Ab initio simulation and optimization of phonon drag
effect for lower-temperature thermoelectric energy
ARCHNES
conversion
MA
1 AT NQTITUTE
by
JUL 3 0 2015
Jiawei Zhou
Submitted to the Department of Mechanical Engineering
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in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
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Ab initio simulation and optimization of phonon drag effect for
lower-temperature thermoelectric energy conversion
by
Jiawei Zhou
Submitted to the Department of Mechanical Engineering
on May 8, 2015, in partial fulfillment of
the requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
In recent years, extensive efforts have been devoted to searching for materials with high
thermoelectric (TE) efficiency above room temperature for converting heat into electricity.
These efforts have led to significant advances with a record-high zT above 2. However,
the pursuit of higher TE performance at lower temperatures for cooling and refrigeration
applications receives much less attention. Today's most widely-used thermoelectric
materials below room temperature are still (Bi,Sb) 2(Te,Se) 3 material system, discovered
60 years ago with a maximum zT around 1. This thesis develops the first-principles
simulation tools to study the phonon drag effect - a coupling phenomenon between
electrons and non-equilibrium phonons - that leads to a large Seebeck coefficient at low
temperatures. Phonon drag effect is simulated successfully from first-principles for the
first time and results compare well with experimental data on silicon. While the common
wisdom always connects a significant phonon drag effect to a high thermal conductivity,
a key insight revealed from the simulation is that phonons contributing to phonon drag
and to thermal conductivity do not spectrally overlap. Even in a heavily-doped silicon
sample with 1019 cm-3 doping concentration, phonon drag still contributes to -50% of the
total Seebeck coefficient. By selectively scattering phonons contributing to heat
conduction but not to phonon drag, a large improvement in thermoelectric figure of merit
zT is possible. An ideal phonon filter is shown to tremendously enhance zT of n-type
silicon at room temperature by a factor of 20 to -0.25, and the enhancement reaches 70
times at lOOK. A practical phonon filtering method based on nanocluster scattering is
shown to enhance zT due to reduced thermal conductivity and optimized phonon drag
effect. This work opens up a new venue towards better themoelectrics by harnessing
non-equilibrium phonons. More material systems can be systematically studied with the
developed simulation tools.
Thesis Supervisor: Gang Chen
Title: Head of the Department of Mechanical Engineering and
Carl Richard Soderberg Professor of Power Engineering
3
4
Dedication
To my beloved family.
5
6
Acknowledgement
This thesis work would be simply impossible without the help and advice from various
people I have met from the first day I entered the graduate school. Firstly, I would like to
thank my thesis advisor, Prof. Gang Chen, who provides me the great opportunity to
work in a wonderful and exciting research environment, points out the possible directions
in the long-term and helps with the tiny problems I met during the research, even though
he has to squeeze out his time from the commitments to the department among other
things. His inspiring vision and scientific rigor always helps me to shape my research
habits. Secondly, I would like to thank Mr. Sangyeop Lee and Mr. Bolin Liao, who are
knowledgeable seniors I have learned from. I have spent many hours bothering them on
the details and understandings of the density functional theory as well as the transport
property calculations based on it. It is hard to imagine how I would move from knowing
nothing to the phonon transport and later the electron-phonon interaction without their
advice and patience. I would also like to thank my labmates in the nanoengineering group
for their generous help, especially Mr. Samuel Huberman, Mr. Lee Weinstein, Dr. Yuan
Yang, Ms. Yi Huang and Dr. Yanfei Xu. I would also like to thank Prof. Mildred S.
Dresselhaus who has pointed to me a different perspective of phonon drag and shared
with me her stories in the thermoelectric research.
Finally, I want to thank my family for their support, and many friends outside my
research field, for who I would not list the names but who have painted wonderful colors
to the sky of my research life.
7
8
Contents
Chapter 1.
Introduction .........................................................................................
1.1.
Coupled electron-phonon Boltzmann equation.............................................................
1.1.1.
Picture of Boltzmann equation ...........................................................................
1.1.2.
Electron-phonon interaction...............................................................................
1.1.3.
Scattering rates due to electron-phonon interaction...........................................
1.2.
Phonon drag effect ........................................................................................................
1.3.
Kelvin relation for phonon drag ....................................................................................
1.4
Experimental investigation into phonon drag ...............................................................
1.5
Organization of thesis....................................................................................................
13
15
15
18
21
26
30
32
34
Chapter 2.
Ab initio approach for transport property calculations ..................
2.1.
Density functional theory ...............................................................................................
2.2.
Electron-phonon interaction...........................................................................................
36
36
39
2.2.1.
Wannier function-based interpolation scheme .................................................
2.2.2.
Electron scattering by phonons ........................................................................
2.2.3.
Phonon scattering by electrons...........................................................................
2.3.
Phonon-phonon interaction ...........................................................................................
2.4.
Impurity scattering ............................................................................................................
