron mass mo. Therefore, it is necessary to include the contributions from the far
bands in calculating the effective mass of each subband. We therefore write the more
complete expression to relate m*,mn to Tinz:
1
Mzmn
-
1
mO
69
1
'Ynmmz
(3.59)
Table 3.4: Calculated effective mass components of each carrier pocket for determining the band structure of Bi nanowires at 77K, based on the values given in Table 3.1.
The z' direction is chosen to be along the wire axis. All mass values in this table are
in units of the free electron mass, MO.
Mass Component
mX'
e- pocket A
e- pocket B
e- pocket C
hole pocket
3.4
Trigonal
0.1175
Binary
0.0023
Bisectrix
0.0023
[0112]
0.0029
[10h1]
0.0024
my/
___
0.0012
0.2659
0.0012
0.0012
0.0012
ns'
0.0052
0.0012
0.2630
0.2094
0.2542
m/,
0.1175
0.0023
0.0023
0.0016
0.0019
my,
0.0012
0.0052
0.1175
0.0012
0.0052
0.0590
0.0590
0.6340
0.0016
0.1975
0.0023
0.0016
0.1975
0.6340
0.0590
0.0590
0.0048
0.0666
0.0023
0.0048
0.0666
0.6340
0.0590
0.0590
0.0125
0.0352
0.0016
0.0125
0.0352
0.1593
0.0590
0.2349
0.0071
0.0526
0.0019
0.0071
0.0526
0.3261
0.0590
0.1147
___
m__
my/
fz/
mX1
my/
mzi
Band Shifting and Semimetal-Semiconductor
Transition in Bi Nanowires
In this section, the band structures of ID Bi quantum wires are calculated for several
selected wire orientations. The results are based on the knowledge of band structure
parameters of bulk bismuth; calculation techniques developed in previous sections are
employed.
As discussed in Sec. 3.3.1, the quantized energy levels in a nanowire are determined
by the mass components mx, and my, perpendicular to the wire axis z' that are given
by Eq. (3.28), while the ID density of states of each subband and the motion along the
wire are governed by the transport effective mass mz, given by Eq. (3.27) or Eq. (3.59).
From the band structure parameters of bulk Bi (Table 3.1), we calculate these effective
mass components for the three L-point electron pockets and the T-point hole pocket.
Table 3.4 lists the calculated results of mr', MY and mz, (or fz,) at 77K for the three
principal crystallographic axes (trigonal, binary and bisectrix directions), and the
preferential growth directions [0112] and [1011] (or (0, 0.8339, 0.5519) and (0, 9503,
70
100
80
60
L e (A,BC)
T holes
40
~320
-o=3
55.1 nm
20
L holes
Eg=
15
-20
_0
-40
0
50
100
150
200
Wire diameter (nm)
Figure 3-6: The subband structure of Bi quantum wires oriented along the trigonal
direction at 77 K, showing the energies of the first subbands for the T-point hole
pocket, the L-point electron pockets (A, B and C), as well as the L-point hole pockets.
The zero energy refers to the conduction band edge in bulk Bi. As the wire diameter
d decreases, the conduction subbands move up while the valence subbands move
down. At d, = 55.1 nm, the lowest conduction subband edges formed by the Lpoint electrons cross the highest T-point valence subband edge, and a semimetalsemiconductor transition occurs.
0.3112) in Cartesian coordinates, respectively) for Bi nanowires. It should be noted
that m,, and my can be interchanged without affecting any physical results.
For
the L-point pockets, Eqs. (3.56) and (3.59) should be used to include non-parabolic
effects into the entries in Table 3.4, while the parabolic approximation of Eq. (3.55)
was used to describe the small energy excursions from k,, = 0 for the dispersion along
the wire axis.
Figures 3-6, 3-7, 3-8, and 3-9 show the calculated subband structure of Bi quantum
wires oriented along the trigonal, binary, bisectrix, and [0112] directions at 77K,
respectively, as a function of wire diameter d. The zero energy in these figures refers
to the band edge of the conduction band in bulk Bi.
For quantum wires oriented along the trigonal axis, the three-fold symmetry in
Bi is preserved.
Thus, the three carrier pockets at the L points remain degenerate
regardless of the wire diameter. The band edge of the lowest subband of the L-point
71
100
-/
80
\
60 -
L
L
E
e~(B,C)
T holes
e~(A)
20
-20(
240 -
.o
.
= 38.
-40
0
100
50
150
200
Wire diameter (nm)
Figure 3-7: The subband structure of Bi quantum wires oriented along the binary
direction at 77K, showing the energies of the first subbands for the T-point hole
pocket, the L-point electron pockets (A, B, and C), as well as the L-point hole
pockets. The zero energy refers to the conduction band edge in bulk Bi. As the wire
diameter d decreases, the conduction subbands move up while the valence subbands
move down. At d, = 39.8nm, the lowest conduction subband edge formed by the
L(A) electrons crosses the highest T-point valence subband edge, and a semimetalsemiconductor transition occurs.
electrons increases with decreasing d, while the highest subband edges of the T-point
and L-point holes decrease in energy (see Fig. 3-6).
At d < 55.1 nm, the lowest
conduction subband edge exceeds the highest valence subband edge, indicating that
these nanowires become semiconducting.
For nanowires oriented along other directions, the subband structures are more
complicated and the degeneracy at the L points will usually be lifted due to the lower
symmetry in these nanowires. For binary quantum wires, the L-point electron pocket
A has larger mass components (mr', my,) in the confined direction than the electron
pockets B and C (see Table 3.4). Therefore, the electron pocket A forms a lower
conduction subband, while the electron pockets B and C form a two-fold degenerate
subband at a higher energy (see Fig. 3-7). It should be pointed out that since the
electron pocket A has a smaller mass component m, along the wire axis than pockets
B and C, the dispersion relation of the L(A) subband will have a larger curvature. We
72
100
\\
80
60
Le (A) L e~(B3,C)
S 40 - - - - - -
T holes
- - - - - -
40
-
48.5 nm
W
-- -------
-20
0 --- ------------L
-40
0
50
holes
100
-- -- - - ----.E9
150
=3
= 15
200
Wire diameter (nm)
Figure 3-8: The subband structure of Bi quantum wires oriented along the bisectrix direction at 77 K, showing the energies of the first subbands for the T-point hole
pocket, the L-point electron pockets (A, B and C), as well as the L-point hole pockets.
The zero energy refers to the conduction band edge in bulk Bi. As the wire diameter d
decreases, the conduction subbands move up while the valence subbands move down.
At dc = 48.5 nm, the lowest conduction subband edges formed by the L(B, C) electrons cross the highest T-point valence subband edge, and a semimetal-semiconductor
transition occurs.
also note that the subband structure for the L-point holes exhibits a similar behavior
as for the L-point electrons except for an opposite sign in the dispersion relations. At
dc = 39.8 nm, the lower lying L(A) conduction subband crosses the highest T-point
hole subband, and the semimetal-semiconductor transition occurs.
For bisectrix wires, the degeneracy at the L point is also lifted, resulting in two inequivalent groups of carrier pockets: a single electron pocket A and two-fold electron
pockets B and C. However, in contrast to the binary wires, the L(A) electrons in the
bisectrix wires form a higher subband due to their smaller m,, and my mass components compared to the L(B) and L(C) electrons (see Table 3.4). The semimetalsemiconductor transition in these wires occurs at dc
48.5 nm, where the lower lying
L(B, C) subbands cross the highest T-point valence subband edge.
The quantum wires oriented along the preferred growth directions of [0112 and
[1011] can be visualized as bisectrix wires tilted towards the trigonal axis by 33.5' and
73
100
80
60
L e~(A)
Le (B,C)
T holes
40
S 20
-
-A,,= 38
W49.0
nm--L holes
-40
0
50
100
E9= 15
150
200
Wire diameter (nm)
Figure 3-9: The subband structure of Bi quantum wires oriented along the [0112]
direction at 77K, showing the energies of the first subbands for the T-point hole
pocket, the L-point electron pockets (A, B and C), as well as the L-point hole pockets.
The zero energy refers to the conduction band edge in bulk Bi. As the wire diameter d
decreases, the conduction subbands move up while the valence subbands move down.
At dC = 49.0 nm, the lowest conduction subband edges formed by the L(B, C) electrons cross the highest T-point valence subband edge, and a semimetal-semiconductor
transition occurs.
18.130 , respectively. Therefore, they have the same symmetry and similar subband
structures to the bisectrix wires (see Fig. 3-9). The critical wire diameters d, for
the semimetal-semiconductor transition for Bi nanowires oriented along these two
directions are slightly larger compared to that of the bisectrix wires (see Table 3.5).
It should be noted that since dc is determined by the band structure of Bi,
which is strongly temperature-dependent for T > 80 K, the critical wire diameter
d, for the semimetal-semiconductor transition will also be temperature-dependent
for T > 80 K.
Figure 3-10 shows the calculated critical wire diameter d, for the
semimetal-semiconductor transition in Bi nanowires as a function of temperature for
different wire orientations. Since the band overlap AO (between the L-point conduction band and the T-point valence band) and the effective mass components of the
L-point electrons both increase with temperature (see Table 3.2),
a smaller wire
diameter is required to achieve the semimetal-semiconductor transition at higher
74
60
50
E
0
40
C
------trigonal--- [0112]
[10 1i]
bisectrix
- - - - binary
30
20
10
0
50
I
150
100
200
250
300
Temperature (K)
Figure 3-10: Calculated critical wire diameter d, for the semimetal-semiconductor
transition as a function of temperature for Bi nanowires oriented in different directions.
Table 3.5: Calculated critical wire diameters d, for the semimetal-semiconductor
transition of Bi nanowires at 771K and 300 K for various crystallographic orientations.
Temperature
77K
300 K
Trigonal
55.1 nm
15.4 nm
Binary
39.8 nm
11.2 nm
Bisectrix
48.5 nm
13.6 nm
[0112]
49.0 nm
14.0 nm
[1011]
48.7 nm
13.6 nm
Table 3.6: Calculated critical wire diameters d, and critical wire widths ac for Bi
nanowires oriented along various directions at 771K, using the cyclotron effective mass
approximation and the square wire approximation, respectively.
Calculation Method
Cyclotron Effective Mass Approx. [33]
Trigonal
33 nm
Binary
20 nm
Bisectrix
44 nm
Square Wire Approx. [45]
52.1 nm
34.5 nm
41.3 nm
75
[1011]
45 urn
temperatures.
Critical wire diameters d, for the semimetal-semiconductor transi-
tion in Bi nanowires at 77K and 3001K for various crystallographic orientations are
summarized in Table 3.5, showing the large T dependence of d,.
Lastly, as a comparison, Table 3.6 lists the calculated critical wire diameters and
critical wire widths at 77K obtained using the cyclotron effective mass approximation
[33] and the square wire approximation [45], respectively, for various wire
orientations. The non-parabolic band structure of the L-point bands was taken into
account in both approximation methods.
However, due to the high anisotropy of
carrier pockets in Bi, a significant discrepancy is noted between results obtained with
these approximation methods (Table 3.6) and those obtained in this study (Table
3.5).
76
Chapter 4
Transport Properties of Te-Doped
Bi Nanowire Systems
Nanowires are of great research interest as one-dimensional (ID) systems.
They
also provide the potential for designing a superior thermoelectric material for power
generation and refrigeration, in view of an enhancement in thermoelectric performance predicted for low-dimensional systems [5, 6].
Since their band structure is
strongly perturbed due to quantum confinement effects, ID nanowire systems should
exhibit very different transport properties from their bulk counterparts. In addition,
due to the long mean free paths of electrons in bulk Bi (-
400 pm at 4 K and 100 nm
at 300 K), single-crystalline Bi nanowires with small diameters should be an excellent
system for studying the classical and quantum finite-size effects in transport phenomena [33, 46].
4.1
Semi-Classical Transport Model
In this thesis, we use the semi-classical model, which is based on the Boltzmann
transport equation, to derive various transport coefficients of interest to Bi nanowires.
The Boltzmann transport equation has been studied for ID systems [6, 32].
