Search for Resonant Impurities in Bismuth and Bismuth-Antimony Alloys: Lithium,
Magnesium, and Sodium
THESIS
Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in
the Graduate School of The Ohio State University
By
Christine M. Orovets
Graduate Program in Mechanical Engineering
The Ohio State University
2012
Master's Examination Committee:
Joseph Heremans, Advisor
Shaurya Prakash
Copyright by
Christine M. Orovets
2012
Abstract
Thermoelectricity, the conversion of heat to electricity and vice-versa, is becoming
increasingly popular in response to the present-day demand for green energy.
Thermoelectric devices have the ability to reduce carbon dioxide and other greenhouse
gas emissions, and have the benefit of requiring very little maintenance. Moreover,
thermoelectric devices have great potential for use in solid state cooling applications,
such as infrared, x-ray, and gamma ray detectors on UAVs and Situational Awareness
Satellites. For this reason, thermoelectric materials for use at cryogenic temperatures are
studied in this thesis.
Bismuth-antimony crystals have been found to be the most efficient n-type thermoelectric
material in the 2 K–150 K range. Theoretical calculations predicted that doping this
material with certain metals would result in a “resonant level”, a mechanism in which
thermopower is increased due to a distortion in the electronic density of states. Both pure
bismuth and bismuth-antimony alloys were doped with lithium, magnesium, and sodium
in search of these resonant levels. The materials were tested for electrical resistivity,
thermal conductivity, and thermopower in order to determine an optimized figure of
merit, used to characterize thermoelectric materials. Further studies were performed to
fully understand the mechanisms of each dopant, such as scanning electron micrographs
as well as Hall and Shubnikov-de Haas measurements.
ii
The data shows that Li and Mg were each electrically active in bismuth and bismuthantimony, while Na likely was not. By comparing the data to the theoretical density of
states, it was found that lithium was a strong n-type dopant that was an interstitial
impurity in the alloys. The eutectic composition formed a secondary “BiLi” phase that
resulted in a lower electrically resistivity and higher thermal conductivity than other
compositions of bismuth-lithium. The magnesium is likely a weak n-type dopant in the
bismuth and bismuth-antimony alloys. Data show that sodium did not dope the alloys.
None of the three impurities showed improved figures of merit at cryogenic temperatures.
However, the Shubnikov de-Haas measurements of the magnesium-doped alloys suggest
that a new carrier pocket was induced with ≥ 0.5% magnesium.
iii
This document is dedicated to my parents and sisters for their support and
encouragement, especially during my time here at Ohio State.
iv
Acknowledgments
I would like to acknowledge Dr. Heremans for his advising during my time here at Ohio
State, both in my undergraduate and graduate studies. His experience and teaching have
been invaluable in helping me research and present my findings.
I would also like to acknowledge Audrey Chamoire for training me in sample
preparation, lab techniques, and publications, as well as Hyungyu Jin for all his help with
the bismuth-antimony alloy work. Thank you also to Bartlomiej Wiendlocha for his
theoretical calculations.
My work was supported by the AFOSR MURI “Cryogenic Peltier Cooling”, Contract
FA9550-10-1-0533.
v
Vita
January 14, 1988 ............................................Born – Cleveland Heights, Ohio
2006................................................................Beaumont School
2010................................................................B.S. Mechanical Engineering, The Ohio
State University
2011 to present ..............................................Graduate Research Associate, Department
of Mechanical and Aerospace Engineering,
The Ohio State University
Publications
Orovets, Christine M., A. Chamoire, H. Jin, B. Wiendlocha, and J. P. Heremans.
"Lithium as an Interstitial Donor in Bismuth and Bismuth-Antimony Alloys." JEM 03615235 (2011). Web. <http://dx.doi.org/10.1007/s11664-011-1861-0>.
Fields of Study
Major Field: Mechanical Engineering
vi
Table of Contents
Abstract ............................................................................................................................... ii
Acknowledgments............................................................................................................... v
Vita..................................................................................................................................... vi
List of Tables ..................................................................................................................... ix
List of Figures ..................................................................................................................... x
Chapter 1: Introduction to Thermoelectrics ....................................................................... 1
1.1
Motivation ............................................................................................................ 1
1.2
Seebeck, Peltier, and Thomson Effects ................................................................ 3
1.3
Magnetic Field Effects ......................................................................................... 5
1.4
Thermoelectric Devices........................................................................................ 7
1.5
Figure of Merit and Thermal Conductivity ........................................................ 10
1.6
Matching n-type and p-type Material in a Device.............................................. 13
1.7
Thermoelectric Material Overview .................................................................... 13
1.8
Shubnikov-de Haas Oscillations and the Fermi Surface .................................... 16
1.9
Resonant Levels ................................................................................................. 22
Chapter 2: Experimental Methods .................................................................................... 26
vii
2.1
Sample Preparation ............................................................................................ 26
2.2
Experimental Setup ............................................................................................ 27
2.3
Sources of Error ................................................................................................. 30
Chapter 3: Bismuth and Bismuth-Antimony Alloys ......................................................... 31
3.1
Introduction ........................................................................................................ 31
3.2
Bi + Li and BiSb + Li ......................................................................................... 34
3.3
Bi + Mg and BiSb + Mg ..................................................................................... 44
3.4
Bi + Na and BiSb + Na ...................................................................................... 52
3.5
Conclusions ........................................................................................................ 59
References ..................................................................................................................... 60
viii
List of Tables
Table 1: Shubnikov-de Haas and Hall data for Bi and BiSb + Mg at 2 K ........................ 50
Table 2: Shubnikov-de Haas and Hall data for Bi and Bi + Na at 2 K ............................. 56
ix
List of Figures
Fig. 1: Comparison of TE devices and steam engines (From Vining2) .............................. 2
Fig. 2: Schematic of a thermocouple .................................................................................. 3
Fig. 3: Hall Effect ............................................................................................................... 6
Fig. 4: Schematic of a thermoelectric device ...................................................................... 8
Fig. 5: Representation of n-type and p-type semiconductor systems ............................... 15
Fig. 6: The Fermi-Dirac function10 ................................................................................... 17
Fig. 7: First Brillouin zone for a square lattice ................................................................. 18
Fig. 8: Orbit of k-vector outlined by intersection of simple Fermi surface and plane
perpendicular to magnetic field (from Ziman10) ............................................................... 19
Fig. 9: Representation of quantization of orbits (From Ziman10) ..................................... 20
Fig. 10: Various mechanisms that distort a system’s density of states ............................. 22
Fig. 11: Definition of resonant levels in terms of dispersion relation and DOS ............... 24
Fig. 12: Experimental setup for PPMS, Thermal Transport Option ................................. 28
Fig. 13: Experimental setup for PPMS, AC transport option for Hall measurement ....... 30
Fig. 14: Binary phase diagram for Bi-Sb system (1992 Feutelais Y.)13 ........................... 32
Fig. 15: Schematic diagram of bands near the Fermi level of Bi1-xSbx alloys as a function
of x at 0 K (From Lenoir14). .............................................................................................. 33
Fig. 16: Theoretical DOS of Bi doped with Li for (a, b) substitutional impurity and (c)
interstitial impurity18 ......................................................................................................... 35
x
Fig. 17: Binary phase diagram for Bi-Li system (1994 Gasior W.)13 ............................... 35
Fig. 18: Secondary phase BiLi structure ........................................................................... 36
Fig. 19: (a) XRD results of Bi100-xLix with BiLi secondary phase peaks identified; SEM
results of (b) eutectic composition Bi86Li14, (c) Bi97Li3, (d) Bi80Li20 ............................... 37
Fig. 20: Electrical resistivity and thermal conductivity of Bi100-xLix versus temperature . 38
Fig. 21: SEM of polycrystalline (Bi88Sb12)86Li14 .............................................................. 39
Fig. 22: Electrical resistivity of Bi88Sb12 doped with indicated amounts of Li, in single
crystals (left) and polycrystals (right); measured along trigonal axis direction of single
crystals .............................................................................................................................. 40
Fig. 23: Thermal conductivity of Bi88Sb12 doped with indicated amounts of Li, in single
crystals (left) and polycrystals (right); measured along trigonal axis direction of single