2.5.
Phonon drag modeling .................................................................................................
Chapter 3.
Simulation of phonon drag .................................................................
3.1.
3.2.
Intrinsic phonon drag ....................................................................................................
Saturation effect ................................................................................................................
3.2.1.
Reduction of phonon drag at high doping concentrations..................................
Cause of saturation effect..................................................................................
3.2.2.
3.3.
Mode contribution to phonon drag................................................................................
3.3.1.
Phonon mode contribution .................................................................................
Electron mode contribution................................................................................
3.3.2.
3.3.3.
Effect of normal scattering and Umklapp scattering .........................................
Chapter 4.
Optimization of phonon drag ............................................................
4.1.
Preferable phonon modes for phonon drag ....................................................................
4.1.1.
Optimization of n-type silicon with ideal phonon filters..................................
4.1.2.
Optimization of p-type silicon with ideal phonon filters....................................
4.2.
Other Phonon filters ......................................................................................................
4.2.1.
Nanocluster scattering for frequency selectivity ................................................
C hapter 5.
5.1.
5.2.
Conclusion............................................................................................
Summ ary ...........................................................................................................................
Future work.......................................................................................................................75
R eferences........................................................................................................................
9
39
42
45
46
48
49
51
51
52
52
55
57
57
59
60
63
63
64
67
69
69
75
75
76
List of Figures
Figure 1-1. (a) Illustration of the collision between electrons and phonons, (b) transitions
shown in the electronic band and (c) corresponding phonon modes in the phonon
dispersion .......................................................................................................................
18
Figure 2-1. Temperature dependence of the calculated intrinsic mobility in n-type and
p-type silicon compared with that of sufficiently pure samples from the experiment. . 45
Figure 2-2. Temperature dependence of the thermal conductivity of pure silicon
com pared with the experim ent...................................................................................
48
Figure 3-1. Intrinsic phonon drag effect for (a) electrons and (b) holes in lightly-doped
silico n .............................................................................................................................
52
Figure 3-2. Calculated Seebeck coefficient with respect to doping concentrations for (a)
n-type silicon and (b) p-type silicon at 300K and 200K on a semilog plot............... 54
Figure 3-3. Phonon scattering rates due to phonon-phonon interaction (red points) and
electron-phonon interaction (blue points)..................................................................
56
Figure 3-4. Phonon mode-specific accumulated contributions to the phonon drag
Seebeck coefficient and the thermal conductivity with respect to (a) phonon frequency,
(b) phonon wavelength and (c) phonon mean free path.. .........................................
58
Figure 3-5. Accumulated contribution to electrical conductivity, diffusive Seebeck
coefficient and phonon drag Seebeck coefficient, with respect to (a) electron band energy,
(b) electron wavelength and (c) electron mean free path.. .........................................
61
Figure 3-6. Electron mode contribution to the phonon drag effect for (a) n-type silicon
and (b) p-type silicon, as well as (c) the phonon mode contribution to the phonon drag
effect for n-type silicon..............................................................................................
62
Figure 4-1. Distribution of preferable phonon modes in wave vector and phonon
frequency........................................................................................................................65
Figure 4-2. (a) Contribution of the most preferable modes to the phonon drag Seebeck
coefficient at different reduced thermal conductivity values and (b) the enhancement of
the factor S 2 /
K
as a function of doping concentration when phonon modes are
selectively scattered ..................................................................................................
67
Figure 4-3. (a) The enhancement of zT compared to bulk crystal achieved by selecting
preferable modes at 300K for n-type silicon with respect to the doping concentration and
(b) the zT enhancement at a doping concentration of 4 x 10' cm-3 as a function of the
tem perature.. ..................................................................................................................
68
Figure 4-4. (a) Contribution of preferable modes to the phonon drag Seebeck coefficient
at different reduced values of the thermal conductivity and (b) the enhancement of the
factor
S2
/
K
as doping concentration for p-type silicon.........................................
69
Figure 4-5. The enhancement of the thermoelectric figure of merit zT with respect to the
10
volume fraction of nanoclusters using phonon frequency selectivity for n-type silicon
with a doping concentration of 1019 cm-3 at (a) 300K and (b) 200K.......................... 73
Figure 4-6. The enhancement of the thermoelectric figure of merit zT if single impurities
74
are used to scatter short-wavelength phonons preferentially.........................
11
List of Tables
Table 1-1. Prefactors for different types of electron-phonon collisions..................... 22
Table 2-1. Scattering mechanisms for electrons and phonons ..................................
49
Table 2-2. Parameters used in determining the electron and phonon relaxation times as
well as in the calculation of the phonon drag effect.. ................................................