For a
simple one-band model for a 1D quantum wire, important transport coefficients,
77
such as the electrical conductivity a, the Seebeck coefficient S, and the thermal
conductivity
Ke
are derived as [32],
-
S
=
L( ,
1 L(1
L
1 (L( 2)
e T
2
(4.1)
(4.2)
(L())
2
)
L(0 )
(4.3)
where T is the temperature, and
L
) = e2
Jr
4dk
2 2
d
(df\dE)
T(k )v(k)v(k)(E(k) - Ef)0 ,
(4.4)
where a = 0, 1, 2, E(k) denotes the carrier dispersion relation, T(k) is the relaxation
time, Ef is the Fermi energy, and f(E) is the Fermi-Dirac distribution function,
f(E) =1 1 +
C(E-Ef)/kBT
(4.5)
We note that the numerical calculation of Eq. (4.4) requires knowledge of the relaxation time T(k), and the calculation of T(k) from the fundamental principles of scattering mechanisms is usually very complicated. Therefore, we have used a simple first
approximation, known as the constant relaxation time approximation, to simplify the
calculations. For more systematic detailed studies, the effect of individual scattering
mechanism should be taken into account, as in Ref. [47]. In the constant relaxation
time formalism, T(k) = r is constant in k and in energy, and the relaxation time
T
can be related to the carrier mobility 1- along the wire by
p*
T
=
C
(4.6)
where m* is the transport effective mass along the wire, and IL can be obtained from
experimental data. Thus, the integration of Eq. (4.4) can be carried out readily as
long as the dispersion relation E(k) is known. Furthermore, the electrical conductivity
78
of the ID system can be written in a familiar form,
o = enL,
(4.7)
where n is the carrier density given by
J 4 dk f (E)
whereby g(E) = ;d
I
g (E)f (E)dE,
(4.8)
is the density of states and d is the wire diameter.
In Bi quantum wire systems, there are many ID subbands due to the multiple
carrier pockets at the L points and the T point; the quantum confinement-induced
band splitting also forms a set of ID subbands from a single band in bulk materials.
Therefore, when considering the transport properties of real ID nanowire systems,
contributions from all of the subbands near the Fermi energy should be included. In
a multi-band system, the L(')'s in Eqs. (4.1)-(4.3) should be replaced by the sum
L ,a=
EL( ) of contributions from each subband (label by i), and the transport
coefficients o, S, and se become
Utotal
=
Stotai
=
Ke,total
-
-,(4.9)
(4.10)
,
2
1(L~
e2 T
total
(4.11)
totLal 2
L(a)
total
where o-i and Si are the electrical conductivity and the thermopower corresponding
to each subband, respectively.
Another physical quantity of interest in the thermoelectric applications of Bi
quantum wire systems is the lattice thermal conductivity
KL
which, together with
the electronic thermal conductivity Ke, determines the total thermal conductivity of
the system. From the kinetic theory, the thermal conductivity of phonons is given
by [48]
u
= 'C ve,
37
79
(4.12)
where C, is the heat capacity per unit volume, v is the sound velocity, and fp is the
mean free path for phonons. We note that for an ideal quantum wire system embedded
in a host material with a large bandgap, the electron wavefunctions are well confined
within the quantum wire, and they can only travel along the wire axis. However, the
host material that confines electrons cannot restrict the phonon paths, and phonons
will be scattered when they move across the wire boundary. This increased boundary
scattering of phonons in the quantum wire system will decrease the phonon mean free
path 4,, as well as the lattice thermal conductivity along the wire. A fair approximation to model the lattice thermal conductivity in the quantum wire system is to
replace the phonon mean free path fp in Eq. (4.12) with the wire diameter d if d
is smaller than the phonon mean free path fp of the bulk material.
It should be
noted that for wire diameters much smaller than the phonon mean free path, the
lattice thermal conductivity is expected to decrease dramatically. This reduction in
thermal conductivity is one of the reasons for the expected thermoelectric performance enhancement in low-dimensional systems.
4.2
Carrier Density in Pure Bi Nanowires
Since most transport coefficients are closely related to the number of mobile carriers
in the system and the density of states near the Fermi energy level, it is necessary to
investigate the band structures, density of states and carrier densities to study the
transport properties of Bi nanowires.
The carrier density in a semimetal exhibits a rather different temperature dependence from ordinary metals or semiconductors. In a semimetal, mobile carriers result
from the band overlap between the conduction band and the valence band. The Fermi
level usually goes through the region of band overlap and is determined such that the
number of electrons in the conduction band equals the number of holes in the valence
band. Therefore, unlike metals which always have a constant number of electrons
or holes, the number of mobile carriers in a semimetal is temperature dependent
1020
E
? 10
18
10
10 17
0
100
200
300
Temperature (K)
Figure 4-1: Calculated total carrier density (for electrons and holes) of bulk Bi as a
function of temperature.
since the temperature-dependent Fermi energy may change the number of electrons
and holes.
However, this temperature dependence for the carrier density n(T) in
semimetals is much weaker compared to that in intrinsic semiconductors, whereby the
mobile carriers result from thermal excitation and the carrier density n(T) increases
exponentially with temperature.
In addition, the most striking difference in n(T)
between a semimetal and an intrinsic semiconductor is that as T approaches 0 K, the
carrier density in a semiconductor drops to zero rapidly, while the number of intrinsic
carriers in a semimetal will still be quite significant.
Figure 4-1 shows the calculated density of carriers (electrons and holes) in pure
bulk Bi as a function of temperature. In this calculation, non-parabolic dispersion
relations are used for the L-point electrons and holes, and the temperature dependence
of band parameters (Table (3.2)) is also taken into consideration. The carrier density
in bulk Bi increases with temperature, and at 300 K the carrier density is ~ 20 times
of that at 0 K. We note that the increase of carrier density in Bi is more gradual with
T than that in a semiconductor, but is greater than that in a metal.
For Bi quantum wires, the band structure and the density of states are rather
81
2e+17
I
I
40-nm Bi nanowire
bulk Bi
L-point electrons
E
E
T-point holes
le+17
.
.
-10
0
10
20
30
40
50
60
Energy (meV)
Figure 4-2: Calculated density of states for 40-nm Bi nanowires oriented along the
[0112] growth direction at 77K (solid lines), compared to the density of states of
bulk Bi (dashed lines). The zero in energy represents the band edge of the L-point
conduction band in bulk Bi.
different from those of bulk Bi, due to the strong perturbation from quantum confinement effects. Figure 4-2 shows the calculated density of states of 40 nm-diameter Bi
quantum wires oriented along the [0112] direction at 77K, compared to the density
of states of bulk Bi, with the zero in energy representing the band edge of the L-point
conduction band in bulk Bi.
Since the critical wire diameter for the semimetal-
semiconductor transition is calculated to be 49.0 nm for Bi nanowires oriented along
the [0112] direction at 77K (Table 3.5), there is a bandgap between the lowest L-point
conduction subband and the highest T-point valence subband for 40-nm Bi nanowires,
as shown in Fig. 4-2. In addition to the bandgap, we also observe the singularities in
the density of states at the subband edge of Bi nanowires, which are signatures of ID
electron systems.
Due to the semimetal-semiconductor transition that occurs at a critical wire
diameter dc, the temperature dependence of the carrier densities for Bi nanowires is
strongly affected by the wire diameter. Figure 4-3 shows the calculated total carrier
densities of Bi nanowires oriented along the [0112] direction as a function of T for
82
different wire diameters. Since dc is temperature dependent, there will be three different types of T dependences for carrier densities in pure Bi nanowires, depending on
the wire diameters. For 10-nm Bi nanowires, which are always in the semiconductor
regime up to 300 K (see Fig. 3-10), the carrier density increases exponentially with T.
On the other hand, for 80-nm Bi nanowires, which remain in the semimetallic regime
even down to 0 K, the carrier density has a similar T dependence as that for bulk Bi.
However, we also note that the carrier density of 80-nm nanowires is smaller than
that of bulk Bi because of the smaller band overlap between the conduction band
and the valence band.
As for 40-nm Bi nanowires, the semimetal-semiconductor
transition temperature is around 180 K (see Fig. 3-10).
Thus, for T < 180K, 40-
nm Bi nanowires are in the semiconductor regime, and the carrier density drops
significantly with decreasing T; whereas for T > 180 K, the nanowires are in the
semimetallic regime, and the carrier density has a similar T dependence as bulk Bi.
It should also be noted that for semiconducting wires, the slope of the curves in Fig. 43 is approximately proportional to the bandgap between the conduction band and the
valence band, which increases with decreasing T for 40-nm Bi nanowires below 180 K.
Therefore, the slope of the curve for 40-nm Bi nanowires increases with decreasing
temperature (see Fig. 4-3).
In addition to the wire diameter, the carrier density in Bi nanowires may also
depend on the wire orientation.
Figure 4-4 shows the calculated carrier densities
for 40-nm Bi nanowires oriented along various directions.
The dependence of the
carrier density on the wire orientation becomes weaker as the temperature or the
wire diameter increases. For larger wire diameters, the quantum confinement is less
effective at lifting the subband energies, and the energy separation between adjacent
subbands is smeared out by the thermal energy kBT at high temperatures. Thus, as
the temperature increases, the carrier density in Bi nanowires will approach that of
bulk Bi regardless of the wire orientation. For 40-nm Bi nanowires, the differences in
the carrier densities for different wire orientations become negligible for T > 150 K
(see Fig. 4-4).
10
Bulk Bi
COE 18
-..-
310
_80 nmn
17
10
16
10
nm
10
10
40 nm
0
200
100
300
Temperature (K)
Figure 4-3: Calculated total carrier density (for electrons and holes) of Bi nanowires
of different diameters oriented along the [0112] direction as a function of temperature.
The carrier density of 10-nm Bi nanowires has a temperature dependence similar to
that of a narrow-gap semiconductor, while 80-nm nanowires behave like a semimetal.
The carrier density of 40-nm Bi nanowires has a semiconductor-like temperature
dependence at low temperatures (T < 180 K), and a semimetal-like temperature
dependence at high temperatures (T > 180 K).
109
10
E
10 1
C:
-
binary
bisectrix
[1011]
-----------
0112]
trigonal
---D-
-[
10
10
0
200
100
300
T (K)
Figure 4-4: Calculated total carrier density (for electrons and holes) of 40-nm Bi
nanowires oriented along various directions as a function of temperature.
The
differences in carrier densities for different wire orientations become negligible for
T > 150 K.
84
4.3
Te Doping of Bi Nanowires
In pure Bi nanowires, there are always equal numbers of electrons and holes. However,
in some applications, it is desired to control the Fermi level so that the transport
phenomena are dominated by a single type of carrier only, i.e. electrons or holes. For
example, in thermoelectric applications, it is necessary to fabricate both n-type and
p-type systems for the thermoelectric device to operate. In addition, to optimize the
efficiency of thermoelectric devices, it is essential to obtain a high Seebeck coefficient
S. However, since the contributions from holes and from electrons have opposite
parities with regard to the Seebeck coefficient (Eq. (4.10)), the magnitude of the
Seebeck coefficient in pure Bi is usually very small, although individual contributions
from electrons and holes can be quite significant.
Therefore, it is expected that
Bi nanowires can be a very promising thermoelectric material if the Fermi level is
adjusted properly so that the contributions from electrons and holes are isolated
without canceling each other.
Since Bi is a group V element, the Fermi level can be increased by introducing a
small amount of group VI element, such as Te, which acts as a monovalent electron
donor in Bi [49, 50]. Group IV elements such as Sn or Pb, on the other hand, act as
an electron acceptor in Bi, and can be used to synthesize p-type Bi [49]. The study
of n-type Bi nanowires is of particular interest because of the very small electron
effective mass and the highly anisotropic Fermi surface of the electron pockets in Bi.
Therefore, in this thesis, we will focus on investigating Te-doped Bi nanowires.