crystals .............................................................................................................................. 41
Fig. 24: Thermopower of Bi88Sb12 doped with indicated amounts of Li, in single crystals
(left) and polycrystals (right); measured along trigonal axis direction of single crystals . 42
Fig. 25: Figure of merit of Bi88Sb12 doped with indicated amounts of Li, in single crystals
(left) and polycrystals (right); measured along trigonal axis direction of single crystals . 43
Fig. 26: Theoretical DOS of Bi doped with Mg for substitutional impurity (from
Bartlomiej Wiendlocha, AGH University of Science and Technology) ........................... 44
Fig. 27: Binary phase diagram for Bi-Mg system (1992 Oh C.S.)13 ................................ 45
Fig. 28: Carrier concentrations of Bi and BiSb samples doped with Mg in single crystals,
obtained by measuring the Hall coefficient along the trigonal axis direction up to 1 Tesla
(Hall Prefactor = 2/3). ....................................................................................................... 46
xi
Fig. 29: Thermal transport properties of Bi and BiSb alloys doped with 0.2% Mg
including (a) Seebeck coefficient, (b) thermal conductivity, (c) electrical resistivity, and
(d) figure of merit vs. temperature. All samples were single crystals measured along the
trigonal axis direction. ...................................................................................................... 47
Fig. 30: Thermal transport properties of BiSb alloys doped with 0.7% Mg including (a)
Seebeck coefficient, (b) thermal conductivity, (c) electrical resistivity, and (d) figure of
merit vs. temperature. All samples were single crystals measured along the trigonal axis
direction. ........................................................................................................................... 49
Fig. 31: Shubnikov-de Haas traces with background subtraction. The left shows pure Bi
with one clear period. The right shows BiSb with a new period beginning near 5 T. ..... 52
Fig. 32: Theoretical DOS of Bi doped with Na for substitutional impurity (from
Bartlomiej Wiendlocha, AGH University of Science and Technology) ........................... 53
Fig. 33: Binary phase diagram for Bi-Na system (1990 Sangster J.)13 ............................. 53
Fig. 34: Carrier concentrations of Bi and BiSb samples doped with Na in single crystals,
obtained by measuring the Hall coefficient along the trigonal axis direction up to 1 Tesla
(Hall Prefactor = 2/3). ....................................................................................................... 54
Fig. 35: Thermal transport properties of Bi and Bi doped with 0.2% Na including (a)
Seebeck coefficient, (b) thermal conductivity, (c) electrical resistivity, and (d) figure of
merit vs. temperature. All samples were single crystals measured along the trigonal axis
direction. ........................................................................................................................... 55
Fig. 36: Shubnikov-de Haas traces with background subtraction..................................... 56
xii
Fig. 37: Thermal transport properties of BiSb doped with Na including (a) Seebeck
coefficient, (b) thermal conductivity, (c) electrical resistivity, and (d) figure of merit vs.
temperature. All samples were single crystals measured along the trigonal axis direction
(Sb contents are unknown)................................................................................................ 58
xiii
Chapter 1: Introduction to Thermoelectrics
1.1 Motivation
Thermoelectric (TE) devices are systems that convert thermal energy into electricity for
power needs, or electricity into thermal energy for heating and cooling purposes. These
devices have great potential as green energy systems; thermoelectricity has the ability to
reduce carbon-dioxide and other greenhouse gas emissions1. Moreover, TE devices are
solid state and require very little maintenance since there are no moving parts, leading to
a virtually unlimited lifespan. Another benefit is that TE devices are scalable (1W –
1kW) so they can be used for a range of purposes, from pocket stoves to automotive
waste heat recovery to cryogenic cooling on satellites. For these reasons,
thermoelectricity is being explored with goals of improving efficiency and reducing
costs.
However, thermoelectric applications are limited presently, the main reason being that
TE devices are much less efficient than steam generators at high power levels:
1
Fig. 1: Comparison of TE devices and steam engines (From Vining2)
As the comparison shows, the overall more efficient steam cycles require larger, heavier
machines, which is why TE devices are beneficial at lower power levels. They are also
useful when placed in conjunction with existing technologies such as in waste heat
recovery systems for automobiles.
The main motivation for this work is the need for solid state cooling materials to be used
by the US Air Force. These materials are to be utilized in Peltier cooling devices, which
are extremely lightweight and portable compared to large Stirling engines. They also
eliminate the need for environmentally unfriendly refrigerants. Cryogenic cooling
materials, namely at the 10K-150K range, are being researched for these devices, to be
used in infrared, x-ray, and gamma ray detectors on UAVs (Unmanned Aerial Vehicles)
and Situational Awareness Satellites. Bismuth-antimony crystals are the best n-type
thermoelectric material in the cryogenic range presently15.
2
1.2 Seebeck, Peltier, and Thomson Effects
Thomas Johann Seebeck discovered the first thermoelectric phenomenon in 1821 which
was later called the Seebeck effect. The Seebeck effect occurs in a material with a
temperature gradient, or when two different materials are joined at both ends. When one
junction between the metals is heated and the other is kept cool, a voltage difference can
be recorded. In wire form, this setup is called a thermocouple, as shown below:
A
Thot
Tcold
B
Fig. 2: Schematic of a thermocouple
The Seebeck coefficient, S, (also called thermopower or thermoelectric power) is defined
as:
(1.1)
where ∆V is the potential difference and ∆T is the temperature difference between the hot
and cold junctions. The temperature difference causes the majority carriers to travel to
the cold junction side, causing that side to be positive if the carriers are holes and
negative if the carriers are electrons. The hot junction side takes the opposite polarity.
Each material has its own Seebeck coefficient, and the Seebeck coefficient of a junction
between two materials is the difference between the Seebeck coefficients of the two
3
materials. Metals have smaller absolute Seebeck coefficients than semiconductors and
oxides since they usually have half filled bands and thus higher carrier concentrations
(Metals have Seebeck coefficients less than 10 μV/K or less, while semiconductors are
usually greater than 100 μV/K). Values for semimetals typically lay in between.
The second thermoelectric phenomenon was discovered in 1834 by Jean-Charles Peltier.
The Peltier effect is basically the reverse of the Seebeck effect: heating or cooling occurs
when a current is passed through a thermocouple as shown in Fig. 2 (whether heating or
cooling takes place depends on the direction of the current). The Peltier coefficient, Π, is
defined as:
(1.2)
where q is the heat flow and I is the electric current. If the junction the current enters is
heated and the opposite junction on the same wire is cooled as the current exits, then Π is
positive.
In 1855, William Thomson (also known as Lord Kelvin) found that the Seebeck and
Peltier effects were related, and determined the following:
(1.3)
where T is the absolute temperature at the junction between the metals. Thomson also
discovered what is now known as the Thomson effect in homogeneous conductors in
which heating or cooling occurs with current flow and a temperature gradient:
(1.4)
4
In equation (1.4),
is the heat absorbed or released, I is electric current,
is the
temperature gradient, and τ is the Thomson coefficient (defined as the rate of heat per
unit length). The Thomson and Seebeck coefficients are also related:
(1.5)
The Seebeck coefficient reflects carrier entropy, which changes with temperature since
various temperatures mean various amounts of heat carried3. From this we know that S =
0 at 0 K.
The Seebeck, Peltier, and Thomson effects are all thermodynamically reversible, but are
always accompanied by joule heating which is irreversible. Joule heating occurs when
heat is released due to the passing of an electric current through a conductor (proportional
to
where R is the material’s resistance). Electrons (or holes) collide with other
carriers or lattice vibrations when moving through the material which produces the heat.
1.3 Magnetic Field Effects
There are also thermoelectric phenomena that occur with application of a magnetic field.
The Hall Effect was discovered by Edwin Hall in 1879 who was studying magnetic field
effects on currents in conductors. Hall found the following relation:
(1.6)
where EH is the electric field in the y-direction (“Hall Field”), RH is the Hall coefficient,
Ix is the current in the x-direction, and Bz is the magnetic field in the z-direction. Thus, an
5
electric field is created that is perpendicular to the magnetic field and the current flow
that are applied, as shown below:
Fig. 3: Hall Effect
This is explained by the Lorentz force pointing in the opposite direction that the electric
field must compensate for, in order for the electrons to travel through the wire:
(1.7)
In the above equation, the entire right hand side is the scalar of the Lorentz force, vx is the
velocity of an electron, and e is the electron charge (equal to ±1.60217733E-19 C, where
the sign depends on the carrier). As shown in equation (1.7), the Lorentz force is the
force on a carrier with charge ±e and velocity vx due to magnetic field Bz. In other words,
the Lorentz force causes the carriers to deflect, causing them to create a balancing electric
field in the opposite direction (
. Solving for EH in equation (1.7) and substituting
into equation (1.6) gives:
6
(1.8)
where
and n is electron density4. This same logic could be used for hole
carrier density, p. This shows how the Hall coefficient can be measured and used to find
carrier density (n or p) of a material. Also, a negative Hall coefficient implies an n-type
material (the majority carriers are electrons) and a positive Hall coefficient implies a ptype material (the majority carriers are holes). It should be noted that equation (1.8)
applies to a one carrier system. Also, when the real band structure of solids is taken into
account, the relation is an initial approximation only.