50
12
Chapter 1. Introduction
Understanding the transport properties of solids has a main focus on the interaction
between various elementary excitations, among which the interplay between electrons
and phonons
has played a
significant
role in particular
phenomena
such
as
superconductivity' and the metal-insulator transition 2 . The electron-phonon interaction
(EPI) problem was first studied by Bloch3 , Sommerfeld and Bethe , who all assumed the
phonons to be in equilibrium (so-called "Bloch condition") when calculating the
scattering rates of electrons caused by EPI, because of the frequent phonon-phonon
Umklapp scattering. This assumption is well justified and widely adopted for the
determination of transport properties of electrons at higher temperatures5 ,6, including the
electrical conductivity and diffusive Seebeck coefficient. Below the Debye temperature,
however, the phonon-phonon Umklapp process is largely suppressed, and this assumption
becomes questionable. The significance of non-equilibrium phonons on the electrical
transport properties and especially the Seebeck coefficient was first recognized by
Gurevich 7 . The experimental evidence given later by Frederikse8 and by Geballe and
Hull9 clearly showed an "anomalous" peak of the Seebeck coefficient at around 40K in
To
germanium.
address
this
unusual
observation,
Herring
proposed
that
the
non-equilibrium phonons can deliver excessive momenta to the electron system via the
EP1'0 ".
This process generates an extra electrical current in the same direction as the
heat flow, as if the electrons were dragged along by phonons. Therefore this effect has
been dubbed "phonon drag"' 0 , which makes itself distinct from the normal diffusive
contribution to the Seebeck coefficient derived from the diffusion of electrons.
Subsequent explorations revealed that this effect exists in various materials systems,
including
simple
semiconductors
newly-emerged ones1'7
systems2 >2
4
19
like
silicon1-14
and
InSb 1'1
among
other
, layered structures like Bi20 and Bi 2Se32 1 , lower-dimension
and even high-T, superconductors 2 5 . In particular, it has been speculated that
13
phonon drag is responsible for the extremely high Seebeck coefficient experimentally
found in FeSb 22 2- 8 . In parallel, the theoretical picture for phonon drag has been
.
confirmed and further refined by more detailed theoretical models 5 ,2 9- 34
The efficiency of thermoelectric materials is characterized by the figure of merit zT,
defined as zT
=
oS 2 T/Ir, where o-,S,T,iK are the electrical conductivity, Seebeck
coefficient, thermal conductivity and absolute temperature, respectively. Thermoelectric
energy conversion at lower temperatures (around and below 300K) can benefit a wide
range of applications including refrigeration, air conditioning and cryogenic cooling 35 but
is also challenging mainly because the normal Seebeck coefficient drops in magnitude
while the thermal conductivity increases as temperature decreases. It is therefore
tempting to make use of the phonon drag effect for higher-efficiency thermoelectrics in
the lower-temperature range via boosting the Seebeck coefficient. Straightforward as it
seems, controversies exist as to whether zT can be enhanced at all by utilizing the phonon
drag effect. Theoretical models 36 concluded that the optimal zT achievable using the
phonon drag effect is much smaller than 1, while the experiment on silicon nanowires has
suggested the possibility of reaching zT of 1 at 200K 37. The major concern lies in the fact
that significant phonon drag requires phonons to be far away from their equilibrium when
subject to a given temperature gradient (or equivalently, with long mean free paths),
which usually implies high thermal conductivity. So far it is not clear whether it is
possible to decouple the contributions, i.e. keeping a high Seebeck coefficient while
reducing the lattice thermal conductivity, the latter of which has recently become a
common strategy in increasing the thermoelectric efficiency 340. A further opposition to
the use of phonon drag comes as increased doping concentration leads to reduced phonon
drag, known as the saturation effect'0 . It has been generally believed that in samples with
high doping concentration (crucial in obtaining high electrical conductivity), the phonon
drag almost vanishes, which however is an incorrect opinion as we will show later.
14
The key information for a better understanding here is a mode-by-mode analysis of
phonon contributions to phonon drag and thermal conductivity. With the recent
development of first-principles simulation tools442 , the mode-specific contributions to
the thermal conductivity have now become accessible. Obtaining the same information
for phonon drag, however, can be exceedingly challenging. Ab initio calculation of the
EPI, even when the Bloch condition is assumed, has already been proven difficult due to
the ultra-dense sampling mesh entailed, only becoming tractable recently thanks to the
invention of an interpolation scheme based on maximally localized Wannier functions4344.
A further step towards the phonon drag calculation requires an accurate description of the
non-equilibrium
phonon
distributions when calculating EPI, thus combining
the
above-mentioned two calculations, in addition to more stringent convergence conditions.
We will undertake the task of examining the detailed phonon mode contributions to the
phonon drag effect in silicon from first-principles and also explore new possibilities for
improving thermoelectric efficiency based on the information thus revealed. In this
chapter the theoretical formalism is given first for the phonon drag effect, which allows
us to distinguish the contributions from different phonon modes. Henceforth, we will
provide the justification of our calculation by looking at the temperature and carrier
concentration dependence of the phonon drag effect compared with experiments. We then
show that the phonon drag effect can be engineered to enhance the Seebeck coefficient
while largely reducing the thermal conductivity by identifying the "preferable" phonon
modes and filtering out others. An ideal phonon filter is demonstrated to increase zT in
n-type silicon by a factor of 20 to -0.25 at room temperature, with the enhancement
reaching 70 times at lOOK.