As discussed in the XRD study in Chapter 2, the crystal structure of Bi is not
changed by introducing a small amount of Te. Thus, we assume the band structure of Te-doped Bi to be the same as pure Bi, except for a Te donor energy level
located below the conduction band edge by Ed. The ionization energy Ed required for
releasing donor electrons from Te atoms can be estimated using the Bohr hydrogenlike model,
Ed
~ 13.6 x
mn*/mo
2
(eV),
(4.13)
where m* is the effective mass at the conduction band edge, mo is the free electron
85
mass, and c is the dielectric constant of Bi in the low frequency limit. Bi has a very
large dielectric constant, c ~ 100 [8, 51]. The value for the effective mass m* is taken
as the average value of the electron effective mass tensor, and at low temperatures,
m*/mo ~_0.002 for bulk Bi. With these values of c and m*, we obtain
Ed-- 3.20 x 10-3 meV
(4.14)
with an effective Bohr radius
a* = 00.5 (A) x
rn*
c 2.5 pm.
(4.15)
Thus, we see that the energy required to ionize a Te atom in Bi is very small because
of the small electron effective mass and the large dielectric constant.
However, it
should be noted that the results obtained in Eqs. (4.14) and (4.15) are only valid
for bulk materials. In Te-doped Bi nanowires where the diameter d is smaller than
the effective Bohr radius a*, the ionization energy will rise due to the confinement of
donor electrons to the impurity atom. Since the Coulomb potential energy is inversely
proportional to the distance between two charged particles, a rough estimation of the
ionization energy Ed in nanowires can be made by multiplying the ionization energy
Ed(B) in bulk materials by the ratio between the effective Bohr radius and the wire
diameter. Thus, for 40-nm Te-doped Bi nanowires, the ionization energy would be
~i 0.2 meV.
Ed~ Ed(B) x
(4.16)
At very low temperatures where the thermal energy kBT < Ed, the donor electrons
will freeze out, and the freeze-out temperature is TFO ~- 2K for 40-nm Te-doped Bi
nanowires. Since the temperatures of interest range from 4 K to 300 K, we can assume
that all the donor atoms are ionized in our transport study of 40-nm Te-doped Bi
nanowires, and that each Te atom donates one electron to the conduction band.
Figure 4-5 shows the calculated carrier densities of 40-nm Bi nanowires oriented
along the [0112] direction as a function of temperature for different donor concen-
86
1020
10
C
10
-18
0
Nd1=1X0C
3-Nd =
7
X1
l
Cm
3
1xN Cmx0 8 c
~~ 1N=
N =lO
cm
10
0
200
100
300
Temperature (K)
Figure 4-5: Calculated total carrier density
(for electrons and holes) of 40-nm Bi
nanowires of different Te dopant concentrations oriented along the [0112] direction as
a function of temperature.
trations Nd.
At low temperatures where the intrinsic carriers are overwhelmed by
the carriers from the Te dopants, the total carrier density is essentially constant
and is approximately equal to the dopant concentration Nd.
As the temperature
rises, the number of intrinsic carriers increases exponentially, and will eventually
exceed the number of dopant carriers above a certain threshold temperature Tth,
so that the carrier concentration will again be dominated by the intrinsic carriers
in this high-temperature regime. This threshold temperature Tth for the extrinsicintrinsic transition increases with dopant concentrations Nd, as shown in Fig. 4-5.
For N
1x 107 cm-3 ,2x 1018 cm-3 and 7 x 1018 cm-3 , the threshold temperatures
are Tth ~ 85 K, 180 K and 250 K, respectively.
87
4.4
Thermoelectric Investigations of Te-doped Bi
Nanowires at 77K
The efficiency of a thermoelectric material is measured by the figure of merit,
Z=
S2o,
K
(4.17)
,
where S is the Seebeck coefficient, a is the electrical conductivity, and
t
is the thermal
conductivity. The thermal conductivity consists of two contributions: lattice thermal
conductivity
'L
and electronic thermal conductivity
Ke,
and
=
KL
+
K,.
In order to
achieve a high ZT, one should choose a material with a large Seebeck coefficient, a
high electrical conductivity and a small thermal conductivity. However, an increase
in the electrical conductivity a generally reduces the Seebeck coefficient S, while
the electronic thermal conductivity K, increases with a as a result of WiedemannFranz law. Therefore, a modification to one of these coefficients would adversely
affect the other two, and the resulting ZT does not vary significantly. Currently, the
materials with the highest ZT are Bi 2 Te3 alloys, such as Bio.5 Sbi.Te3 with ZT ~1
at 300 K [52]. It has been shown recently that low-dimensional systems could provide
another approach to increase ZT because of the enhanced density of states at the
subband edge, and the reduction of lattice thermal conductivity KL by limiting the
phonon mean free path f, [53, 54]. In this section, modeling of the thermoelectric
coefficients of Te-doped Bi nanowires will be presented at 77 K, which is a temperature
of great interest for cryogenic cooling applications. Empirical results for the carrier
mobility tensors y, the lattice thermal conductivity tensor
KL,
the heat capacity
C, and the sound velocity v of Bi are used for the modeling of the thermoelectric
properties of Bi nanowire systems.
In Bi, the mobility tensor for each carrier pocket is highly anisotropic. For the
88
Table 4.1: The values of mobility tensor elements for electron and hole pockets of Bi
at 77 K [55]. The values are given in cm 2 Vls- 1 .
Pei1
6.8 x10 5
p42
1.6 x10 4
pe3
3.8 x10 5
/e4
Phl
Ph3
-4.3x10 4
1.20x10 3
2.1x10 4
L-point electron pocket A (see Fig. 3-2), the mobility tensor has the form
pe",
A e(A)
0
0
0
Pe2
Pe 4
0
Pe4 Pe3
(4.18)
The mobility tensors for electron pockets B and C can be derived by rotations of
pe(A) by ±1200 along the trigonal axis. For the T-point holes, the mobility tensor
has the form
A=
phi
0
0
0
P1i
0
0
0
(4.19)
AU
The values for these mobility tensor elements at 77K are listed in Table 4.1 [55].
Since the mobility tensors are anisotropic, the mobility p7
1 for carriers traveling along
the wire will depend on the wire orientation, and is given by
pg = (i. -
1
1)- i)
(4.20)
where 1 is the unit vector along the wire axis. Equation (4.20) follows from the general
definition of the carrier mobility in terms of
ft
er/m* and from Matthiessen's rule
summing 1/Ti for each scattering process i. Table 4.2 lists the calculated mobility for
each carrier pocket in Bi nanowires of various wire orientations at 77K.
89
Table 4.2: Calculated mobility along the wire for each carrier pocket in Bi nanowires
of various wire orientations at 77K. The values are given in m 2 V-Is- 1 .
Mobility
e- pocket A
e- pocket B
e- pocket C
hole pocket
Trigonal
25.93
25.93
25.93
2.10
Binary
68.11
1.42
1.42
11.84
Bisectrix
1.08
4.11
4.11
11.84
[12]
1.32
7.59
7.59
4.91
[10h]
1.10
5.21
5.21
8.17
The lattice thermal conductivity in bulk Bi is also anisotropic, and has the form
ut1 0
KL
where KL,I and
KL,I
0
0
K",-
0
0
0
KL,11
(4.21)
are the thermal conductivities parallel and perpendicular to
the trigonal axis, respectively.
By extrapolating the experimental data measured
between 100 K and 300 K [41], the lattice thermal conductivity tensor elements at
77 K are estimated as rL,I = 13.2 W/mK and rL,I = 9.9 W/mK, respectively. For
Bi nanowires oriented in directions other than the three principal axes, the lattice
thermal conductivity along the wire is given by
=
KLI
cos 2
0+ ILL sin2 0,
(4.22)
where 0 is the angle between the wire axis and the trigonal axis.
The phonon mean free paths in Bi nanowires are estimated by the heat capacity
C,, the sound velocity v and the thermal conductivity rL,g via Eq. (4.12). At 77K,
the value for the heat capacity of Bi is measured as Cv ~ 1.003 JK--Icm
3
[56]. The
sound velocity of Bi is determined by the slope of the acoustic branch, and has a minor
temperature dependence. The measured sound velocities along the selected directions
at 1.6 K [57] and 300 K [58] are listed in Table 4.3, whereby the interpolated values
for the [0112] and [1011] directions are also given. Using Eq. (4.12), the phonon mean
90
Table 4.3: The sound velocities v of Bi nanowires oriented along various directions.
The values at 77K are interpolated from the experimentally measured results at
1.6 K [57] and 300 K [58].
v (105 cm/s)
T (K)
1.6
77
300
Trigonal
2.02
2.01
1.972
Binary
2.62
2.60
2.540
Bisectrix
2.70
2.67
2.571
) ]_[0112]
2.15
2.26
2.24
2.13
2.18
2.082
(0,,
[1011]
2.58
2.45
2.375
Table 4.4: Calculated phonon mean free paths tp at 771K for Bi crystal orientations
along the three principal axes and the [0112] and [1011] directions.
Nanowire orientation
Trigonal
f, (nm)
14.7
Binary
15.2
Bisectrix
14.8
[0112]
15.3
[1011]
15.7
free paths f, of bulk Bi at 771K are calculated in Table 4.4 for Bi crystals oriented
along the three principal axes, as well as the [0112] and [1011] directions. As the wire
diameter becomes smaller than the phonon mean free path for bulk Bi, the scattering
at the wire boundary becomes the dominant scattering process for phonons, and
the phonon mean free path in these nanowires is approximately limited by the wire
diameter, or fP
d. Thus, the lattice thermal conductivity in Bi nanowires will
decrease significantly as wire diameters decrease below 15 nm (see Table 4.4).
Figure 4-6 shows the calculated ZlDT for Te-doped Bi nanowires oriented along
the trigonal axis at 77K as a function of Te dopant concentration for three different
wire diameters. We note that the value of ZlDT for a given Te dopant concentration increases with decreasing diameter d, and the maximum ZlDT for each diameter occurs at an optimized Te dopant concentration Nd,(pt). For 5-nm Te-doped Bi
nanowires oriented along the trigonal axis at 771K, the maximum ZlDT is - 6 with an
optimized electron concentration Nd(,pt) ~ 1018 cm-3. The value of ZlDT also strongly
depends on the wire orientation due to the anisotropic nature of Bi. Figure 4-7 shows
the calculated ZlDT at 771K as a function of Te dopant concentration Nd for 10-nm
Te-doped Bi nanowires oriented in different directions. We note that for 10-nm Te91
7
6
5 nm
5
F- 0
4
N 3
nm
2
1
40nm
1719
0
20
Dopant Concentration (cm-)
Figure 4-6: Calculated ZlDT for Te-doped Bi nanowires of three different wire diameters oriented along the trigonal axis at 77 K as a function of Te dopant concentration.
doped Bi nanowires at 77 K, the trigonal nanowires have the highest optimal ZlDT
of ~ 2.0, while the bisectrix nanowires have the lowest optimal ZlDT of ~ 0.4. This
optimal Z1DT increases as the wire orientation moves away from the bisectrix axis to
the trigonal axis (note: [1011] and [0112] nanowires are angled at 71.90 and 56.5' to
the trigonal axis, respectively). The optimum dopant concentrations Nd(0 pt) and the
corresponding ZlDT values for n-type Bi nanowires at 77 K are listed in Table 4.5 for
various wire diameters and orientations.
It is noted that although binary wires have a smaller critical wire diameter for the
semimetal-semiconductor transition than bisectrix wires (see Fig. 3-10), they have a
higher optimal ZlDT than bisectrix wires of the same diameter. This finding is quite
different from the previous results calculated with the square-wire approximation [45],
whereby the bisectrix wires were found to have a higher ZlDT value than the binary
wires of the same width.
This significant discrepancy is found to result from the
approximation used in the previous calculations to compute the carrier mobilities
along the wire axis.