The Nernst effect is a transverse voltage that appears in the presence of a magnetic field,
but as a result of a thermal gradient that arises in the x-direction (in the direction of
current flow). The Nernst coefficient is given by:
(1.9)
A thermal gradient also appears perpendicular to the current flow (in the y-direction) in
the presence of a magnetic field in what is called the Ettingshausen effect. Both the
Nernst and Ettingshausen effects are discussed further in several texts3,5.
1.4 Thermoelectric Devices
Thermoelectric devices can include generators as well as heat pumps and coolers. First,
we will look at thermoelectric generators. In a TE generator, a temperature gradient is
7
applied to a p-type/n-type configuration to produce electricity, and a heat flux is applied
to maintain this temperature gradient, as shown here.
Fig. 4: Schematic of a thermoelectric device
In Fig. 4, the top of both materials is the hot side (Thot), while the bottom is the cold side
(Tcold). The materials are connected electrically in series and thermally in parallel. An
analysis will be performed, assuming no contact losses and a single thermocouple as
shown, to determine the power delivered to the load and the device efficiency. We know
from Ohm’s Law that V = IR, and for this analysis we can say that the total resistance of
the materials and load is R = Rn+Rp+RL. Combining this with the Seebeck effect from
equation (1.1), the relation for the current flow is:
(1.10)
From this, we can find the power to the load:
8
The maximum power occurs when load resistance (RL) is equal to generator resistance
(Rn+Rp). Next, we know that the total rate of heat flow includes Joule heating: that from
the Peltier effect (the heat balances Peltier cooling from current flow), and that due to
thermal conduction along the materials, shown respectively:
(1.11)
where Kp and Kn are the thermal conductance of the materials. The efficiency is defined
as the power delivered to the load divided by the total rate of heat flow:
(1.12)
An increase in RL overall reduces P but increases η. The maximum efficiency is given
by:
(1.13)
where ZT is the thermoelectric figure of merit (see section 1.5 below) and Tm is the mean
temperature of the hot and cold sides. As ZTm goes to infinity, the efficiency approaches
∆T/Thot, which is the Carnot cycle efficiency3.
Analysis can also be done for refrigerators and heat pumps (again, single thermocouples
are considered). The quantity used to determine device efficiency is the coefficient of
performance, φ (COP). Similar logic as that used for generators gives:
9
(1.14)
(1.15)
These equations apply to thermoelectric refrigerators. The power P in equation (1.14)
has a term not seen previously (
. This is the rate of power working to
overcome the thermoelectric voltage. Heat pump analysis is discussed further in several
references, along with derivations for multi-stage devices1,3. Multistage devices allow a
way to further maximize the temperature difference between Thot and Tcold, though in
practice this is limited by contact resistances. The best commercial cascaded Peltier
cooler has 6 stages. So far TE refrigerator devices are only about 25% as efficient as
conventional coolers that use R-134A as a refrigerant. The lower efficiency is why TE
devices are not used more frequently, except in applications where there is a premium for
their compactness1.
1.5 Figure of Merit and Thermal Conductivity
The figure of merit of a material (n or p-type) is defined by:
(1.16)
where σ is the electrical conductivity, S the Seebeck coefficient, κ is the thermal
conductivity, and ρ is the electrical resistivity. Also,
is referred to as the power
factor (PF). The figure of merit is what is most commonly used to characterize TE
devices and materials, since this value can be maximized at different temperature ranges
depending on the application.
10
From equation (1.16), we can see that a high performance thermoelectric material has
high thermopower, low electrical resistivity, and low thermal conductivity. The material
must have low electrical resistivity; otherwise joule heat is produced throughout the
entire thermocouple due to electron scattering. It must have low thermal conductivity to
decrease heat loss from conduction. In order to optimize all three properties, one must
take into account that they affect each other. For instance, in materials with low carrier
concentrations, thermopower is increased but electrical conductivity is decreased, so
there is power factor optimization involved. Also, if we add scattering, thermal
conductivity (
– see below) is decreased, but so is mobility, and thus electrical
conductivity; we must optimize the mobility to lattice thermal conductivity ratio by
scattering only phonons and not electrons1. Lastly, increasing electrical conductivity
increases total thermal conductivity, as described below:
Thermal conductivity describes how well a material can conduct heat, either through
phonons (traveling lattice vibrations), photons, electrons, or holes as heat carriers. The
thermal conductivity of a system is the sum of the electronic thermal conductivity and
lattice thermal conductivity:
(1.17)
where L is the Lorentz factor, σ is electrical conductivity,
is the specific heat per unit
volume, l is the mean free path of the phonons, and v is the sound velocity. The equation
is called the Wiedemann-Franz Law and shows that electrical and thermal
11
conductivities of a material are dependent on one another. The Lorentz factor of a free
electron in a vacuum is given by
but needs to be
calculated per material.
We can further define
in a two carrier system (that with both holes and electrons) by
the following:
(1.18)
(1.19)
where
is the electronic thermal conductivity due to ambipolar effects, and the
remaining terms in equation (1.18) are thermal conductivities due to holes and electrons.
In two carrier systems (such as BiSb alloys), both the electrons and holes can travel in the
same direction so that heat is transported with no electric current. These ambipolar
effects are most significant at high temperatures, when thermally excited carriers form
electron-hole pairs at the Thot end of a material with a temperature gradient across it.
Carriers then diffuse towards Tcold, and the electron-hole pairs recombine and release
heat. Thus, the overall thermal conductivity is increased, while thermopower is
decreased. The lattice thermal conductivity
is generally negligible in metals but
significant in semiconductors. This value can be limited in order to decrease the figure of
merit by introducing phonon scattering with alloys, impurities, and lattice defects
(vacancies, dislocations, etc).
12
1.6 Matching n-type and p-type Material in a Device
Consider the setup shown in Fig. 4. One can optimize the figure of merit of a
thermoelectric device (ZT) by minimizing
where K is thermal
conductance and R is resistance. This occurs when:
(1.19)
where L is the length of the branch and A is the cross-sectional area. The figure of merit
for a pair of materials is then given by3:
(1.20)
The best ZT commercially available in cooling/heating modules is 1.0. It would need to
be about 9 to be equally efficient as more frequently used non-TE systems. However, an
average ZT of just 2 (with more comparable costs) would allow TE systems to be of
practical use due to low maintenance, very high power per unit volume, and elimination
of greenhouse gasses1.
1.7 Thermoelectric Material Overview
In the past few decades there have been numerous initiatives, both theoretical and
experimental, to improve material zT for heating and cooling applications. In 1993, a
theoretical model by Hicks and Dresselhouse predicted that confining electrons to 2D
quantum wells would increase thermopower and decrease thermal conductivity12. This
began many experimental achievements such as a zT of 1.5 at 500°C in 2008. The details
of these high zTs and other recent developments can be found in Ref. 1 and 6-8.
13
Next, we will look at why given systems are able to be used in thermoelectrics. The
transport of electric charge in both semiconductors and metals occurs due to conduction
electrons (or holes) in these crystalline solids, which carry both charge and thermal
energy. The electrons interact with the potential that is in the crystal lattice, and the
energy of the electrons occurs in discrete bands. Each band can hold two electrons of
opposite spins. In insulating materials, the valence band is full and the conduction band
is empty, so there is no electron flow and thus low electrical conductivity. Metals have
semi-filled and/or overlapping bands, and therefore high electrical conductivity (as well
as high thermal conductivity as seen in equation (1.17)).