1.1. Coupled electron-phonon Boltzmann equation
1.1.1. Picture of Boltzmann equation
The dynamics of a system relies on the "particle" (for example, electrons and phonons)
15
distributions in the real space and momentum space 45 . A major approximation is made
when the single particle distribution function is used to represent all the particles in the
system, therefore simplifying the problem into a differential equation known as the
Boltzmann equation. Simply speaking, Boltzmann equation describes the balance of the
transition of one particle between various available states. There are two major causes for
these transitions. One is coming from the forces (or potential gradients in a more general
sense) acting on the particles, with examples including electric fields for electrons and
temperature gradients for both electrons and phonons. The other originates from the
interaction between the particles, also called scatterings. For convenience, the terms of
the former type in the Boltzmann equation are often called drift terms while those of the
.
latter type are referred as collision terms 46
In equilibrium, the distribution functions for electrons and phonons are described by
45
Fermi-Dirac and Bose-Einstein statistics respectively
. As the macroscopic fields start to
drive the system, electrons and phonons will move away from the equilibrium, and the
degree of the non-equilibrium will be determined by the strength of the scatterings 5 . In
general, the dynamics of electrons and phonons are coupled to each other due to the
electron-phonon
interaction which will become clear in later chapters,
and the
corresponding coupled Boltzmann equations in the steady-state can be expressed as46
f(k)f (k) (fa(k)
a,(k)
VT -
(k)f
aT
V,()-I(q)q) VT = - na
aT
where T is the temperature,
f(k)
aE
T(k)
&at
(1.1)
at(q
()
rA(q)
at
)e,,
e is the electron charge,
p is the electrochemical
potential (incorporating electric potential and chemical potential) and the velocity vectors
v for electrons ( v, (k)) and for phonons (V A(q)) are specified with wave vectors k (for
electrons) and q (for phonons), as well as band number a and branch number A (in
the following we will restrict our discussion to periodic solids where the Bloch band
16
theory applies).
f
and N represent the distribution functions for electrons and phonons
respectively, with equilibrium state (labeled with a superscript 0) described by
11
fO(k) =
+1(1.2)
-A)IkBT
1
N"(q) =
For electrons, the external driving forces include the electrochemical potential gradient
,
VV and temperature gradient VT, each of which leads to a current of electron flow 4 5
while for phonons the only external driving force comes from the temperature gradient.
of
Besides
the
drift
terms
on
f,(k)-f
VT eva( k)a(k)vP
,()
aT
aE
E
{Va~)Af
the
T
r*(q)
of
Eq.
(k) 9fa(k)
*(k) k~k~e~ph
+
Ta
n,(q)-nO(q) +rT)
8n (q)
left-hand-side
t
)e_,h
(1.1), there
(q)
at
e-ph
are also collision terms on the right-hand-side, which describe the various scattering
events experienced by electrons and phonons. For both electrons and phonons, there are
various mechanisms that can make transitions from one state to another. Among these
processes, the electron-phonon interaction is most important for our problem because it
couples the electron and phonon systems, which essentially leads to the interesting
phonon drag effect as we will see below. For this reason, we have separated them out
from other terms contributing to the scattering events. Apart from the electron-phonon
interaction, we have also considered electron-impurity scattering, phonon-impurity
V,(k)-
V
as
aT
q)
nt (q)-
aT
well
as
VT -ev, (k)=
phonon-phonon
aE
VP=
n.(q)-no(q) +
rz(q)
scattering
fa(k>- ,f +k)
11)&
a(1.1).
in
Eq.
Note
ot
n (q)1.
at
processes
,
scattering
e-ph
that the combined effect of these scatterings has been summarized into a-mode-specific
17
variable ( * (k) for electrons and
* (q) for phonons), called mode-dependent relaxation
time. For electrons, z*(k) only includes electron-impurity scattering, while for phonons
r*(q) considers both phonon-impurity scattering and phonon-phonon scattering, added
together according to Matthiessen's rule. Previously it has been pointed out that in certain
materials4 7' 48 (for example diamond and graphene) such relaxation time model is not
accurate enough to describe the thermal transport properties. However, generally this
relaxation time model leads to good agreement for the thermal transport across a wide
range of materials4 9- 53, and has been recently used to study the electrical transport in
silicon4'4 , the material we will focus on in this thesis. The use of the relaxation time
model will be further justified as we compare the simulation results with the experiments.