In contrast to Eq. (4.20), mobilities along the wire axis were
92
2.0
trigonal
binary
1.5
IN 1.0
[1011]
[0112]
bisectrix
0.5
0.0
16
10
17
10
10
18
10
19
Dopant Concentration (cm
10
20
3)
Figure 4-7: Calculated ZlDT at 77 K as a function of Te dopant concentrations Nd for
10-nm Te-doped Bi nanowires oriented along trigonal, binary, bisectrix, [0112] and
[1011] directions.
approximated by pi ~
f
p - in previous calculations, which is only valid for nearly
isotropic mobility tensors. For carriers with very anisotropic mobility tensors, such
as in Bi, the discrepancy caused by this approximation can be quite significant. For
example, for nanowires oriented along the bisectrix axis, the mobility along the wire
axis given by this approximation method is 58 m 2 V-IS-1 for electron pockets B and
C, which is more than 10 times larger than the value of 4.1 m 2 V-s- 1 obtained with
Eq. (4.20). Since electron pockets B and C have conduction subbands with the lowest
energy in bisectrix wires, their subbands determine predominately the thermoelectric
properties of the system. Thus, the much larger mobility value used in the previous
calculations would overestimate the ZIDT values for bisectrix wires, and gave rise to
a higher optimal ZlDT value for bisectrix wires than for binary wires.
In addition, the dependence of the optimal ZlDT on the wire orientations can be
qualitatively explained by considering the dependence of ZlDT on the effective mass in
a ID system. First, we note that the optimum Fermi energy level for ZlDT is usually
slightly below the lowest conduction subband edge. For semiconducting Bi nanowires
93
Table 4.5: The optimal carrier concentrations Nd(opt) (in 1018 cm-3) and the corresponding ZlDT at 77K for Te-doped Bi nanowires of various wire diameters and
orientations.
Wire
Orientation
Trigonal
Binary
Bisectrix
[0112]
[1011]
5 nm
10 nm
40 nm
Nd(,pt)
ZlDT
Nd(opt)
ZlDT
Nd(0,pt)
ZDT
0.96
0.35
4.10
2.07
3.21
6.36
3.68
2.21
3.41
2.69
0.81
0.28
1.78
1.33
1.57
2.00
1.14
0.40
0.70
0.51
0.38
0.56
4.97
2.73
2.57
0.31
0.13
0.03
0.06
0.04
with the Fermi energies close to the optimum Fermi level, the system can be approximated by a one-band model at low temperatures, whereby the thermal energy kBT is
much smaller than the bandgap and adjacent subband separations. Since the Seebeck
coefficient in a one-band system is fairly independent of the band structure and is
determined by the position of the Fermi energy level only, the dependence of ZIDT on
the carrier effective mass can only be associated with the electrical conductivity u-and
the thermal conductivity K. At this Fermi energy range, the electronic contribution
Ke to the thermal conductivity is usually small due to the low carrier densities; the
total thermal conductivity is dominated by the lattice thermal conductivity
a ID system, the carrier density n is proportional to
m*
1
L. For
and the degeneracy g of
the lowest subband, so that the conductivity, which can be written as o- = nT/m*
in a one-band system, will be proportional to (g - m*- 1/ 2 ). Therefore, the optimal
ZIDT
is roughly proportional to ( - (g - m*- 1 / 2 ) in a ID transport system. In Bi
nanowires, the degeneracy factor is equal to the number of degenerate pockets that
form the lowest-energy subband, and g = 3, 1 and 2 for nanowires oriented along
the trigonal, binary and bisectrix axes, respectively. The relative values of ( for Bi
nanowires oriented along the trigonal, binary, and bisectrix directions are calculated
to be 1 : 0.69 : 0.19, which agrees quite well with the relative values of the calculated
optimal ZIDT of 1 : 0.57 : 0.2 for 10-nm Bi nanowires oriented along these three
directions (see Table 4.5).
94
Table 4.6: The optimum acceptor concentrations Na(opt) (in 1018 cm- 3 ) and the corresponding ZlDT at 77K for p-type Bi nanowires of various wire diameters and orientations.
Wire
Orientation
Trigonal
Binary
Bisectrix
[0112]
[1011]
5 nm
10 nm
40 nm
Na(opt)
ZlDT
Na(opt)
ZlDT
Na(opt)
ZlDT
0.96
0.79
0.74
6.36
1.78
0.32
12.9
10.3
0.19
0.72
0.16
0.40
6.2
7.9
0.50
0.17
0.05
0.07
2.59
1.04
2.46
1.16
0.58
0.43
0.18
0.19
0.75
0.63
0.03
0.05
As a comparison, the optimal acceptor concentration Na(opt) and the corresponding ZlDT for p-type Bi nanowires are calculated and listed in Table 4.6 for various
wire diameters and orientations. Compared with the results in Table 4.5 for n-type
Bi nanowires, we note that p-type Bi nanowires in general have a much lower ZlDT
for the same diameter. This asymmetric behavior will be discussed in Sec. 4.5.
4.5
Effect of the T-point Holes on the Thermoelectric Properties of Bi Nanowires
In Bi nanowires, due to the presence of the T-point holes that usually dominate the
transport phenomenon for the holes, the transport properties display an asymmetric
behavior for n-type and p-type systems. The T-point holes, which are also anisotropic,
have a larger effective mass component (~
0.634no) along the trigonal direction,
and smaller effective mass components (- 0.059mo) along the binary and bisectrix
directions (Table 3.4). The different anisotropy of the T-point holes from the L-point
carriers further complicates the dependence of thermoelectric properties on crystal
orientations.
To reveal the effect of T-point holes on the thermoelectric properties of Bi nanowires, we first present results for trigonal Bi wires, whereby the three electron pockets
95
become degenerate at the L points. The solid curves in Figure 4-8 show the calculated
ZlDT at 77 K as a function of Fermi energy for 40-nm, 20-nm, 10-nm and 5-nm trigonal
Bi nanowires, respectively, with the zero in energy referring to the conduction band
edge of the L-point electrons in bulk Bi. The highest subband edges of the T-point
(0)
holes and L-point holes are denoted as E()h(T) andE h(L)
, respectively, and the lowest
subband edge of the L-point electrons is labeled as E5.
The calculated ZDT curves
usually have two or more extreme values: the extremal ZlDT with the higher Fermi
energy corresponds to n-type nanowires, while the extremal ZlDT with lower Fermi
energies corresponds to p-type nanowires. The curves of 40-nm, 20-nm and 10-nm
nanowires show an expected asymmetric behavior for n-type and p-type nanowires
(see Figs. 4-8(a)-(c)).
5-nm Bi nanowires, on the other hand, exhibit a symmetric
behavior for n-type and p-type nanowires (see Fig. 4-8(d)), and will be discussed
below. In addition, calculated ZlDT curves assuming that there are no T-point holes
are shown as dashed curves in Fig. 4-8. They are symmetrical for both n-type and
p-type components in all four nanowire systems due to the symmetric band structures
of the L-point electrons and holes.
For 40-nm trigonal nanowires, the highest subband edge of the T-point holes E
(0)T
is only ~ 15 meV below the lowest conduction subband edge E (0.
The optimum
Fermi energy for n-type systems is located within the conduction band, and not far
from
8
(see Fig. 4-8(a)), compared to the thermal energy kBT (-
6 meV). There-
fore, the presence of the T-point holes partly cancels the contribution of electrons
to the thermopower S, and the optimal ZlDT is slightly reduced with respect to the
dashed curve (see Fig. 4-8(a)). We also note that although both the dashed and solid
curves decrease as the Fermi energy is lowered from its optimum value, the solid curve
decays faster due to the increasing influences of the T-point holes. The adverse effect
of the T-point holes on ZlDT for n-type Bi nanowires becomes less significant as the
wire diameter decreases, so that E
is pushed further away from E (0)
As shown
in Figs. 4-8(b)-(d), the T-point holes have essentially no effect on ZlDT for 5-20-nm
n-type Bi nanowires.
The T-point holes play a more important role in determining the behavior of ZlDT
96
(b)
(a) 0.5
1.0
(0)
(0):
(Eh(T)
0.4
0.8
/
-\
0.3
N
0.6
no T-point holes
I-
N
0.2
0.1
0.0
no T-point holes
0.4
0.2
100
50
0
Energy (meV)
0.0
-1 50
50
(c)
-100
0
Energy (meV)
-50
50
100
200
400
(d)
2.5
8
nh(T) -(h
l
Eo)h(l
7
)
2.0
6
5
1.5
no T-point holes
4
N
N
1.0
3
2
0.5
u.
-300
-200
-100
0
Energy (meV)
100
200
-600
-400
-200
0
Energy (meV)
Figure 4-8: Calculated ZlDT as a function of the chemical potential at 77 K for (a)
40-nm, (b) 20-nm, (c) 10-nm and (d) 5-nm Bi nanowires oriented along the trigonal
direction. The zero in energy refers to the conduction band edge in bulk Bi. E()
and E(O) denote the highest subband edges of the T-point holes and L-point holes,
.
respectively, and the lowest subband edge of the L-point electrons is labeled as E
The solid curves are the Z1DT calculated with both the T-point holes and L-point
holes present, and the dashed curves show the Z1DT calculated assuming there are
no T-point holes. For 5-nm Bi nanowires, the solid and dashed curves coincide with
each other, indicating that the T-point holes have a negligible effect on the transport
properties.
97
for p-type nanowires than for the n-type nanowires. For p-type Bi nanowires of 40 nm,
20 nm and 10 nm, there are two Z1DT extrema (see the solid curves in Figs. 4-8(a)-(c))
near the subband edges E()
and E(0,
which are mainly attributed to the T-point
holes and L-point holes, respectively. In trigonal nanowires, the T-point holes are
undesirable in p-type thermoelectric applications due to a large transport effective
mass.
For example, in 10-nm Bi nanowires (Fig. 4-8(c)), the ZlDT variation with
Fermi energy near E h(T)
(0) arises essentially only from the T-point holes. The value of
this ZlDT extremum is only
-
0.14, which is much smaller than the optimal value of
~ 2 for n-type nanowires. Like the L-point electrons, the L-point holes are favorable
for p-type thermoelectric materials, but the presence of heavy-mass T-point holes
reduces the overall thermoelectric performance significantly.
8(a)-(c), the optimal ZDT near
As shown in Figs. 4-
L) are only about 0.17, 0.35 and 0.70 for 40-nm,
20-nm and 10-nm p-type nanowires, respectively, which would have been about 0.32,
0.74 and 2.0 if there were no T-point holes.
For 5-nm Bi nanowires, however, the thermoelectric performance is essentially
independent of the presence of T-point holes, as shown in Fig. 4-8(d) where the solid
curve coincides with the dashed curve.
The dramatically different behavior in the
nanowires is due to the subband crossing of E h(T) and 8E()h(L)
ZIDT curve of 5-nm
ZlD Bicuve
in trigonal Bi nanowires with very small diameters. For Bi nanowires oriented along
the trigonal direction, the T-point holes have the smallest effective mass components
(-
0.059mo) in the confined directions (the binary and bisectrix directions), while
the carriers in the L-point pockets have one of the largest effective mass components
(~
0.12mo) in the confined directions.
Since band shifting due to the quantum
confinement effect is approximately proportional to the inverse of mass components
will move downwards faster than E()h(L) as the diameter
in the confined directions, E0)
h(T)
decreases. In addition, the non-parabolic band structure also tends to reduce the band
shifting of the L-point holes because of the increasing effective masses away from the
band edge.
Therefore, at a certain wire diameter (< 6nm), E4
crosses E(0),
and
the highest valence subband is formed by the L-point holes. The suppression of the
T-point holes with respect to the L-point holes results in nearly symmetric conduction
98
2.0
/*
\
8
5 nm
6
1.5
--
4
N
-
10 n
2/
N 1.0
0
10
15
N21/m17
16
10
10
10
N. (1/cm')
10
2lnm
/
19
18
20/
I20m
0.5
4Onm
0.0
10
\
10
-
10"
10"
10
Acceptor Concentration (1/cm 3 )
Figure 4-9: Calculated ZDT for p-type Bi nanowires of different diameters oriented
along the trigonal axis at 77 K as a function of acceptor dopant concentration N.
The dashed curves represent the results obtained assuming there are no T-point holes.
The inset shows ZlDT calculated for 5-nm nanowires.
and valence bands, and the influence of the T-point holes becomes negligible for the
range of Fermi energies of greatest interest (see Fig. 4-8(d)).