Semiconductors are materials that have electrical conductivity between that of metals and
insulators, typically 103 to 10-8 (Ω cm)-1. Semiconductors are defined by their finite
energy band gap between the valence and conduction bands. An intrinsic (or “pure”)
semiconductor has charge carriers that are created by thermal excitations of electrons
across the energy gap, from the valence to the conduction band. An extrinsic
semiconductor is one with impurities added to change the conductance. These can be
either n-type or p-type, as shown in Fig. 5. An n-type semiconductor has electrons as the
majority carrier (“donors”). In this case, semiconductors are doped with impurities that
allow electrons to be excited into the conduction band. A p-type semiconductor has
positive holes as the majority carrier (“acceptors”). Vacant bands are introduced near the
valence band through doping. Electrons pass into the vacant spaces in the opposite
14
direction of an applied field, causing the positive holes to travel in the field’s direction.
+
+
Energy
Gap
Energy
Gap
Energy
Conduction Band
Valence Band
n-type
+
+
+
+
+
+
+
p-type
Fig. 5: Representation of n-type and p-type semiconductor systems
Electrical conductivity (σ) of a semiconductor can be expressed as:
(1.21)
Where n is the carrier density, e is the electron charge, and μ is the mobility of the charge
carrier. It should be noted that typically, in a semiconductor, electrical conductivity
increases as temperature increases due to carrier density being extremely temperature
dependent. The opposite is true for metals, which have nearly constant carrier
concentration versus temperature. For a two carrier system, one can add the
conductivities of each carrier type:
(1.22)
The mobility of a charge carrier (μ) is defined as:
(1.23)
15
Where τm is the average relaxation time of electrons (the average time an electron is free
before it is scattered) and m* is their effective mass. We can also define current density
( ) as
(1.24)
Where
is the drift velocity of the electrons and E is the electric field applied4.
1.8 Shubnikov-de Haas Oscillations and the Fermi Surface
The Shubnikov-de Haas (SdH) effect is one that was discovered in 1930 by L.
Schubnikov and W. J. de Haas9. Starting with their experiments, it has been found that
oscillations in electrical resistivity occur with changes in magnetic field at low
temperatures for single crystal samples. In the Bi-band alloys in this thesis, the magnetic
field is applied along the trigonal axis of the single crystal, and the electrical resistivity is
measured along the trigonal axis as well. The conduction electrons and/or holes act as
simple harmonic oscillators. One can then use their period of oscillations (equal to the
inverse of the applied magnetic field H) to determine the cross-sectional area of the
Fermi surface (A) using equation (1.32), explained below.
First, we will look at the statistics of a given system. The probability that a given state of
energy E is occupied is given by the Fermi-Dirac distribution:
(1.25)
where EF is the Fermi energy and k is Boltzmann’s constant (1.3806503 × 10-23 m2 kg s2
K-1). The Fermi energy level of a given system is the highest occupied energy band at 0
16
K. Thus, it is a function of carrier density. The Fermi energy level is typically on the
order of 5 eV in metals. As seen below, as the energy state approaches zero, the
probability of occupation approaches 1.0.
Fig. 6: The Fermi-Dirac function10
As the energy state increases past EF, the probability of occupation approaches 0 for a
given temperature. Only those electrons near the Fermi level can be thermally excited
into higher states at a given temperature. Those in lower energy levels cannot be excited
into higher states since the states above them are full.
Before we can properly define the Fermi surface, we look at Brillouin zones. A wave
vector or “wave number” is used to show electron states in 2D and 3D and is known as k,
where
and
is the momentum vector of the electron. The first Brillouin zone
consists of the primitive unit cell with k extending from –π/a to +π/a (where a is the
lattice parameter):
17
ky
π/a
k
π/a
-π/a
kx
-π/a
Fig. 7: First Brillouin zone for a square lattice
This primitive unit cell is defined in the reciprocal space lattice, also known as “k-space”
or “momentum space”.
A Fermi surface is a boundary that is mathematically calculated for a given system and is
“related to the dynamical properties of the conduction electrons in a metal”10. This
surface is made up of the states near the Fermi level at constant energy in k-space,
defined by:
(1.26)
This shows us all the electron states that take part in the equilibrium thermodynamic
properties (only electrons within a range of energies ±kT around EF take part in transport
properties). The Fermi surface of a free electron gas is just a sphere. Metals and
semiconductors become more complicated: the surface is “carved” out by zone
boundaries and energy band gaps. Because of the shape of a Fermi surface, the
thermodynamic properties mentioned above are anisotropic depending on which direction
they are measured with respect to the Fermi surface.
18
An electron experiences velocity in k-space of
force
and acceleration with
, so that when a magnetic field H is applied, the Lorentz force is given by:
(1.27)
From this, the equation of motion of a representative point is:
(1.28)
where ħ is reduced Planck’s constant (1.05459E-34 J*s). Focusing on equation (1.28),
we see that in a given magnetic field H, the k-vector of a given electron on a Fermi
surface follows a path outlined by the intersection of the Fermi surface and a plane
perpendicular to H:
Fig. 8: Orbit of k-vector outlined by intersection of simple Fermi surface and plane
perpendicular to magnetic field (from Ziman10)
19
Thus, a magnetic field does not change the energy of an electron; it deflects the electron
into a helicoidal path described above. As long as it is not scattered, this electron will
make one orbit in the following period (or “time”, T):
(1.29)
where v is the velocity component of the electron that is normal to the magnetic field
and
is called the cyclotron frequency. Using single crystals with low carrier
concentration and at low temperature prevents scattering. The cyclotron frequency for a
free-electron Fermi surface (a sphere) is given as:
(1.30)
We can rearrange equation (1.29) to find the general equation for cyclotron frequency for
any Fermi surface. Now we reference Fig. 9:
Fig. 9: Representation of quantization of orbits (From Ziman10)
Previously, it was stated that v is a function of
energy (∆E) occurs, the point moves a distance of
on a new orbit (E+∆E). We can see from Fig. 9 that
20
. So, when a small change in
outward. Now the point is
where
is the area
between the old and new orbits. Substituting the above information into equation (1.29)
and rearranging, we get:
(1.31)
Moreover, the permitted energy levels must be quantized in units of ħωH, which means
that the area of an orbit must be quantized in units of 2πeH/ħ. Thus, applying a magnetic
field quantizes the electron gas differently than with basic ħk units. A magnetic field
divides k-space into “tubes” with constant cross-sectional area which correspond to states
of constant magnetic quantum number. Without the magnetic field, the energy is
distributed uniformly in k-space. With magnetic field, each state increases or decreases in
energy so as to be near to the surface of the closest “tube”. At room temperature and
weak magnetic fields, this effect is not noticed since states moved up are compensated by
those moved down. However, at high magnetic fields (observed in pure single crystals),
the tubes move outward more. Sometimes a tube breaks away from the Fermi surface at
a maximum-area cross-section, increasing the energy of the states near here. Then the
next tube moves out and enough levels become available on it so that the energy
decreases below the average value. Thus, an energy oscillation occurs, with the period
expressed as the inverse of the change in magnetic field:
(1.32)
Therefore, we can use the period of oscillations to determine the cross-sectional area of
the Fermi surface. One must determine whether the oscillations are due to holes or
electrons. This becomes more complicated depending on the system and is discussed for
this work in section 3.3.
21
1.9 Resonant Levels
There are two major approaches to improving the figure of merit of a thermoelectric
material: decreasing the lattice thermal conductivity using phonon scattering and
increasing the thermopower (and therefore, the power factor). The latter is the focus of
this thesis, and the mechanism used here to increase thermopower is through the
distortion of the electronic density of states (DOS). This was recently reviewed and
confirmed experimentally by Heremans7,11. According to Mahan-Sofo theory, the
materials with the highest zT have a delta-like DOS; the larger the DOS and the stronger
its dependence on energy E, the larger the thermopower. The four recognized distortions
with this shape are those due to quantum size effects, correlated systems (though this is
technically inherent and not a distortion), Kondo systems, and resonant levels, shown
below:
Fig. 10: Various mechanisms that distort a system’s density of states
22
If we can increase the DOS in the above ways, we can increase the Seebeck coefficient
almost independently of temperature. Another result that may occur is a resonant
scattering that increases thermopower at cryogenic temperatures where electron-phonon
interactions are weak. The factors that limit this goal include inhomogeneities in sample
composition that broaden the peaks, as well as electronic states that arise due to other
atoms contributing to “background DOS”.
The DOS is also called function g(E), which is the number of electron states in a unit
volume of a material at energies from E to E+dE. The unit of g(E) is number of states
per eV per cm3 and is defined as:
(1.33)
This involves the transformation of integrals over k-space defined in the Brillouin zone
of a 3D solid to scalar integrals over energy E. Also, S is the Fermi surface of energy E.