1.1.2. Electron-phonon interaction
The electron-phonon interaction couples the electron and phonon systems, and is the
key factor that leads to the phonon drag effect5 . In the lowest order approximation,
electrons and phonons lie in their eigenstates, described by the electronic band and
phonon dispersion respectively. Because electrons sit in an environment of atoms, the
atomic vibrations (phonons) will therefore affect the movement of electrons. In the
particle language, this means electrons can collide with phonons, making transition from
one state to another (Figure 1-1). During these processes, energy and crystal momentum
conservation need to be satisfied, which impose the conditions on which processes are
allowed.
(b)
(a)
k
2
ek e+hk+qW22
18
Ii
(c)
Figure 1-1. (a) Illustration of the collision between electrons and phonons, (b) transitions shown in the
electronic band and (c) corresponding phonon modes in the phonon dispersion. In part (a) k1 is the initial
state of the electron, which is scattered to the final state k 2 via the absorption of a phonon mode q.
As we see from the illustration in Figure 1-1, the electron-phonon interaction is a
three-particle process and therefore the corresponding term in the Boltzmann equation
va(k)-
afa(k) Vfe(k).(k)
VT--eva(k)-
c
f, (k)f-f,(k)
zVCP=-
aT
(Eq.
aT
a
ep
(k)(1.1)
an,(q)
n,(q)-no(q)
T
VT=(q)
v(q). -
(fa(k)
+
at
r*(q)
)_,i
involves distribution functions in three different states: the initial electron state, the final
electron state and the participating phonon, each characterized by
fka,
fk,,'
and n,,,
respectively. The rate of the transition processes (also called the scattering rate) depends
on the occupation numbers of these states and typically involves prefactors like
fk,
(1 - fk,,),2
(k -* k' via phonon absorption), which means for an electron to absorb
one phonon, the probability of the scattering is proportional to the probability of finding
the electron at the initial state, the probability that there is position available for the
electron at the final state and the number of phonons, which essentially derives from the
quantum statistics of electrons and phonons.
The coupling between electrons and phonons is described by the coupling matrix
element in the language of quantum mechanics. The eigenstates mentioned before for
electrons and phonons are only approximations to the true states of the system. This is
because in the real system, the Hamiltonian will deviate from the ideal case and
eigenstates assumed for each are not true eigenstates of the whole system5 . However,
since usually the deviation is small, we can regard this as a perturbation to the original
solutions and therefore use perturbation theory to determine the real dynamics of the
system. Electron-phonon interaction is one of such perturbations, which means that the
Hamiltonian, which represents the total energy of the system, has non-zero overlap
19
between the electron and phonon states. In an intuitive way, Bloch wrote down the
contribution to the total energy due to the combined effect of electrons and phonons54
SE =
urR -E
rR
(1.3)
_11 r,R
0
where u is the displacement of the atom, r describes different sub-lattice atomic sites,
R distinguishes different unit cells, and E is the total energy of the system. A simple
explanation of this equation is that, as the atoms in the system move around, the electrons
will adjust themselves to find the lowest energy for that atom configuration and therefore
,
the total energy also changes. Born-Oppenheimer approximation is often assumed5
which states that as the atoms vibrate, electrons respond so fast that they almost see a
static atomic configuration and therefore finds the lowest energy corresponding to that
configuration. This simplifies the problem because electrons and phonons still have their
own eigenstates, and are only coupled to each other via coupling matrix element. Models
go beyond this approximation often consider "vibron" which is a combined state of
electron and phonon in a general sense5 5 (Note that there are also different types of
vibrons suited for different problems). For our problem, the Born-Oppenheimer
approximation applies well and we will restrict our discussion to this picture.
The
explicit
coupling
term,
as
given
in
Eq.
SE =
IUrR
rR
aE
r,R
(1.3), will not be useful for the transport property until we consider how it is incorporated
into the Boltzmann equation. This is accomplished by applying the Fermi's golden rule,
which determines the transition probability (or scattering rate) when the perturbation
induces coupling between different eigenstates (electrons and phonons in our case). As a
first step, Eq. SE
IUr,R
r,R
-
(1.3) needs a more
rR
explicit expression. As shown by Ziman56 , the intuitive expression given by Bloch can
exactly match the results obtained from more rigorous quantum mechanics derivation, if
20
we regard uR as operator acting on the phonon eigenstates while
F
as operator
OI"r,R
acting on the electron states. This simple interpretation clearly couples the electron and
phonon system as we expect, and the electron-phonon coupling matrix element can be
explicitly described by
(n,,
1u n)-(k'aq,,Vjka)
(1.4)
where we have transformed the change of the energy due to the displacement of one
OE
single atom
to the potential change due to a collective atomic motion
ar,R
corresponding to a certain phonon mode with wave vector q and branch number A
(defined as @qV ), which varies in the real space and will be referred as the
(phonon-induced) perturbed potential.