Figure 4-9 shows the ZlDT curves calculated as a function of acceptor dopant
concentration at 77 K for trigonal Bi nanowires of different diameters. The results for
the 5-nm p-type Bi nanowires are depicted in the inset for clarity. We note that with
the presence of T-point holes, the acceptor dopant concentrations Na required for
the optimal ZlDT for p-type Bi nanowires are much higher than those for the n-type
counterparts (see Fig. 4-6). The optimized Na are
-
1.3 x 1019 cm- 3 , 8.3 x 1018 cm-3
and 6.2 x 1018 cm- 3 for 10-nm, 20-nm and 40-nm p-type nanowires, respectively.
For Bi nanowires oriented along the binary and bisectrix directions, the T-point
holes have identical subband structures due to symmetry. In both orientations, the
T-point holes have one of the smallest effective mass components along the transport
direction and the largest mass component lying in the confined plane.
thermoelectric performance for one band is roughly proportional to (m*)-
Since the
1
/2 ,
where
mI* is the transport effective mass, the T-point holes are more favorable for the thermo-
99
(a)
(b)
0.2
0.5
h(LoET
no T-point holes
(n)
)
-
0.4
no T-point holes
0.3
0.1
N
N0.
0.2
0.1
0.0
0
Energy (meV)
-50
0.0. 150
50
100
-50
0
50
Energy (meV)
100
150
(d)
(c)
1.5
4
E(( 1)
Eh(L
1.0
h
E)h(T)
h
no T-point holes
3
r o T-point holes
H
N-
rK
N
2
0.5
-
-
1
U.U
-200
-100
0
Energy (meV)
100
,-350
200
-250
-150
-
-
-50
50
Energy (meV)
150
250
Figure 4-10: Calculated ZDT as a function of the chemical potential at 77 K for (a)
40-nm, (b) 20-nm, (c) 10-nm and (d) 5-nm Bi nanowires oriented along the binary
direction. The zero in energy refers to the conduction band edge in bulk Bi. Eh(T)
0)
and (0)
denote the highest subband edges of the T-point holes and L-point holes,
respectively, and the lowest subband edge of the L-point electrons is labeled as E )
The solid curves show the ZlDT calculated with both the T-point holes and L-point
holes present, and the dashed curves show the ZlDT calculated assuming there are
no T-point holes.
100
(a)
(b)
0.10
0.10
ch(L)
EN)(
no (L
4
ho((4T-pon)
no T point hoies
-
no T-point hole
0.05
N0.
05
0.00
50
-100
0
0.00-2(
00
160
50
-100
Energy (meV)
0
Energy (meV)
100
21
(d)
(c)
0.5
4
0.4
Eh(T)
no T-point holes
3
0.3
-
no T-point holes
i
N~
2
It
0.2
1
0.1
0.0
200
-100
100
0
Energy (meV)
200
'-400 -300
300
-200
100
0
-100
Energy (meV)
200
300
400
Figure 4-11: Calculated ZlDT as a function of the chemical potential at 77K for (a)
40-nm, (b) 20-nm, (c) 10-nm and (d) 5-nm Bi nanowires oriented along the bisectrix
direction. The zero in energy refers to the conduction band edge in bulk Bi. E()
and (0)
h(L) denote the highest subband edges of the T-point holes and L-point holes,
respectively, and the lowest subband edge of the L-point electrons is labeled as E .
The solid curves show the ZlDT calculated with both the T-point holes and L-point
holes present, and the dashed curves show the ZlDT calculated assuming there are
no T-point holes.
101
1.5
5
4
5nm
N2
//
1.0 -
\
0115
10
10
16
17
018
10 10
SNa(1/cm')
10
19
N7
/20
0.5
40 nm
40nm
0.0
5 16
\
nm
10nm
~-
110
10
10
Acceptor Concentration (1/cm")
Figure 4-12: Calculated ZlDT for p-type Bi nanowires of different diameters oriented
along the binary axis at 77 K as a function of acceptor dopant concentration N. The
dashed curves represent the results obtained assuming there are no T-point holes.
The inset shows Z1DT calculated for 5-nm nanowires.
electric applications in the binary and bisectrix wires compared to the trigonal wires.
Figures 4-10 and 4-11 show the calculated
ZlDT
as a function of Fermi energy for
Bi nanowires oriented along the binary and bisectrix axes. We note that unlike the
trigonal wires, the subband crossing between E
and E(0) is not observed for the
h(L)
ntosre
o h
binary and bisectrix wires for diameters down to 5nm. This is due to the smaller
()
band shifting of E (0) in binary and bisectrix wires because of the heavier mass in the
confined directions. Therefore, for 5-nm binary and bisectrix wires, the T-point holes
still play a crucial role in determining the overall
ZlDT
in the p-type range, as shown
in Figs. 4-10(d) and 4-11(d). For binary wires, the band crossing of ()
and(T) will
occur for diameters slightly below 5 nm, so that n-type and p-type binary nanowires
with ultrafine diameters will exhibit a symmetric thermoelectric behavior.
As for
bisectrix wires, such band crossing is very unlikely because the very small effective
mass component of the L-point pockets
(~ 0.0023mo) in the confined directions will
move E(L) downwards
much faster than Eo)
as the wire diameter decreases.
h(L)
h(T)
102
0.5
3
0.4
2
5nm
/
-0
0.3
\/
0 5
10
1016 1017 10'1 1019
Na (1/cm 3)
/
10
lnm
N\
0.2
40 nm
0.1
0.0 1
101
61
1
81
3
10
Acceptor Concentration (1/cm)
Figure 4-13: Calculated ZlDT for p-type Bi nanowires of different diameters oriented
along the bisectrix axis at 77 K as a function of acceptor dopant concentration Na.
The dashed curves represent the results obtained assuming there are no T-point holes.
The inset shows ZlDT calculated for 5-nm nanowires.
Although the T-point holes in general are undesirable for both n-type and ptype thermoelectric applications of Bi nanowires, and the optimal Z1DT for p-type
Bi nanowires is usually smaller than their n-type counterparts, there are also some
exceptions. In 40-nm and 20-nm bisectrix Bi nanowires (Figs. 4-11(a) and (b)), the
optimal ZlDT is found to occur in the p-type range with the presence of the T-point
holes. While the bisectrix wire has the least favorable direction for L-point carriers
in terms of thermoelectric applications, it has one of the most favorable orientations
for T-point holes. Therefore, the T-point holes are more desirable than the L-point
carriers for thermoelectric applications of 40-nm bisectrix wires.
Figures 4-12 and 4-13 show the calculated ZlDT as a function of acceptor dopant
concentration for p-type binary and bisectrix Bi nanowires, respectively.
For Bi
nanowires oriented along the preferred growth directions observed experimentally
([flol] and [0112]), the ZlDT calculated as a function of Fermi energy are shown
in Figs. 4-14 and 4-15, respectively. The dashed curves in these figures (Figs. 4-14
103
(a)
(b)
0.10
.
.-
-0.10
~(0)
(0)
01.
h(T)0 )
h(L)(T)
(
no T-point holes
no T-point holes
N00
N
0.05
0.00
-150
(c)
-100
0.6 r, __II
-50
0
50
Energy (meV)
o
Eh(
0.00
100
-150
ET)
-50
0
Energy (meV)
50
100
3.0
Eole)
Sno T-pomnt holes
0.4
-100
(d)
_
___I ___.
. ___._I_._I_.
. .
i
.
,
___._TI
0.05 -
no T-point holes
2.0
N
N
0.2
on0
1.0
/~\'/
-200
-100
0
100
Energy (meV)
0.0
200
-400
-200
0
Energy (meV)
200
400
Figure 4-14: Calculated ZlDT as a funct ion of the chemical potential at 77 K for (a)
40-nm, (b) 20-nm, (c) 10-nm and (d) 5-nm Bi nanowires oriented along the [10 f 1]
direction. The zero in energy refers to the conduction band edge in bulk Bi. E (T)
and E 4)
denote the highest subband ec ges of the T-point holes and L-point hol es,
respectively, and the lowest subband edg e of the L-point electrons is labeled as E
The solid curves show the ZlDT calculat ed with both the T-point holes and L-po int
holes present, and the dashed curves sh ow the Z1DT calculated assuming there ire
no T-point holes.
104
(a)
(b)
0.10
0.2
Eh(L)
Eh(T)
(o
E
no T-point holes
no T-point holes
N
0. 0 5
0.00
N
A-
-150
-100
-50
0
50
100
I.-
0.1
0.U
-150
150
-100
-50
Energy (meV)
(c)
0
50
Energy (meV)
100
150
(d)
1.0
4.0
(0)
(0)
()
)
E(h))
(T)
0.8
3.0
no T-point holes
no T-poit holes
-H 0.6
3:-
NH
2.0
0.4
1.0
0.2
nn
-200
-100
0
100
.
)n
200
Energy (meV)
-400
-200
0
200
400
Energy (meV)
Figure 4-15: Calculated ZlDT as a function of the chemical potential at 77K for (a)
40-nm, (b) 20-nm, (c) 10-nm and (d) 5-nm Bi nanowires oriented along the [0112]
direction. The zero in energy refers to the conduction band edge in bulk Bi. 6 ()
and E(0) denote the highest subband edges of the T-point holes and L-point holes,
Ch(L)
respectively, and the lowest subband edge of the L-point electrons is labeled as E()
8
e(L)
The solid curves show the ZlDT calculated with both the T-point holes and L-point
holes present, and the dashed curves show the Z1DT calculated assuming there are
no T-point holes.
105
and 4-15) show that if there are no T-point holes, the Fermi energies for the optimum
thermoelectric performance of p-type nanowires are located near (
instead of E
,
and the optimal Z1DT are dramatically enhanced.
In conclusion, our calculations show that the T-point holes will significantly
reduce the thermoelectric performance of n-type Bi nanowires of diameters > 40 nm.
For smaller wire diameters where the bandgap is large compared to kBT, the adverse
effect of the T-point holes on the ZlDT of n-type Bi nanowires becomes negligible.
For p-type Bi nanowires, the T-point holes are always undesirable.
A significant
enhancement in ZlDT can be achieved if the T-point holes can be removed or suppressed to lie below the L-point holes. One possible approach to this is to introduce
an appropriate amount of antimony (Sb) into bismuth to form Bii_,Sb, alloys. It
has been observed that for 0.075 < x < 0.18, the T-point holes will move below the
L-point valence band edge
[59],
and the desired properties of the L-point holes can be
utilized. The suppression of T-point holes in Bi nanowires can also be accomplished
by quantum confinement along some wire orientations. As shown in Fig. 4-8(d), for
trigonal Bi nanowires with very small wire diameters (< 6 nm), the T-point holes will
cross the L-point holes due to quantum confinement effects, and doping of Sb is not
needed. However, for Bi nanowires that can be synthesized at the present stage, due
to the limit in wire diameters (7 nm - 200 nm) and the preferred orientations (not
along the trigonal direction but along the [0112] or
[1011]
direction), the addition
of Sb is necessary to help suppress the T-point holes to improve the thermoelectric
performance in practical applications.
4.6
4.6.1
Two-Point Resistance Measurements
Experimental Set-up
In the fabrication techniques described in Chapter 2, Te-doped Bi nanowires prepared
by the pressure injection method are embedded within the anodic alumina template,
106
Silver Paint
Gold Wires
Figure 4-16:
ments.
Schematic of the experimental set-up for pseudo-four-point measure-
and only the two ends of each wire are exposed.
The anodic alumina template
protects the Te-doped Bi nanowires against oxidation, and offers a convenient system
package for handling the nanowires. However, this particular geometric configuration
is difficult for some transport property measurements. For example, a conventional
four-probe measurement cannot be performed to obtain the absolute resistivity value
of a single nanowire without breaking or dissolving the alumina template. Therefore,
a pseudo-four-probe method was used to study the resistance or magneto-resistance
of Bi nanowires embedded in alumina templates [22, 33].
Figure 4-16 shows the schematic set-up for the pseudo-four-point resistance measurements.