In 3D, the DOS is proportional to square root of E.
Resonant levels, or using “resonant impurities” as dopants, will be briefly discussed.
This was the mechanism used to predict high figures of merit for the materials reported
on in Chapter 3. Resonant levels occur when electrons in a dilute impurity couple with
those in the conduction or valence band of a host system. See below in Fig. 11, which
shows a conduction band on the left (dispersion relation) and the available DOS on the
right. The dotted line shows a typical material, while the solid line shows one with a
resonant level.
23
Fig. 11: Definition of resonant levels in terms of dispersion relation and DOS
The “hydrogenoid” model shown (in red) is the classical case of a donor impurity in a
semiconductor. An example of this is phosphorus (P) in silicon (Si), where P shares four
electrons with Si atoms, leaving 1 electron weakly bound to P. The donor energy ED
(also called the effective Rydberg, R*) is the energy needed to overcome this weak bond
and release the electron into the conduction band. As shown in the figure above, the
energy level of this fifth electron is ED below the conduction band (in the energy gap).
Resonant levels, on the other hand, fall inside the conduction band (or valence band) to
coincide with energies of extended states, producing the sharp spikes as illustrated (in
blue). This state will “resonate” with the extended states, creating two new extended
states of slightly different energies, which resonate in turn. ED, along with the width of
the resonant state, Γ, are both the main design parameters for thermopower optimization.
The wave functions that contribute to the DOS distortions can be “localized states”, in
which they behave like atomic orbits, or “extended states”, in which they behave more
like free electrons. In the “localized states”, the energy levels of the impurity electrons in
24
the host band scatter the conduction electrons of the host material resonantly, causing
these electrons to have an energy-dependent mobility. In the “extended states”, electrons
are plane-wave-like and carry charge and heat that contribute to thermopower. These two
types of states affect each other, so one must find a balance between the two in order to
optimize the thermopower of a material.
There are certain criteria that have been found needed to increase the benefits of resonant
levels in thermoelectric materials. These include: 1) optimum carrier density, 2) having a
Fermi energy level close to ED, 3) narrow width Γ, 4) good charge and heat conductance,
and 5) as small of a “background DOS” as possible. It has been found that these criteria
are most often met in s and p level electrons; these are mostly associated with the
“extended states” mentioned above, while the “localized states” are most often d level
electrons. Moreover, resonant levels occur relatively often in narrow-gap semiconductors
and semimetals. This is because the electrons in these types of materials have small
effective masses which mean long de Broglie wavelengths compared to the interatomic
distances. Therefore the electrons are more likely to hybridize with the impurity electron
wave functions, and so a resonant level is more likely to occur. For a more in depth
discussion of resonant level theory and experimental work, refer to Ref. 11.
25
Chapter 2: Experimental Methods
2.1 Sample Preparation
Polycrystals
For the polycrystalline samples discussed in section 3.2, stoichiometric amounts of the
pure elements (5N or greater) were placed in carbon-coated ampoules. The dopants were
loaded into the ampoules in a glovebox under argon to prevent oxidation in air. The
ampoules were then sealed under high vacuum (2E-6 to 5E-6 mbar), heated in the vertical
position at 400°C, and then water-quenched.
Single Crystals
All other experiments were performed on single crystal samples since it has been found
that the best n-type thermoelectric performance of BiSb alloys can be obtained along the
trigonal axis direction15,16. The Bi was further purified by a vertical Bridgman method,
passed one time at 274 mm/h through the furnace. Just as for the polycrystalline samples,
the correct amounts of the pure materials were loaded into the ampoules and then sealed
under high vacuum. Then, a modified horizontal Bridgman method was used: the sealed
ampoules were placed horizontally in a furnace, heated to above the liquidus for one
hour, cooled very slowly (2°C/hour), and then annealed for 11-13 days at temperatures
indicated by the respective phase diagrams (see Chapter 3).
26
2.2 Experimental Setup
Crystallographic analysis was performed on each sample using powder x-ray diffraction
(XRD) by Rigaku Miniflex. Bi and Sb compositions were determined using wavelength
dispersive x-ray fluorescence (WDXRF). The dopant concentrations were undetectable
due to the resolution limit of the instrument. Microstructure is discussed in section 3.2
for Bi100-xLix samples. This was observed with a Quanta200 scanning electron
microscope (SEM) with a solid-state back-scattered electron detector (BSD). The Bi and
Sb compositions observed with SEM were determined using energy-dispersive x-ray
analysis (EDX).
For transport property measurements, parallelepipeds approximately 5mm x 1mm x 1mm
were cut using a wire saw from the ingots. They were mounted with gold-plated copper
leads and a heat sink using silver epoxy as shown in Fig. 12:
27
Qin
x
y
Trigonal Axis
z
Thot
Vtop
Tcold
Vbottom
Heat Sink
Fig. 12: Experimental setup for PPMS, Thermal Transport Option
Note that the sample is cut so that the trigonal axis is parallel to the length of the sample.
The sample was measured in a Physical Property Measurement System (PPMS) using the
thermal transport option (TTO, Quantum Design). The electrical resistivity (ρ), thermal
conductivity (κ), and Seebeck coefficient (S) were recorded using a quasistatic heaterand-sink method over the temperature range 300K to 2K. As shown in Fig. 12, the top
lead measures one voltage and temperature (Thot and Vtop) and the bottom lead measures
the voltage and temperature at a different spot (Tcold and Vbottom) . The temperatures are
obtained from the resistances of the attached thermometers. The lead on the very top of
the sample is connected to a heater for the heat flow (Qin) to enter. The SdH
measurements were also recorded with this setup.
28
Next, we can show how the above setup is used to find the transport properties. We will
define the x-direction as the trigonal-axis direction; this is also the direction the current
(I) is passed through the sample. First, we can define the heat flux (q), current density
(jx), and electric field (Ex):
(2.1)
(2.2)
(2.3)
Note that Ac is the cross-sectional area of the sample, w is the width, th is the thickness,
and L is the length between the top and bottom leads. One more term that needs defining
is the temperature gradient in the x-direction:
(2.4)
Now we can use equations (2.1) – (2.4) to specify how we find ρ, κ, and S using the
measurements taken:
(2.5)
(2.6)
(2.7)
The Hall measurements were taken on the same parallelepipeds. These were mounted in
the PPMS (using the AC transport option, in the horizontal position) as shown below:
29
z
Vb+
y
x
B
Trigonal Axis
I-
I+
Va+
Va-
Fig. 13: Experimental setup for PPMS, AC transport option for Hall measurement
The wire (copper, 0.001” diameter) was attached with silver epoxy. A transverse
magnetic field (Bz) was passed through the sample as shown in Fig. 13. Measurements
were taken at temperature intervals from 300K to 2K by sweeping the magnetic field
from -1 to +1 Tesla. The Hall coefficient was obtained as follows:
(2.8)
2.3 Sources of Error
The main source of experimental error is due to geometry (of the sample itself and of the
length measured between the leads). The ρ and κ measurements have an error of about
7% while S is approximately 3%. Radiation effects may also contribute to κ at T > 150K.
Error of about 5% may also arise during Hall measurements due to geometry as well as
wire misalignment. The latter was reduced by sweeping the magnetic field in both the
positive and negative directions and taking the odd portion of the signal. Error in zT was
also minimized since all transport measurements were taken simultaneously; the
geometrical errors in κ and ρ cancel each other out.
30
Chapter 3: Bismuth and Bismuth-Antimony Alloys
3.1 Introduction
Bismuth (Bi) and antimony (Sb) are both Group V elements, which indicates they have
five valence electrons. They also both crystallize into a structure with two atoms per unit
cell, meaning there are ten total electrons to place in bands. Five energy bands are filled.
Of these, the bands due to the 6s electrons of Bi or 5s electrons of Sb are at very low
energy, and do not contribute to transport. There is a slight overlap of the Bi or Sb pelectron valence bands into a sixth band. This makes both Bi and Sb have a very small
number of free electrons (1 per 105 for Bi atoms or 1 per 104 for Sb atoms) and they are
thusly classified as semimetals. Electrons in Bi and Sb have the smallest effective
masses of any solid and therefore high mobilities due to a small energy gap at the L point
of the Brillouin zone. Moreover, both Bi and Sb have similar rhombohedral crystal
structures (
) and lattice parameters; as a result Sb goes substitutionally into Bi, and
BiSb can form complete solid solution. The phase diagram for BiSb is shown here:
31
Fig. 14: Binary phase diagram for Bi-Sb system (1992 Feutelais Y.)13
As Fig. 14 shows, there is a large temperature difference between the liquidus and
solidus. This large segregation makes it very difficult to prepare uniform crystals and is
the reason we used such slow cooling rates for BiSb alloys, as described in section 2.1.