Ika) and Ik'f8) describe different states that are
coupled through the perturbation, and
jnq,)
represents the phonon state (Note that it is
characterized by the number of phonons and the final state can only differ by one). The
first term in Eq. (n, I u In,,)-(k'p8jjVjka)
(1.4) essentially
leads to the proportionality to the number of phonons as we mentioned before for the
scattering rate. The second term, which includes electron states and perturbed potential, is
the key element that is required for describing the scattering rates in the Boltzmann
equation and thus the calculation of transport properties. For convenience, we define the
electron-phonon interaction matrix element as
j
gf6A(k,k,q) = (
(k',6 IOqV Ika)
(1.5)
2mco,1/
which only differs from Eq. (nqA
1 li n,) - (k'pJ
V ka)
(1.4) by
a prefactor inversely proportional to the square root of the phonon frequency. In Eq.
21
C+()f
+h
EEs
G)
q+
k+
6)(k'-(1-f
n
+1g(n(kk',q)
n(k,)k2,cq)
h
g,6(k,k',q)=
(1.5)
-(k'fiVaqAVka)
h is the Planck
is the electron mass.
constant and m
1.1.3. Scattering rates due to electron-phonon interaction
With the electron-phonon matrix element written down, now we can apply the Fermi's
golden rule to calculate the scattering rates due to the electron-phonon interaction. More
the
()"
V(k)a
T
EPI-induced
v,(k)
scattering
terms
(last
f,(k)-f(k)
(k)
V,(q)(q)q)VT=-n
aT
in
Eq.
8fa(k)
VT ev(k)aE ((k)
+
n(I.
z(q +
terms
at
)
specifically,
are
a
t
rj(q) &)e-ph
described by
(af1(k)
at
2rr
,_-h
-
nqAfka0-
+ 1)fka(1
A(kk',q)2(n.,
h k'#lqA + ga,,, (k',k,q)
+
ga,, (k',k, q)1
( a- fk,A,,(k
(nqA +
-Iga, (k~k',q)
anA(q)
at
2;r I E
)ep,
h 2 kak'fl
fkf,)S(k'-k - q + G)6(E,, -
Eka -
hoqA)
fk,,)S(k'--k + q + G)8(Ek,, - Ek, + hpqA)
-k'-q+G)(E, - E, - hcq2
)
-jga,(k,k',q)j
1)(1 - fka)fkg(k -k'+ q + G)S(Eka - Ek,, + hcoq)
nff(1 -fk)(k'-k -q+G)(E
- Eka -
ho)
)qA)
12ka
+
g-,4(k',k, q) n1a(1- fkG)fkS(k -k'
+ga,(k',k,q)1
(nq, +1)(1
-
fka)f,6S(k
-
q+ G)(Eka - Ek, - hcoqA)
-k'+ q + G)6(Eka
- E,, + hcq)
(1.6)
where the electron-phonon interaction matrix element g,6A(k,k',q)
gafpA(k,k',q)=
2mocWo
.(k'fiaqAVjka)
is defined in Eq.
(1.5). In Eq. (1.6), Ekc and
Ek,, describes the energy of electron states and c,, describes the phonon frequency. The
energy conservation is imposed by the last delta function while the crystal momentum
22
conservation is included by the delta function next to it, which describes either Umklapp
process or normal process depending on whether extra reciprocal lattice vectors are
required to bring the difference of the wave vectors to zero. Note that for the scattering
rate for phonons there is an extra 1/2. This is because when k and k' go over the
Brillouin zone each k, ->k
2
process is counted twice (let k = k, and k'=k2 or vice
versa). Here we want to make simple comments on the physical meanings of these
complicated expressions. As we expect, the scattering rates involve the distribution
functions of three different states, and the dependences are clearly different for different
types of processes. For the electron as an example, we list in the following the
corresponding prefactors for different processes, which are similar to the phonon
absorption process (k --+ k' via phonon absorption) we mentioned above.
Table 1-1. Prefactors for different types of electron-phonon collisions
qk
q
k
k
q
k -> k'
phonon absorption
fka
(1-
fk'/O)qA
kq
k
k' -> k
phonon absorption
k -> k'
phonon emission
fA
(1- fki)(qI
+
(1
1)
f ka )fkyJOnq
k' -> k
phonon emission
(1
f ka ) fk'pO(qA +
1)
Besides, the energy and crystal momentum conservation are seen as the delta functions,
automatically imposed when applying the Fermi's golden rule. The scattering rate is
proportional to the magnitude squared of the electron-phonon coupling matrix element
and therefore the coupling matrix serves as a quantitative parameter that determines the
strength of the interaction between electrons and phonons.
23
The
solution
of
"kVT - ev, (k) -
V, (k)-
VT
V
the
Boltzmann
"
aE
a(q)n()VT=-na
-n
aT
rA(q)
equation
described
f, (k) - fa (k) +f()
(k)
*a(k)
+
&t
by
Eq.