Silver paint (Ladd Research Industries) was applied onto both sides of
the template to make ohmic contacts, to which two gold wires were attached on each
side. One pair of the gold wires was connected to a current source, and the voltage
drop was measured across the other pair of gold wires. The current was applied both
forward and backward to eliminate any thermoelectric effects. Since the size of the
silver paint particle is on the order of 1 pm, it is difficult for the silver particles to be
in good contact with each individual nanowire in the template. However, the mushrooms on the back side of the nanowire template (see Fig. 2-11) may be helpful in
making electrical contacts due to the larger extension of these nanowires. To improve
the contact between the silver particles and the nanowires, the top and bottom sides
of the alumina template were partially dissolved before making electrical contacts
107
by dipping the nanowire template in a 5 wt% H3 PO 4 solution for ~ 2 hours. This
exposed a larger portion of the nanowires so that the contact area for the nanowires
was increased. It was found that the total resistance of the nanowire array could be
significantly reduced using this technique, indicating that more wires in the template
were connected to silver particles after this treatment. Since the number of wires connected by the silver paint is unknown, we cannot obtain an absolute value for resistivity. However, the T dependence of resistivity can be studied by normalizing the
resistance R(T) to the resistance at a fixed common temperature (e.g. 300 K). Typical
resistance values for the samples ranged from a few hundred ohms to several thousand
ohms. Recently, a preliminary resistivity measurement on a single Bi nanowire was
carried out using the four-probe method, and the results showed that the resistivity
for 70-nm Bi nanowires at 300 K was 7.8 x 10-4 Q-cm [29], about six times larger
than the resistivity of bulk Bi at 300 K. Assuming this measured resistivity value
for our samples, the number of nanowires connected in each sample is on the order
of 102_103, which is rather small considering the high density of wires in the array
(-
1.2 x 108 wires/mm 2 ). Although the fraction of wires connected is not a signi-
ficant problem for studying the temperature dependence of transport properties, this
fraction has to be dramatically increased to optimize the operation efficiency in practical applications.
Measurements on the temperature-dependent transport properties of Te-doped
Bi nanowires have been performed in a SQUID magnetometer (Quantum Design,
MPMSXL), which is capable of operating in the temperature range of 2-300 K with
a magnetic field of 0-5.4 T. Automatic control and data acquisition with the SQUID
unit, voltage meter (Keithley 181 Nanovoltmeter), and DC current source (Keithley 220 Programmable Current Source) are provided by a central computer, and the
individual devices are connected to the computer through an IEEE 488 port.
108
4.6.2
Transport Measurements of Bi Nanowires
Measurements on the temperature dependence of resistance R(T) of Bi nanowire
arrays have been carried out on samples prepared by the liquid-phase pressure injection
[20] and vapor-phase deposition [60] methods.
Figure 4-17 shows the measured
R(T)/R(300 K) of Bi nanowire arrays of various diameters prepared by the vapor
deposition process, and measured by our collaborator, Dr. J. Heremans of Delphi
Research & Development [221. The vapor deposition process is capable of introducing Bi into anodic alumina templates with smaller pores.
The resulting finer Bi
nanowires have the same preferential crystal orientation as those prepared by the
pressure injection process.
As shown in Fig. 4-17, the temperature dependence of resistance of Bi nanowires
is very different from that of bulk Bi, and is very sensitive to the wire diameter.
At high temperatures (T > 70 K), the resistance of all nanowire arrays shown in
this figure increases with decreasing T.
For T < 70 K, the resistance of the Bi
nanowires of smaller diameters (7 nm, 28 nm, 36 nm, and 48 nm) continues to increase
with decreasing T, while R(T) decreases with decreasing T for Bi nanowire samples
with larger wire diameters (70 nm and 200 nm).
The general trend observed here
is consistent with the previous results on single-crystal Bi wires of larger diameters
(d > 200 nm) [59, 61]. Recently, Huber et al. also reported on the fabrication and
resistance measurements of 200-nm and 30-nm Bi nanowires [21]. They employed a
similar pressure injection technique with a different experimental set-up that could
achieve a higher pressure (by ~ 10 times) than the one described in Chapter 2.
Although the filling factor of Bi nanowires could be increased with this set-up, it was
not clear if the alumina template remained intact under such a high pressure. Due to
the different sample configurations, most of the wires were connected to give a much
smaller sample resistance [21]. Huber et al.'s resistance measurements for 30-nm Bi
nanowire arrays showed qualitative agreement with our results given in Fig. 4-17.
The striking difference in the temperature dependence of resistance between Bi
nanowires arrays and bulk Bi can be explained qualitatively on the basis of the T
109
2.0 -
*
*+ +
n-
.
48 nm Wire diameter
28 nm
1.5 -
-
7 nm
0.
20 nm
1
CI:
+
+
44+
70 nmn
0,5
Bulk Bi-
0.0
'.
10
100
T (K)
Figure 4-17: Temperature dependence of resistance for Bi nanowire arrays of various
diameters prepared by the vapor deposition method, compared to bulk Bi [22].
110
4
36 nm
3
2
1
70 nm
0
10
100
Temperature (K)
Figure 4-18: The calculated temperature dependence of the resistance for Bi
nanowires of 36 nm and 70 nm, using a semi-classical transport model.
dependence of the carrier density n(T) and the mobility [t(T).
For both the Bi
nanowires and bulk Bi, n(T) increases with increasing T, while p(T) decreases with
increasing T. For the nanowires of smaller diameters (< 48 nm in Fig. 4-17), the
increase in n(T) with T outweighs the decrease in p(T), and therefore, the resistance
R(T) decreases with increasing T. However, for semimetallic Bi nanowires of larger
diameters (e.g. 70 nm and 200 nm), the increase in n(T) with increasing T is less
than that for the semiconducting wires of smaller diameters, especially at low temperatures (< 100 K, see Fig. 4-3). Therefore, for bulk Bi or Bi nanowire arrays of
larger diameters (70 nm and 200 nm in Fig. 4-17), the T dependence of p(T) becomes
more important due to the weaker T dependence of n(T), and R(T) increases with
increasing temperature for T < 100 K. For T > 100 K, the increase in n(T) with
increasing T overwhelms the decrease in p(T) for all Bi nanowire samples measured
(7 nm-200 nm), resulting in the decreasing R(T) with increasing T. For instance,
from theoretical calculations, the carrier density of 28-nm and 70-nm Bi nanowires
increases by about 58 and 16 times, respectively, when the temperature increases from
100 K to 300 K. However, since the mobility of bulk Bi decreases only by a factor of
13 from 100 K to 300 K, the resistance of 28-nm and 70-nm Bi nanowires decreases
with temperature for T > 100 K.
111
Based on the transport model established for Bi nanowires (see Sec. 4.1), we calculated the normalized T-dependent resistance R(T)/R(300 K) for 36-nm and 70-nm Bi
nanowires, and the results are shown in Fig. 4-18. The wire diameters of 36 nm and
70 nm are chosen for the calculation here to represent semiconducting and semimetallic Bi nanowires, respectively. In the modeling of Fig. 4-18, some assumptions are
made to account for the discrepancies between an ideal ID Bi quantum wire and a
real Bi nanowire. First, in a perfect single-crystalline Bi quantum wire, there is no
scattering at the wire boundary since the electron density is zero at the boundary,
and possible scattering mechanisms are electron-phonon and electron-electron interactions. However, in a real Bi nanowire sample, the boundary conditions are far from
ideal, and the energy barrier at the boundary is finite instead of infinite. In addition,
real Bi nanowires may have a higher defect level in a thin layer at the boundary
than in the interior of the nanowires due to the conditions at the Bi/A12 0
3
interface.
Therefore, the electrons will experience substantial boundary scattering, due to the
finite-amplitude wave functions at the boundary, and this boundary scattering effect
has been observed in magneto-resistance measurements [22, 31, 33].
In addition to
the wire boundary scattering, the electrons can also be scattered at grain boundaries within real Bi nanowire samples, whereby a domain size on the order of the
wire diameter is observed [62]. However, since the domains in Bi nanowires possess
the same crystal orientation along the wire axis, the small-angle scattering at the
grain boundary would be a minor scattering mechanism in determining the transport
properties. Another important difference between the ideal Bi nanowire and the real
sample is the uncontrolled impurities that act as dopants in Bi nanowires.
For an
ideal semiconducting Bi nanowire, the carrier density should decay exponentially with
decreasing T at low temperatures, resulting in a dramatic increase in the resistance
at low T. Instead, even for the 7-nm Bi nanowires, the measured resistance increases
slowly and steadily with decreasing T (see Fig. 4-17). This effect can be attributed
to the uncontrolled impurities in the Bi nanowires, which give rise to a finite carrier
density at low temperatures. The uncontrolled impurities not only alter the carrier
density, but also decrease the carrier mobility by ionized impurity scattering. The
112
relaxation time TtOt that takes into account all the scattering mechanisms in the real
Bi nanowire is given by Matthiessen's rule,
1
TtOt(T)
where
Tbulk
1
1
_
±
-TbIk(T)
1
ounnary
(
Timp(T)
is the total relaxation time in bulk Bi, and
Tboundary
and Timp are the
relaxation times for boundary scattering (including wire and grain boundary scattering) and ionized impurity scattering, respectively.
The boundary scattering and
ionized impurity scattering are only important at low temperatures (< 100 K), while
phonon scattering becomes the dominant scattering mechanism at higher temperatures (> 100 K). The relaxation time for the ionized impurity scattering in a crystalline semiconductor is approximately proportional to T 3 / 2 [63], while boundary scattering is much less T-dependent; for simplicity,
Tboundary
is assumed to be constant in
T. Since the mobility t is proportional to the relaxation time T, a good approximation
for the mobility is given by,
1
Ptot (T)
where
tbulk(T)
_
1
P/buk(T)
+
1
Pboundary
+
1
Pimp(T)
,(4.24)
can be found in the literature [42], Pboundary is assumed to be constant
in T, and Pimp ~ T 3 /2. The curve for the 36-nm Bi nanowire array in Fig. 4-18 is fit
by pboundary = 33 m 2 V-is-' and pimp = 0.2 x T3/ 2 m 2 V-IS-1, and the carrier density
due to uncontrolled impurities is fitted to Nimp ~_5 x 1016 cm-3. The data for the 70nm Bi nanowires are fit by Pboundary ~ 50 m 2 V-is-
and pimp - 1.0 x T3/ 2 m 2 V-s-1.
Since 70-nm Bi nanowires are semimetallic, the carrier density contribution due to
the very small amount of uncontrolled impurities Nimp can be neglected. The smaller
values of
p-1sdr
for the 70-nm wires relative to that for the 36-nm wires is attributed
to a larger cross-section so that a smaller fraction of atoms is on the nanowire surface.
The trends of the calculated R(T)/R(300 K) curves in Fig. 4-18 are consistent with
the experimental results in Fig. 4-17.
Another interesting feature in the experimental results shown in Fig. 4-17 is the
dependence of R(T)/R(300 K) on the wire diameter d for T < 10 K. We note that for
113
200-nm Bi nanowires, the ratio 0 -- R(4K)/R(300 K) is about 1.0, and decreases to
about 0.9 for 70-nm nanowires. As the wire diameter decreases from 70 nm to 48 nm,
the ratio 13 rises to more than 2, but then decreases again when d is further decreased.
This non-monotonic dependence of /(d)
on the wire diameter d is, however, not
predicted by the present form of our theoretical model. Therefore, more experiments
and theoretical investigations have to be performed to gain insights into this behavior.
It should also be noted that because of the very small geometric dimensions of
the nanowires, the number of carriers is different from that in a bulk material by
many orders of magnitude. For example, for a typical carrier density of
l0l
cm- 3 ,
the number of carriers in a 40-nm nanowire would be only ~ 1200 per pum in length,
and the number decreases quickly as the wire diameter decreases. Therefore, a more
detailed study of the transport properties of Bi nanowires with small diameters should
be undertaken more explicitly, taking into account the small number of carriers in
these wires.
4.6.3
Experimental R(T) Results for Te-doped Bi Nanowires
For this study, we prepared three Bi nanowire samples with different Te dopant
concentrations and one pure Bi nanowire sample for comparison. The wire diameter
of the four samples were all ~ 40 nm.