The slow cooling is meant to reduce problems due to super cooling effects, mainly
inhomogeneity of ingots3. This is important because the transport properties of BiSb
alloys depend strongly on Sb concentration, as we will show.
A review is given in Lenoir’s summary of Bi-rich BiSb alloys14,15.
32
Fig. 15: Schematic diagram of bands near the Fermi level of Bi1-xSbx alloys as a function
of x at 0 K (From Lenoir14).
The figure above shows a positive direct energy gap of 10 meV for pure Bi (x=0). The T
heavy hole band (valence band) overlaps the conduction band by 40 meV at this point.
As more antimony is added, the light hole band (Ls) and the heavy electron band (T)
decrease in energy and the light electron band (La) increases. A band inversion occurs
between the Ls and La bands at 4% Sb. At 7% Sb, T crosses La, and the material behaves
as a semiconductor from 7% until 22% Sb. At this point, La crosses another heavy hole
band (H). The maximum energy gap of the semiconductor region occurs between 15%
and 17% Sb. Because BiSb alloys in the semiconducting region have thermal gaps less
than 30 meV, they are classified as narrow band-gap semiconductors. There is a strong
temperature dependence of BiSb transport properties, especially above 80 K, due to both
a very small direct gap at the L point and temperature dependence of the band structure14.
33
At room temperature, Bi and BiSb alloys have lower figures of merit than bismuth
telluride. Other disadvantages include the cleaving that occurs easily along trigonal
planes in addition to homogeneity issues as discussed above. Moreover, in order to
obtain the highest figure of merit, the elements must be as pure as possible15. However,
there is potential at cryogenic temperatures due to lower values of lattice thermal
conductivity and a positive energy gap3. For more past experimental and theoretical
work on Bi and BiSb alloys, refer to Ref. 3, 14-17. Various studies have been performed
successfully, such as using tellurium and tin as dopants, varying Sb concentration, and
the effects of a magnetic field. Here, we search for resonant level dopants in Bi and BiSb
to improve its thermoelectric properties.
3.2 Bi + Li and BiSb + Li
The first impurity tested for resonance for this thesis was lithium (Li), an alkali metal
with bcc crystal structure. Li was predicted to be either a donor if it was an interstitial
impurity in Bi or a resonant level if substitutional in Bi. If substitutional, the band
structure calculations showed the sharp DOS peaks near the Fermi level, suggesting
resonance, as shown below (theoretical calculations of band structure were performed by
Dr. Bartlomiej Wiendlocha using the Korringa-Kohn-Rostoker (KKR) Green function
method18):
34
Fig. 16: Theoretical DOS of Bi doped with Li for (a, b) substitutional impurity and (c)
interstitial impurity18
It should also be noted that the case of interstitial Li in Bi results in a strong n-type
dopant, whereas substitutional should result in p-type or mixed carriers. It was necessary
to use the phase diagram of the Bi-Li system to determine which compositions to test as
well as which melting temperatures to use during synthesis:
Fig. 17: Binary phase diagram for Bi-Li system (1994 Gasior W.)13
35
As Fig. 17 shows, there is no known information on the solubility of Li in Bi. Thus,
polycrystalline samples of Bi doped with 0.3%, 0.7%, 1.5%, 3%, 14% (the eutectic
composition), and 20% Li were prepared and tested, as described in Chapter 2. Then,
single crystal and polycrystalline samples of Bi88Sb12 were doped with Li and tested to
further enhance thermal and thermoelectric transport properties. As seen in the phase
diagram, we can expect to see a secondary “BiLi” phase in the first set of samples (Bi100xLix).
BiLi is a tetragonal intermetallic compound (CuAu type, P4/mmm, no. 123):
Fig. 18: Secondary phase BiLi structure
First, we will look at the XRD and SEM results of the polycrystalline Bi100-xLix samples:
36
Fig. 19: (a) XRD results of Bi100-xLix with BiLi secondary phase peaks identified; SEM
results of (b) eutectic composition Bi86Li14, (c) Bi97Li3, (d) Bi80Li20
The XRD confirms the existence of the secondary BiLi phase: additional peaks are seen
at the eutectic composition (14% Li) as well as the hypereutectic composition (20% Li).
The other peaks are the Bi phase. The SEM results in Fig. 19(b) show a random
distribution of BiLi (100 nm to 300 nm particles) in the Bi matrix of the eutectic
composition, as expected. The Bi97Li3 shown in Fig. 19(c) has a dendritic structure
similar to pure Bi with a very dilute distribution of particles. This is consistent with the
37
XRD findings – presumably, no secondary phase appears in dilute alloys (less than 1%
Li). Moreover, Fig. 2(d) shows Bi80Li20, which has non-uniform 1μm to 5 μm particles
as expected from the phase diagram.
Next, we can look at the electrical resistivity and thermal conductivity of the Bi100-xLix
polycrystalline samples:
Fig. 20: Electrical resistivity and thermal conductivity of Bi100-xLix versus temperature
In general, for the hypoeutectic compositions, the low-temperature electrical resistivity
increases with increasing amounts of Li. However, the 14% eutectic sample reverts back
to the behavior of pure Bi. Unusual behavior is also observed in the eutectic composition
for thermal conductivity; thermal conductivity generally decreases with increasing
amounts of dilute Li, but the eutectic sample has the highest value at low temperatures
despite the nanometer-sized inclusions. The 20% Li sample has the lowest thermal
conductivity as expected due to large inclusions shown by SEM. Overall, thermal
38
conductivity of the Bi-Li system is dominated by phonon conduction at low temperatures
since the extra Li atoms in Bi increase the lattice portion of thermal conductivity.
Next, we studied both single crystal and polycrystalline samples of Bi88Sb12 with various
amounts of Li (all amounts are mole percents, nominal). The figure below shows the
SEM of polycrystalline (Bi88Sb12)86Li14:
Fig. 21: SEM of polycrystalline (Bi88Sb12)86Li14
This image shows large-scale regions of about 20 μm with approximate composition
Bi20Sb80 in a Bi-rich matrix. We can assume the Li reduces the solid solubility of Sb in
Bi. For this reason, the transport properties of the high concentrations of Li (>1%) were
not a focus of this study.
We now look at dilute amounts (≤1%) of Li in Bi88Sb12 in both single crystal and
polycrystalline form. In general, the polycrystalline and single crystal samples have the
39
same trends, though the thermoelectric properties are enhanced in single crystal form.
First we look at electrical resistivity:
Fig. 22: Electrical resistivity of Bi88Sb12 doped with indicated amounts of Li, in single
crystals (left) and polycrystals (right); measured along trigonal axis direction of single
crystals
The undoped Bi88Sb12 has an electrical resistivity that decreases as temperature increases
up to 100 K as thermally excited carriers increase in concentration. Then, acoustic
phonon scattering starts limiting the carrier mobility, resulting in the slight upturn after
100 K. Both Li-doped samples show linear, increasing electrical resistivity as
temperature increases, which is indicative of semi-metallic behavior (as opposed to the
semiconducting behavior of the undoped samples where resistivity overall decreases as
temperature increases). Thermal conductivity, specifically the single crystal data, is
consistent with this finding:
40
Fig. 23: Thermal conductivity of Bi88Sb12 doped with indicated amounts of Li, in single
crystals (left) and polycrystals (right); measured along trigonal axis direction of single
crystals
The thermal conductivity is relatively dominated by electronic contributions in the doped
sample (as the electrons carry heat), while the undoped sample is most dominated by the
lattice portion of thermal conductivity, especially below 20 K. The increase in thermal
conductivity of the undoped sample after 60 K can be explained by ambipolar effects due
to multiple carriers at high temperatures.