)1_
na
(
at
e-ph
combined with Eq. (1.6) is not as straightforward as it seems. This is because it is a
non-linear equation and it is generally harder to find its solutions in a robust way than
solving linear equations. However, we note that in equilibrium, all distribution functions
should take their equilibrium values and the scattering rates therefore vanish, because
otherwise the state of the system will move away from the equilibrium (see Eq.
F
fa(k)-f0 (k)
k + r8fa(k)~
8k' VT - ev, (k)
V(P=
Va (k)-A,(k)vTev\.fa(k)v
T
an( (q) VT
f,
(k)
=
+(q)
an
(q) -n
A(q)
+
at
)e-ph
(1.1)).
This
t(q)
_ph
-t
inspires one to take the lowest order approximation for Eq. (1.6), or the first order
deviation of the distribution functions, which gives rise to the widely-used linearized
Boltzmann equation:
I,,
f
([Ik):[ -
F (k', q A) -Afk, + (ka,qA)- Af + F (ka,k'8)- An
)-ph
e~
(q)
k',,qA
=I
S
e-ph
I-
k',,qA
[Gk.(k'/, qA) -Afka +Gk,,(ka,qA)- Afk
k',,qA
Gq (ka,k'fl) -Anq
]ka,k'f
ka.k'l
(1.7)
where the coefficients
F and G only depends on the equilibrium distribution functions:
24
+-
_(n,,
F..,,(ka, qA) = [no, + I- fo,,) r_ + (nq I +
F,,(ka,
f") +(n,,2+-f")
(k)
F(k'p,qA)
Afka
[Fk
,(ka,qx)
Afkfl]
+
[Fq,(ka,k'p)
-Anqx]
k'8)=
Gk,(k'#, qA)=
Gs,,(ka~q)
f
0-
0a)-f9 -f~ U
[(Akfl
- n
n
G,fl(ka,k'p) = [(n
fkg"l
+
Fk, (k'#, qA) = [n,,+ fk",fl
f,0a
Aka + (k~fl- Aa
+(1.8)
+
+-
fk' )H -
-
0,H
n4+
- (no -
-fk)1
0a
with Hr
S= hga,,(k,k,q) -g(Ek,
S=
Eka
hIg,(k,k',).5(E,, -E
-h2)-.5(k'-k -q)
+ h coq,2)-.(k'-k+q)
denoting processes due to the absorption of a phonon
respectively.
The
AnqA = n2 - n",
first-order
deviations
Afa
Wq2
fk
and the emission of a phonon,
Af =fk', - f and
,, -
characterize the non-equilibrium state of electrons and phonons. Note
Eq.
in
that
k)
[ZF(k'6,qA)
Ff
-Afk, + Z [F,,(ka,qA)-Afki]+ I
e-ph
-
k', qA
anG(q) ),(k'_,q)Af,+Gk,,(ka,q)-Af,]-eat
-ph
[F,(ka,k',6)-An,
k# qA
k',,q
Gqjka,k'p)
kak',
An,
I
_kak',
(1.7) we have re-grouped the contributions so that the dependence of the scattering rates
on each non-equilibrium state is clearly seen. For normal electrical property calculations,
the relaxation time approximation is often used. This approximation naturally arises if we
assume
that
for
the
electron
scattering
(first
in
line
_
a
-p h
t ()
e-ph
k',,qA
ka,k',
k ,q
k',,q
[G a(k'/,q).Afka +Gk.,(ka,q2). Afk,]-
25
_ka,k',
Gq (ka,k')] Anfq2
I
Eq.
[k',
(1.7)) only the distribution of the initial state of the electron deviates from the equilibrium
( Afgk
# 0 ) and that of the final electron state and of phonons remain at equilibrium
(Afk,'a =0, An, =0 ), which is essentially the Bloch condition. In this case, the prefactor
before Afk,
(k). The relaxation time
can be defined as 1/
ir.-P(k)
is what is
usually called the electron-phonon relaxation time (for electrons), which determines the
intrinsic mobility of one material. We should also note that Afkf,
because the terms containing Afkf
is essentially neglected
sum up to approximately zero. In metals and for
elastic scattering with impurities, this approximation is not valid and therefore an extra
correction term (1- cos 0) is often added to the electron-phonon relaxation time, which
6
is called the momentum relaxation time . In semiconductors, however, it is proved, based
on deformation potential models, that for nearly isotropic scattering, the neglect of AfJ,
will not cause much difference. It has also been shown 7 58 that without considering
Afk,,, good agreement for the electrical properties in silicon with experiments can be
achieved, justifying the approximation that terms containing Afk,'O can be neglected.
A more important perturbation term from equilibrium comes in the evaluation of the
assumption Anfq =0. It is clear that this assumption makes non-equilibrium phonons
have no effect on the electron system. When phonons are far away from the equilibrium,
assuming Anq,, to be zero is no longer valid. These non-equilibrium phonons described
by non-zero
(k)F k', A
at
),-ph
On, (q) )
Anq 2 in the electron system (the last term in the first line of Eq.