Prior to the transport measurements, the
nanowire arrays were thermally annealed at 2000 C under vacuum to release the high
residual stress within the nanowires.
Figure 4-19 shows the experimental results on the T dependence of resistivity
for the four Bi nanowire samples. The Te dopant concentrations (0.025 at%, 0.075
at% and 0.15 at%) noted here were based on the amounts of Bi and Te that were
introduced into the quartz tube to form the alloy that was later used in the nanowire
synthesis.
To compare the results of the different samples, the resistance of each
sample was normalized to R(270 K). Figure 4-19 shows that the T dependence of
resistivity exhibits a very different behavior for Bi nanowires with different Te concentrations. For all four samples, the resistance increases as the temperature decreases.
114
2.4
0
2.2
0
0
0undoped Bi
0.025 at% Te
0
0
AA
A
2.0
0
0
1.8
0 0000
000
0.075 at% Te
0
1.6
R~
0
0.15 at% Te
1.4 -O
1.2
1.0
0.8
10
100
Temperature (K)
Figure 4-19: Temperature dependence of resistance for 40-nm pure or Te-doped Bi
nanowire arrays. The resistance of each sample is normalized to R(270 K).
However, the pure Bi nanowires show the strongest T dependence, with R(4 K) being
~ 2.3 times the resistance at 270 K. The T dependence of resistance weakens as the
dopant concentration Nd increases. For the sample with the highest dopant concentration (0.15 at% Te), resistance is almost constant between 2 K and ~ 100 K, while
the resistance of the pure Bi nanowire sample decreases monotonically with increasing T over the entire temperature range measured (2-300 K). The very different T
dependences of resistance can be qualitatively explained in term of the difference in
Nd
in the four Bi nanowire samples.
The temperature dependence of resistivity p is determined by both the carrier
concentration n(T) and the carrier mobility pu(T) via a relation of p oc (ny)
1
.
According to the theoretical calculations in Sec. 3.4, the 40-nm Bi nanowires will
undergo a semimetal-semiconductor transition at T
~_ 180 K. Therefore, n(T) is
strongly perturbed by the presence of Te dopants, especially at low temperatures
(Fig. 4-5). For Te-doped Bi nanowires, the carrier density is approximately equal to
the Te dopant concentration Nd up to a certain threshold temperature Tth, which
increases with Nd. At high temperatures (> 200 K), phonon scattering is usually the
115
dominant scattering mechanism that limits the carrier mobilities, and this mechanism has the same effect on samples of different doping levels. Thus, the resistance
for different samples will exhibit a similar T dependence, since they all have roughly
the same T dependence of n(T) and p(T) at high temperatures, as shown in Fig. 4-5
and Fig. 4-19 for T > 200 K. In this high-temperature regime, the transport properties
of Te-doped Bi nanowires are mainly determined by the intrinsic thermally excited
carriers and by phonon scattering. On the other hand, in the low-temperature regime,
ionized impurity scattering plays a more important role in limiting the mobility of
Te-doped Bi nanowires than for undoped Bi nanowires, so the mobility of Te-doped
Bi nanowires tends to be smaller than that of pure Bi nanowires. However, since the
mobility usually has a much weaker T dependence than n(T), the overall (normalized) resistance will be smaller for the Te-doped Bi nanowires at lower temperatures,
and the normalized resistance R(T)/R(270 K) at low temperatures decreases with
increasing Te dopant concentration (see Fig. 4-19). Thus, Te-doped nanowires are
expected to have a T-dependent resistance similar to the intrinsic nanowires at high
T, and to behave like an extrinsic semiconductor at low T.
The exact donor concentration in the Bi nanowires, which is not known presently,
is likely smaller than the Te concentration in the alloy melt due to possible segregation
of Te atoms during alloy solidification. To a first approximation, we assume that
about 10% of the Te dopant in the alloy melt is present in the final nanowire product,
and each Te atom will donate one electron to the conduction band when ionized.
With this assumption, the donor concentrations Nd for the 0.025 at%, 0.075 at% and
0.15 at% Te-Bi alloy are 6.67 x 1017, 2.0 x 10'" and 4.0 x 1018 cm- 3 in the respective
resulting nanowires. Based on the measured resistance in Fig. 4-19 and the calculated
carrier density, we can obtain the temperature dependence of the average mobility
[tavg
for Bi nanowires with different dopant concentrations. Figure 4-20 shows the T
dependence of [L-1 for the three Te-doped Bi nanowires, in comparison to that for
the pure Bi nanowires. Since the total mobility is approximately determined through
Eq. (4.24), the inverse mobility is more appropriate for describing contributions from
different scattering processes when there are more than one scattering mechanism.
116
An expression similar to Eq. (4.24) for the average mobility
['doped(T)
of the Te-doped
Bi nanowires can be written as,
where
Lpure
1
_
Pdoped(T)
-
1
+
[pure(T)
1
p['imp(T)
+
1
,(4.25)
['defect
is the average mobility of the pure Bi nanowire with the same diameter,
[limp is the contribution from the increased ionized impurity scattering in Te-doped
Bi nanowires, and Pdefect is the contribution from the possibly higher defect level in
Te-doped Bi nanowires. We note that at high temperatures (> 200 K), the curves of
the three Te-doped Bi nanowire samples in Fig. 4-20 all have a similar temperature
dependence as that of the pure Bi nanowire sample, indicating that
major contributor to p
ed
[pure
is the
at high temperatures. The average mobility of pure Bi
nanowires can be well fitted by a
[pure
~'-T- 2 9. dependence for T > 100 K, as shown
by the dashed line in Fig. 4-20. The T-2 9. dependence of Apure is believed to arise
predominantly from phonon scattering at high temperatures.
For most materials,
phonon scattering gives rise to a T-dependent mobility Pphonon - M *-5/2 T -3/2
T-'- [63]. However, Pphono, in Bi has a stronger T dependence than T-"
a
because
the effective mass m* of Bi increases with increasing T, resulting in a T-29 dependence
for Ppure.
The general trends of the curves for the Te-doped Bi nanowires can be qualitatively explained by the T dependence of Ppure,
['imp,
and
['defect.
Since the scattering
effect due to defects is similar to the neutral impurity scattering, which is essentially
temperature independent, Adefect may be treated as T independent compared to other
scattering mechanisms. As T decreases, the contribution of
the contributions of pconcentrations. Since p
minimum in
[-t- I,
[p-ire
drops rapidly, and
and p-eiject begin to dominate, depending on the Te dopant
generally increases with decreasing temperature, we see a
for Te-doped Bi nanowires.
However, we note that the T 3 / 2 dependence for pimp, which is a common temperature dependence for ionized impurity scattering in most materials, can hardly account
for the T dependence of
['deieI
for Te-doped Bi nanowires in Fig. 4-20, which shows a
117
1.0 -Nd
4xl 1'
cm-
0.8 -
/
_=_0.6 -N
= 2x18"CM3
cj 0.4
cm
0.2 -Nd= 6.67x1 0
,0
®&
c
0.0
. p p-0
0
N
=-0
'1
300
200
100
0
Temperature (K)
Figure 4-20: Calculated temperature dependence of Ip-Yjre and f-doped for 40-nm pure
and Te-doped Bi nanowires, respectively. The calculation is based on the measured
T-dependent resistance in Fig. 4-19 and the calculated carrier concentration similar
to Fig. 4-3. The dashed line is a curve fitting of the mobility of pure Bi nanowires to
a T-2.9 dependence for T > 100 K.
much weaker temperature dependence than T-3/
for the anomalous T dependence of
jpimp
2
at low temperatures. The reason
for Te-doped Bi nanowires is explained as
follows. First, we recall that the scattering time Timp for the ionized impurity scattering is proportional to m*2v 3 where v is the carrier group velocity [63]. For an ordinary
2
material with a parabolic dispersion relation E cx k /m*,rimp
dependence of pimp is then recovered by averaging
(Timp)
c m*1/ 2 E 3/ 2 . The T 3 / 2
with a proper distribution
function. However, the dispersion relations in both bulk Bi and in Bi nanowires are
non-parabolic (Eq. 3.52), and the group velocity of electrons at the (n, m) subband
is calculated as
I
Vnm(k)
1
h
Enm(k/)
)
&k
2
2E
'rni
\
1 +
rz E
2
2h 2 k2
1/2
(4.26)
(.6
whereby 7ynm and Enm(k) are defined in Eqs. (3.54) and (3.52), respectively. We note
that for electrons located very close to the subband edge, Eq. (4.26) can be written as
118
the familiar relation Vnm =hk/m*m; while for electrons far from the subband edge,
the second term in the bracket of Eq. (4.26) becomes negligible, and the group velocity
Vnm
~
2EgL/rl,
which is independent of the electron energy and the subband index.
This very different relation between the group velocity v and the electron energy E
will result in a different T dependence for [Limp in Te-doped Bi nanowires than for
ordinary materials. In Te-doped Bi nanowires, due to the large increase of electrons,
the Fermi energy will move into the conduction band, and those electrons that are
responsible for the transport phenomena will have an approximately E-independent
group velocity v ~~
2EgL/qTz as discussed above. It follows that the relaxation time
of ionized impurity scattering Timp, which is still proportional to m*2 v 3 , would be
approximately E independent, resulting in an almost T-independent (Timp).
There-
fore, the T dependence of [imp for Te-doped Bi nanowires should be much weaker,
and this conclusion is consistent with the results in Fig. 4-20, which shows an almost
constant mobility for doped samples at low temperatures (T < 100 K). However,
although the mobility [Limp = e KTimp) /m* oc m*-1/2EL does not have an explicit T
dependence, it is in fact T dependent because EgL(T) and m*(T) in Bi are highly T
dependent at higher temperatures (T > 80 K). Table 3.2, which gives the T dependence of band structure parameters of Bi, shows that EgL and m* both increase with T,
and the resulting [imp c m*1/2Eg
increases with T almost linearly in the interme-
diate temperature range (100 K < T < 250 K). At low and intermediate temperatures
(T < 200 K), p-1
is the dominant term in determining
[pLJed,
while at high temper-
atures (T > 200 K), y,e becomes more important. Therefore, a minimum in
peLd1
is observed around 200 K for the more heavily doped Bi nanowires (see Fig. 4-20).
4.7
Seebeck Measurements
In order to study the thermoelectric properties of Bi nanowire systems, it is necessary
to measure the thermoelectric power, or the Seebeck coefficient S. Unlike the resistivity measurement, whereby a single wire measurement with a four-probe set-up
119
is preferred, the Seebeck coefficient measurement is essentially independent of the
number of nanowires connected because S only depends on the temperature gradient
AT and the resulting thermoelectric voltage AV. Therefore, the Bi/A12 0 3 nanocomposite provides a convenient system for the direct measurement of thermoelectric
power for Bi nanowires. In addition, a careful measurement of the Seebeck coefficient can provide valuable information about the position of the Fermi energy, so as
to verify our theoretical model. However, due to the geometric limit (small thickness) of the sample, the Seebeck coefficient measurement has been very challenging.
Substantial advance in the measurement technique was made recently to successfully
characterize the Seebeck coefficient of Bi nanowire arrays reproducibly. In the following sections, the detailed experimental set-up for the Seebeck coefficient measurement
will be presented, and the experimental results will be discussed.
4.7.1
Experimental Set-up
Schematic of the experimental set-up developed for the Seebeck measurement of Bi
nanowire arrays is shown in Fig. 4-21. Two thermocouples were attached to each side
of the nanowire array by silver paint, which provides a good electrical and thermal
contact between the thermocouples and the nanowire array.
Since the sample is
usually only several mm 2 in area with a thickness < 60 ,um, a thermocouple with
a size on the order of several pm or less is required to reduce the measurement
error. The smallest commercially available thermocouples are Omega's "cement on"
type-E (Chromel-Constantan) thin foil thermocouples with a thickness of 0.0005 inch
(12.7 pm). The sample, along with these thermocouples, are then placed on a metal
substrate that acts as a heat sink. A thin mica film is inserted between the metal
substrate and the lower thermocouple to avoid shorting of the thermocouple, at the
same time it provides good thermal conduction to the heat sink. A small foil strain
gage with a resistance of 120 Q (Omega Engineering Inc.) was attached to the top of
the sample as a heater. The temperature gradient AT across the sample was measured
from the voltage difference between the two thermocouples, and the thermoelectric
120
Thermocouples
Current Source
,
_Heater
Ag
Mica Fil/
Metal Substrate
Figure 4-21: Schematic of the experimental set-up used for the Seebeck coefficient
measurement of Bi nanowire arrays.
voltage was measured through the two Chromel wires in the two thermocouples.