41
Fig. 24: Thermopower of Bi88Sb12 doped with indicated amounts of Li, in single crystals
(left) and polycrystals (right); measured along trigonal axis direction of single crystals
Next, we look at the absolute value Seebeck coefficient. The Seebeck itself, displayed in
Fig. 24, is negative. The undoped samples increase in magnitude as temperature
decreases, until a peak Seebeck at about 50-70 K. This behavior is indicative of a
narrow-gap semiconductor. The decrease above 70 K occurs due to thermally excited
electrons and holes that compensate each other. In both Li-doped samples, there is a
nearly monotonic increase in magnitude as temperature increases. This is characteristic
of a degenerately doped n-type semiconductor. Because the dopant causes the electrical
resistivity to decrease and the thermopower to also decrease in magnitude (showing semimetallic behavior once again), Li is clearly a conventional donor. We also show the
figure of merit:
42
Fig. 25: Figure of merit of Bi88Sb12 doped with indicated amounts of Li, in single crystals
(left) and polycrystals (right); measured along trigonal axis direction of single crystals
Both sets of data show that doping with Li significantly decreases the figure of merit.
Also, this clearly shows how measuring the properties along the trigonal axis of a single
crystal can improve thermoelectric properties as opposed to polycrystalline samples: the
peak zT increases from 0.2 to almost 0.5 in the undoped sample.
The data is consistent with Li acting as an electrically active n-type donor. The
experimental results agree with the theoretical model of Li going interstitially into Bi.
Thus, it is very likely that Li enters both Bi and Bi-Sb alloys as an interstitial impurity,
rather than a substitutional one, and it is not a good choice to improve zT.
43
3.3 Bi + Mg and BiSb + Mg
Next, we look at Mg as a dopant in Bi and Bi-Sb alloys. This dopant is an alkaline earth
metal, and it has an hcp crystal structure. Band structure calculations show the sharp
density of states peak near the Fermi level of Mg-doped Bi, assuming the Mg substitutes
for Bi:
Fig. 26: Theoretical DOS of Bi doped with Mg for substitutional impurity (from
Bartlomiej Wiendlocha, AGH University of Science and Technology)
Substitutional Mg should give a p-type dopant according to the DOS since the Fermi
level is located towards the valence band. It was also calculated that interstitial Mg in Bi
is predicted to be a strong n-type dopant as was Li in Bi.
The phase diagram is shown below, and was once again used to find the melting
temperature for synthesis of various compositions:
44
Fig. 27: Binary phase diagram for Bi-Mg system (1992 Oh C.S.)13
Since no solubility information is known for the Bi-Mg system or the Bi-Sb-Mg system,
multiple compositions were prepared and tested. These single crystal samples included
Bi doped with 0.2% Mg in addition to Bi90Sb10 doped with 0.1%, 0.2%, 0.5%, and 0.7%
(nominally). The WDXRF was used to determine the Sb concentrations of each sample;
these values are used in the legends of the plots that follow.
Fig. 28(a) presents evidence that Mg dopes Bi, since the doped sample has an increased
carrier concentration at low temperatures compared to pure Bi. In Fig. 28(b), two
samples with different amounts of Sb and equal amounts of Mg are compared. As
expected, a greater amount of Sb decreases the carrier concentration at low temperatures.
A Hall prefactor of 2/3 was used to calculate the carrier concentration for a ρ32(B1)
configuration from Hall data19. Because the Hall measurements with magnetic field
45
parallel to the trigonal axis only measure the electrons in Bi, an increase in carrier
(electron) concentration indeed corresponds to an n-type doping action.
(a)
(b)
Fig. 28: Carrier concentrations of Bi and BiSb samples doped with Mg in single crystals,
obtained by measuring the Hall coefficient along the trigonal axis direction up to 1 Tesla
(Hall Prefactor = 2/3).
Next we look at the transport data for both the Bi system doped with Mg and the Bi-Sb
system doped with Mg:
46
(a)
(b)
(c)
(d)
Fig. 29: Thermal transport properties of Bi and BiSb alloys doped with 0.2% Mg
including (a) Seebeck coefficient, (b) thermal conductivity, (c) electrical resistivity, and
(d) figure of merit vs. temperature. All samples were single crystals measured along the
trigonal axis direction.
First we consider the simple Bi system. Reinforcing the Hall data, Fig. 29 shows
multiple signs of n-type doping, such as a lower absolute thermopower of the doped
sample in Fig. 29(a). The electrical resistivity is overall lower for the doped sample,
except at low temperatures which can be explained by dominating impurity scattering.
Lastly, the Bi doped with Mg has a higher thermal conductivity than Bi, suggesting an
increase in κe of equation (1.17). Thus, though the n-type doping is much weaker than we
47
saw in the Bi-Li system, there are multiple signs that this is the mechanism occurring. It
is of course surprising that Mg, whose valence state is usually 2, could be a donor if it
substitutes for Bi, which typically has an effective valance of 3 (the two 5s electrons not
contributing to conduction). We do not understand the origin of this moment.
Looking at the Bi-Sb systems in the above figure, we can see that the Mg does increase
the thermopower below 100K, though this is likely due to Sb contributions. Fig. 29 also
shows how adding Sb to the system overall enhances the thermoelectric properties by
decreasing thermal conductivity and increasing thermopower. We further realize the
effects of Sb in the following results:
48
(a)
(c)
(b)
(d)
Fig. 30: Thermal transport properties of BiSb alloys doped with 0.7% Mg including (a)
Seebeck coefficient, (b) thermal conductivity, (c) electrical resistivity, and (d) figure of
merit vs. temperature. All samples were single crystals measured along the trigonal axis
direction.
In Fig. 30, we compare the transport properties of the samples seen in Fig. 28(b). The
22% Sb sample displays semiconducting behavior, evidenced by electrical resistivity that
increases as temperature decreases. The 3% Sb sample, however, is semimetallic, with
behavior similar to that of pure Bi. Its electrical resistivity increases with temperature.
Moreover, semimetallic behavior is evidenced by the thermopower, which is more linear
then the 22% Sb sample and increases as temperature increases. This is all consistent
with Fig. 15, which shows that the Bi-Sb system changes from semimetallic to
semiconducting at an Sb concentration of about 7%. It is also noteworthy that the sample
49
with a greater amount of Sb has a peak thermopower and figure of merit at lower
temperatures than that with less Sb.
In an attempt to elucidate the origin of the electrical activity of Mg in Bi as well as to
ascertain carrier concentrations, Shubnikov-de Haas (SdH) measurements taken at 2 K
were compared to Hall data, as shown below:
Table 1: Shubnikov-de Haas and Hall data for Bi and BiSb + Mg at 2 K
Composition
SdH (T)
A (m-2)
If electrons If holes
(cm-3)
(cm-3)
Hall
(cm-3)
Bi
(1) 6.49 ←holes
Bi+0.2%Mg
(1) 4.08 ←holes
3.89E+16
(2) 7.51
7.17E+16
(3) 10.2 ←electrons 9.77E+16
(1) 2.15 ←electrons 2.10E+16
(2) 5.00
4.77E+16
(3) 11.0
1.05E+17
SECOND HARMONIC
Bi99Sb1+0.1%Mg
(1) 2.81 ←electrons 2.68E+16
(2) 5.20 ←holes
4.96E+16
5.3E+16
1.3E+17
8.6E+16
2.2E+17
5.6E+16
Bi91Sb9+0.5%Mg
(1) 0.76 ←electrons 7.22E+15
(2) 17.4
1.65E+17
7.4E+15
1.2E+16
NEW PERIOD
8.0E+15
Bi98Sb2+0.7%Mg
(1) 1.08
1.03E+16
(2) 3.19
3.04E+16
(3) ~8 ←electrons
1.3E+16
6.4E+16
3.0E+17
(1) 6.16
(2) 34.2
1.7E+17
2.8E+17
NEW PERIOD
Bi95Sb5+0.1%Mg
Bi97Sb3+0.7%Mg
6.19E+16
5.88E+16
3.26E+17
-
3.0E+17
2.3E+17
9.2E+16
2.3E+17
3.7E+17
1.5E+17
3.8E+17
6.0E+17
4.7E+17
3.5E+16
1.3E+17
5.8E+17
2.0E+17
2.1E+16
1.0E+17
3.8E17
3.6E+16
2.9E+17
2.9E+17
The Hall carrier concentration, which measures only the concentration of electrons, at 2
K was compared to that calculated by the SdH data for the cases of both electron pockets
50
and hole pockets. Because pure Bi has an equal concentration of electrons and holes, we
can confirm with Hall data that there is a band (pocket) due to holes around 6.5 T. Thus,
we were able to look at each sample and look for periods around this value to find the
bands due to holes, as labeled in Table 1. In the Bi + 0.2% Mg sample, there is a 4 T
period and in the Bi99Sb1 + 0.1% Mg, there is a 5.2 T period. Both of these are slightly
shifted from that seen in pure Bi, as are the hole concentrations calculated from the
oscillations (1.5E17 cm-3 and 2.2E17 cm-3, respectively as compared to 3E17 cm-3 of pure
Bi). The shifts are presumably due to the change in Fermi surface from the addition of
Mg. Then, we could compare the Hall concentration to appropriate SdH concentrations
to determine the bands due to electrons (these are the highlighted concentrations). The
bands found due to electrons are also labeled in Table 1.
From this data, it seems that a new period is formed with higher concentrations of Mg,
observed in two samples with a period of 17.4 T (Bi91Sb9 + 0.5% Mg) and 34.2 T
(Bi97Sb3 + 0.7% Mg). Because new bands are unexpected, we further illustrated this by
the showing the raw traces of Shubnikov-de Haas oscillations in Fig. 31 below. Even the
theoretical calculations in Fig. 26 suggest the existence of this new period: the DOS of
Mg goes with
which is indicative of conduction electrons.
51
Bi97Sb3 + 0.7%Mg
Resistivity
(Background Subtraction)
Resistivity
(Background Subtraction)
Pure Bismuth
1
2
4
3
Magnetic Field (T)
5
6
1
2
3
4 5
Magnetic Field (T)
6
New period
Fig. 31: Shubnikov-de Haas traces with background subtraction. The left shows pure Bi
with one clear period. The right shows BiSb with a new period beginning near 5 T.
Both experimental and theoretical work suggests that Mg is inducing a new carrier pocket
in Bi-Sb alloys. This is extremely surprising because the Mg atoms are quite distant from
each other and their electrons are not expected to interact over such distances. We cannot
conclude one way or another if this is a substitutional or interstitial impurity. This dopant
does not overall improve thermoelectric properties as compared to pure Bi and BiSb.
3.4 Bi + Na and BiSb + Na
Sodium (Na) is the last dopant tested, also an alkali metal with a bcc crystal structure.
Again, band structure calculations of substitutional Na in Bi predicted a resonant level
due to sharp DOS peaks near the Fermi level:
52
Fig. 32: Theoretical DOS of Bi doped with Na for substitutional impurity (from
Bartlomiej Wiendlocha, AGH University of Science and Technology)
As shown below, the phase diagram of the Bi-Na system shows no information on the
solubility of Na in Bi, so a variety of compositions were synthesized. Single crystal
samples of Bi doped with 0.2% Na as well as Bi90Sb10 doped with 0.1%, 0.2%, and 0.5%
Na were prepared and tested as described in Chapter 2.
Fig. 33: Binary phase diagram for Bi-Na system (1990 Sangster J.)13
53
As previously, transport data as well as Hall and SdH measurements were obtained for
each sample. The following data in Fig. 34(a) shows that the Na-doped sample has
carrier concentrations only increased very slightly compared to those in as pure Bi at all
temperatures:
(a)
(b)
Fig. 34: Carrier concentrations of Bi and BiSb samples doped with Na in single crystals,
obtained by measuring the Hall coefficient along the trigonal axis direction up to 1 Tesla
(Hall Prefactor = 2/3).
Also, as with the BiSb doped with Mg, Fig 34(b) shows that greater amounts of Sb
decreases the carrier concentration at low temperatures as expected. To further explore
whether or not Na is doping Bi, we look at the transport data:
54
(a)
(c)
(b)
(d)
Fig. 35: Thermal transport properties of Bi and Bi doped with 0.2% Na including (a)
Seebeck coefficient, (b) thermal conductivity, (c) electrical resistivity, and (d) figure of
merit vs. temperature. All samples were single crystals measured along the trigonal axis
direction.
Again, the transport data of the doped sample is very similar to the undoped sample. The
slight differences can be explained by misalignment of the sample along its trigonal axis
during measurement. However, there is still uncertainty as to whether or not the material
is being doped, so we also look at the SdH oscillations at 2 K:
55
Bi + 0.2% Na
Resistivity
(Background Subtraction)
Resistivity
(Background Subtraction)
Pure Bismuth
1
2
4
3
Magnetic Field (T)
5
6
1
2
3
4 5
6
Magnetic Field (T)
Fig. 36: Shubnikov-de Haas traces with background subtraction.
As discussed, pure Bi has one clear period in the raw data traces. The Na-doped sample
has additional oscillations as shown in Fig. 36. The SdH data was used to calculate
carrier concentrations which were then compared to the Hall data:
Table 2: Shubnikov-de Haas and Hall data for Bi and Bi + Na at 2 K
Composition
SdH (T)
A (m-2)
If electrons
(cm-3)
If holes
(cm-3)
Hall
(cm-3)
3.02E+17
2.3E+17
Bi
(1) 6.49 ←holes
6.19E+16
Bi+0.2%Na
(1) 5.88 ←holes
5.61E+16
1.6E+17
2.6E+17
(2) 7.91 ←electrons 1.19E+17
2.5E+17
4.1E+17
-
2.9E+17
The electron concentration and first period of the doped sample is nearly equal to that of
the undoped sample, and the Hall data confirms this. There is a period at around 6 T in
the doped sample, which likely corresponds to the 6.5 T period found in pure Bi. The
second period (which shows up in the additional oscillations shown in Fig. 36) is
56
probably a harmonic of the first, unlike the case of the Mg-doped samples. The pure Bi
gives a hole carrier concentration of 3.0E17 (confirmed by Hall data since hole and
electron concentrations are equal in Bi), while the sampled doped with Na has a hole
concentration of 2.6E17. Thus we can conclude that Na does not dope Bi.
Similar analysis was performed on the Na-doped BiSb samples. Again changes in
oscillations as compared to pure Bi were either too subtle or non-existent; the 6 T period
was not visibly shifted, nor were obvious new periods observed. One difficulty we had
was determining the effects of Sb on the sample. At this point in our measurements, we
were unable to determine the Sb concentrations in the samples tested due to
instrumentation issues (the WDXRF instrument broke down). Still, we can guess the
relative amounts by thermal conductivity data: as thermal conductivity increases, Sb
concentration decreases. We look at the following samples doped nominally with 0.1%
and 0.5% Na and compare them to an undoped BiSb sample:
57
(a)
(c)
(b)
(d)
Fig. 37: Thermal transport properties of BiSb doped with Na including (a) Seebeck
coefficient, (b) thermal conductivity, (c) electrical resistivity, and (d) figure of merit vs.
temperature. All samples were single crystals measured along the trigonal axis direction
(Sb contents are unknown).
All samples were made with a nominal 10% Sb, but inhomogeneity throughout the ingot
and instrumentation problems prevent us from knowing the exact concentration. We can
hypothesize from Fig. 37(b) that the undoped sample and that doped with 0.1% Na have
equal amounts of Sb, while the 0.5% Na sample has less Sb than the other two samples.
First we will compare the two samples with similar amounts of Sb. It may be that the
addition of the right amount of Sb greatly increased the magnitude of the Seebeck, up to
200 μV/K. Not only that but the temperature that the maximum Seebeck occurs shifts
from 50 K to 20 K, which is convenient for cryogenic cooling purposes. However, this
58
sample also has the highest electrical resistivity, so the figure of merit is not improved.
The sample with the presumably lowest Sb and highest Na concentrations has the overall
lowest electrical resistivity. However, these concentrations are not optimum for Seebeck
or thermal conductivity; the figure of merit falls below that of the undoped sample below
230 K.
Overall, Na does not enhance the figure of merit of either Bi or Bi-Sb alloys. It does not
seem to dope the Bi system, suggested by the Hall and transport data and further
confirmed by the SdH measurements. More work would need to be done on Bi-Sb alloys
doped with Na to conclude if doping takes place as well as optimum Sb and Na
concentrations.
3.5 Conclusions
It is clear from the experimental results that Li and Mg are each electrically active in Bi
and Bi-Sb alloys, while Na is not. We can conclude by comparing the data to theoretical
work that Li is a strong n-type dopant that is an interstitial impurity. The next set of
results suggests that Mg is not only is it a weak n-type dopant, but it likely induces a new
carrier pocket in the band structure, as observed in SdH oscillations. None of these
metals improved the figure of merit at cryogenic temperatures.
59
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