Due to the small heat capacity of the sample and the thermocouples, a constant
temperature gradient was established shortly after the heater was connected to a
constant current source. To produce a continuous temperature gradient AT across
the sample, a current source with a triangular waveform output was used.
Figure 4-22 shows the typical results for the measured AV as a function of AT for
a 40-nm Te-doped Bi nanowire array at room temperature. In spite of the thinness of
the nanowire arrays, an appreciable temperature gradient of ~- 3 K was achieved. The
data sets exhibit good linearity and a very small offset (-0.26 pV), indicating good
contacts to the Bi nanowires. By performing a linear regression on the AV-AT data
sets, the slope was determined to be -58.4 pV/K. Since AV was measured through
Chromel wires, which have a Seebeck coefficient of 23.0 pV/K at 300 K, the Seebeck
coefficient of the Bi nanowire array was calculated as S = -58.4+23.0 = -35.4 pV/K.
It should be noted that since the thermocouple size was comparable to the sample
thickness, there might be a substantial discrepancy between the measured temperature gradient AT and the true temperature gradient across the nanowire array. The
Seebeck coefficient obtained likely underestimated the true Seebeck coefficient of the
Bi nanowire array. A slightly different technique for measuring the Seebeck coefficient of Bi nanowire arrays has been developed by Dr. J. Heremans
[64],
whereby
two thermocouples are mounted differentially across the sample to measure AT. The
121
0
ElE
AV = -58.4AT - 0.26 (pV)
-50
-
-l -100
>
Elio
ED
ElD
E
-150
=-200
E
LI LI
ED E
E
-200
l
D]E
I
0.0
0.5
1.0
1.5
AT (K)
2.0
2.5
3.0
Figure 4-22: Measured thermoelectric voltage AV as a function of the temperature
gradient AT of a 40-nm 0.075 at% Te-doped Bi nanowire array at 300 K. The Seebeck
coefficient of the Bi nanowire array is calculated as S = -58.4 + 23.0 = -35.4 PV/K.
results derived from the two experimental set-ups are in good agreement, and they
both have comparable experimental errors.
4.7.2
Experimental Results and Discussion
Figure 4-23 shows the measured temperature dependence of Seebeck coefficient for 40nm Bi nanowire arrays with different Te dopant concentrations. The measurements
began at 300 K, and the samples were steadily cooled down to 4 K with data taken
at discrete temperatures.
The process was then repeated in a slow heating cycle
to 300 K. The good reproducibility of the measurement could be inferred from the
negligible hysteresis between the cooling and heating curves for each sample. For the
40-nm pure Bi nanowire array, however, the electrical contact was lost during the
heating cycle when T > 100 K, and the results above 100 K were therefore not shown.
The Seebeck coefficient of bulk Bi at 300 K is about -100 IV/K along the trigonal
axis, and about -50 [tV/K for directions perpendicular to the trigonal axis [41]. For
122
-10
>
-20
Cl)
-30
-40
0
100
200
300
Temperature (K)
Figure 4-23: Measured temperature dependence of Seebeck coefficient for 40-nm Bi
nanowire arrays with Oat% (0), 0.075at% (A) and 0.15at% (0) Te. The cooling
and heating curves are shown as solid and dashed lines, respectively.
the 40-nm Bi nanowires measured here and the 200-nm Bi nanowires measured by
Heremans et al. with a slightly different method
[64],
a lower thermoelectric power
than bulk Bi was obtained. This could be due to the underestimation of the Seebeck
coefficient of Bi nanowire arrays associated with the experimental set-ups, as discussed
in Sec. 4.7.1.
In Fig. 4-23, we note that for the Bi nanowire array with the highest Te dopant
concentration (0.15 at% Te), the Seebeck coefficient is approximately linear with T
throughout the temperature range measured (4-300 K). For the 0.075 at% Te-doped
Bi nanowire array, the Seebeck coefficient is approximately proportional to T at low
temperatures (T < 100 K), and becomes fairly constant for T > 150 K. In addition,
a local minimum in the Seebeck coefficient is noted at ~ 150 K. For the undoped
Bi nanowire array, the Seebeck coefficient is also linear with T at low temperatures
(T < 100 K) before the electrical contacts are lost. According to Fig. 4-23, the thermoelectric power of Bi nanowires decreases with increasing Te dopant concentration,
and at low temperatures (T < 100 K), the slopes (dS/dT) in the linear region are
123
0
o
11100 ppm
A
o
7000
3750
.
40
-50-
-100.
0
100
200
300
Temperature (K)
Figure 4-24: Thermoelectric power of 1 [m-thick BiO. 91SbO.09 films with different Te
dopant concentrations as a function of temperature (Cho et al. [65]).
-0.53,
-0.26 and -0.11
pV/K 2 for undoped, 0.075 at% and 0.15 at% Te-doped Bi
nanowire arrays, respectively.
The temperature dependence of Seebeck coefficient for the 40-nm Bi nanowires
with different Te dopant concentrations is consistent with the results for Te-doped
Bio. 91SbO.09 thin films reported by Cho et al. [65] (Fig. 4-24). The addition of a small
amount of antimony (Sb) to bismuth causes the T-point holes in Bii_,Sb, to move
down, and at x ~_0.07, the overlap between the T-point holes and L-point bands disappears, resulting in a semiconducting film. Cho et al. found that the thermoelectric
power of BiO. 91 SbO.09 thin films decreases with increasing Te dopant concentration.
They also observed that for the samples with the highest Te dopant concentrations
(11100 ppm and 7000 ppm), the Seebeck coefficients were linear with T for temperatures up to 300 K. For the BiO. 91 SbO.0 9 samples with lower Te dopant concentrations,
the Seebeck coefficients were linear with T only up to a certain temperature, and
then the thermoelectric power varied slowly with increasing T.
For both the 40-nm Bi nanowires and thin BiO.9 1 SbO.o films, experimental results
show that the thermoelectric power decreases with increasing Te dopant concentra124
tion. However, for 200-nm nanowires, Heremans et al. reported a higher thermoelectric power for Te-doped Bi nanowires than for the undoped Bi nanowires
[64].
The
different effect of Te doping on the 40-nm and 200-nm Bi nanowire arrays can be
attributed to the semimetal-semiconductor transition that occurs in the former. 200nm pure Bi nanowires are semimetallic with equal numbers of electrons and holes; the
contributions from both carriers would cancel each other, resulting in a small Seebeck
coefficient. The introduction of Te dopants into 200-nm Bi nanowires would increase
the n-type carriers, and therefore, increase the thermoelectric power by reducing
the opposite contribution from the holes.
On the other hand, for a typical semi-
conducting system, the thermoelectric power always decreases with increasing Fermi
energy near the conduction band edge. Therefore, for both 40-nm Bi nanowires and
thin Bio. 91Sbo.o9 films, the thermoelectric power decreases with increasing Te dopant
concentration due to an increased Fermi energy. The similarity in the Te dopant
effect on the thermoelectric power for 40-nm Bi nanowires and thin Bi 0 .91 Sb 0 .0 9 films
is interesting, and provides a strong evidence for the semimetal-semiconductor transition in the 40-nm Bi nanowires.
125
Chapter 5
Conclusions and Future Directions
5.1
Conclusions
In this thesis, techniques for fabricating self-assembled Te-doped Bi quantum wire
arrays are presented, and thermoelectric-related transport properties of the iD systems
are extensively studied both theoretically and experimentally.
The fabrication of Te-doped Bi nanowires is based on the processes established
recently for synthesizing Bi nanowires in porous alumina templates.
The quality
of the porous anodic alumina templates has been significantly improved by using a
two-step anodization method, which produces a highly ordered hexagonal array of
pores with uniform diameters. Highly crystalline, dense, and continuous Bi nanowire
arrays are fabricated by pressure injecting molten Bi into the evacuated channels of
the porous alumina templates, and similar techniques have also been used to produce
Bi nanowires with 0.025-0.15 at% Te dopants.
XRD studies show that the pure
and Te-doped Bi nanowires possess the same rhombohedral structure as bulk Bi.
The nanowire arrays exhibit a preferred wire orientation along the [1011] and [01i2]
directions when the wire diameters are > 60 nm and < 50 nm, respectively.
Substantial improvements on the theoretical modeling of the electronic band
structure of Bi nanowires have been made to examine the transport properties of
Bi nanowires. Circular wire boundary conditions, along with anisotropic and nonparabolic dispersion relations for carriers, are explicitly taken into account in deter126
mining the band structure of Bi nanowires. The semimetal-semiconductor transition
that is found to occur in Bi nanowires is very important to the transport properties
of these ID systems. Theoretical investigation of thermoelectric properties indicates
that Bi nanowires with very small diameters (< 10 nm) are very promising for thermoelectric applications.
The performance of p-type Bi nanowires, which are usually
inferior to the corresponding n-type nanowires of the same diameter, can be dramatically enhanced if the T-point holes are removed or suppressed. Various transport
properties of pure and Te-doped Bi nanowires have also been measured experimentally. The very different behavior of R(T) observed for pure Bi nanowires of different diameters provides a strong evidence for the semimetal-semiconductor transition
in these systems. The experimental results are in good agreement with theoretical
modeling. The theoretically predicted effects of Te doping are confirmed by resistance
and Seebeck coefficient measurements of 40-nm Bi nanowires with various Te dopant
concentrations. In addition, comparison of the Seebeck coefficient measurements for
Bi nanowires and thin BiO. 09 1 Sbo.og films corroborates the Te doping effect and the
semimetal-semiconductor transition in Bi nanowires.
5.2
Suggestions for Future Studies
Bi nanowires presented in this thesis represent an interesting system for studying ID
systems, and a promising material for thermoelectric applications. This work has
stimulated several research directions and application opportunities that are worthy
of further pursuit.
First, in order to fully understand the transport phenomenon in Bi nanowires, a
few more experimental and theoretical studies are needed. To improve the theoretical
model of transport properties for Bi nanowire systems, various scattering processes
in the 1D system should be included explicitly, and the small number of carriers
(< 100 electrons/pm in length) in the nanowires also has to be taken into consideration. In addition, several more measurements on pure and Te-doped Bi nanowires
127
would be useful to reveal their band structure and carrier concentrations, and to
obtain conclusive comparisons with the theoretical predictions. Preliminary magnetoresistance measurements have been performed, and in conjunction with models yet
to be developed, these measurements can provide information on carrier mobilities
and scattering processes. Optical measurements can be used to determine the direct
bandgap, and possibly, the Fermi energy and carrier densities. While two-point resistance measurements have been performed on nanowire arrays, the absolute resistivity
o-
of a single nanowire is necessary to quantitatively evaluate the overall transport
behavior and thermoelectric performance ZT of these new materials.
Secondly, as discussed in Sec. 4.5, the thermoelectric performance of Bi nanowires
can be considerably enhanced by removing the T-point holes, especially for p-type
nanowires. The introduction of Sb into Bi nanowires provides a promising approach
to achieve this goal by lowering the T-point holes in energy. Bi-,Sb, nanowires may
be fabricated based on the techniques developed in Chapter 2.
Thirdly, the unique characteristics of the self-assembled pore structures in anodic
alumina templates may be useful for various other applications that require a high
degree of pore regularity, such as photonic bandgap materials, and host materials for
nanoscale magnetic storage media and field emitter arrays. The packing order and
uniformity of pore channels in anodic alumina may be further enhanced by subjecting
the aluminum substrate to photolithographic processes, which can create a desired
pattern on the metal film prior to anodic oxidation. The aluminum film can then
be anodized under appropriate conditions to induce the formation of pore structures
that replicate the lithographic patterns.
128
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