=f
j(k'p,
q Af +I Fk,,(ka, g)- Af + F,(ka,k'p)-Anq, ]
_k',,A
I
~=ph [kG,(k'?, qA) -Afk, + Gk, (ka,q)- fl.J -
(1.7)) are responsible for the phonon drag effect.
26
Gq,(ka,k'8)j- An,
The above picture describing the phonon drag effect is based on the Seebeck effect,
where a temperature gradient induces a phonon heat flow (characterized by non-zero
An,), which delivers part of its momenta to the electron system and gives rise to an
extra current. Because of the Kelvin relation
H = TS, an extra contribution to the
Seebeck coefficient also implies an extra Peltier coefficient. This is manifested by the
first
two
af (k)
S
terms
-[
Fk,(k'pf,qqA)
=I
at )-ph
ka,k'$
the
second
line
Afk, + Z [F,,(ka,qA)-Afk,]+
j
k6qA
he-ph
an, (q)
in
k',qA
I
of
Eq.
[F,(ka,k'p)8 -qAn]
k',6qA
[Gka(k'p,qA)- Aft. + G.,(ka, q)- fAfk,]- Ga(ka,k'p)- An,
2
_ka,k'#
(1.7), which transfer the momenta of the electron system to phonons when an electrical
current is applied through the system. We will discuss the Kelvin relation later with more
details.
we
look
at
Fk (k'8, qA)j-
-k -I
at
-ph
an,, (q)
at
k'#,qA
I
),_ph
the
last
term
Afka +Z
[F
in
the
second
,O(ka,q2)- Af
k'p,qA
of
Eq.
F (ka,k'p) -An
k'p~qA
[Gka (k'f, qA) -Afk, + Gk. (ka,qA)- Afk,]-
Gq,(ka,k'p) -An,
ka,k#
ka,k'l
+ I
line
,
If
I.
(1.7), the prefactor of AnqA in the phonon Boltzmann equation can be readily written as
1/
r 7 -Ph(q)
(just as the definition of electron-phonon relaxation time for electrons),
which denotes the process that phonons are scattered by electrons at equilibrium. As a
result, higher doping concentrations (more free electrons) lead to stronger scattering for
phonons. Normally when considering the phonon relaxation times, the electron-phonon
scattering is neglected because phonon-phonon scattering dominates the scattering
processes. However, as the doping concentration increases, there will be increasing
probability that phonons get scattered by electrons. It is found from the simulation that
this phonon scattering by electrons account for some fraction of the reduction of the
thermal conductivity in heavily-doped materials59. Furthermore, as we will see in later
27
chapters, it is also responsible for the reduction of the phonon drag effect in
heavily-doped samples compared to lightly-doped samples (the saturation effect).
af"(k) VT -ev,(k) aT
=-nA(q) -n (q)
r(q)
t
'an 2 (q)
at
term
in
Eq.
f,(k)- f(k)
at
a
+(
r (k)
e(k)
)ep/
(1.1) and Eq.
ang(q) N
at
Fa(k'#, qA) -Afk + I
Ia
k'$,,A
I-Z
e-ph
each
+
=-[
)e-ph
of
)e_,h
[Fk(ka,q)-Afkj+
k',,A
k
F
k'#,,A
[Gk (k'#,qX)-Afka+ Gk.,(ka,qA)-Afk,,
Gq,(ka,k'8) -An
-
,
I
aT
af, (k)
K
E
v (q)- an(q) VT
[pt(ka,k'fl)-An,,
meaning
the
)
t
Va(k)-
discussed
"
Having
ka,k'#
ka,k'p
I
(1.7), now we want to make the inclusion of these scattering terms more compact by
rearranging them. If we incorporate the first term of the right-hand-side in the first line of
tf
Eq.
(k)
a
-[ I: Fk,, (k'8,qA) -Afka + I
e- ph
an(q)
at
t
(1.7)
Z [G,(k'8,q
e-ph
and
[F,,,(ka,q2)-Afk,]+
)Afa+
Gk,,(ka,q)-AfkJ-ka,k'
last
term
[F 2(ka,k'6)- An,,
k',Aq
ka,k'
the
I
k'6,qA
I
in
the
Gq,(ka,k'p)]-An,2
I
second
line
of
Eq.
in
Eq.
[F,,(ka,qA)-Afkf]+ [F(ka,k'p)-Anq1
[
)e-ph
k'qA
at
k',,A
[Gk (k',q2 )-Afa +G,, (ka,qA)),-ph
(1.7)
k',,A
into
v,(k)
VA(q)-n
the
2
ka,k'6I
relaxation
'(kVT-eva(k)J-
aT
Af
ka,k'#
'
(k) -Z
F(k'8,qA) -Af + Z
_k',,A
times
"V(
VT=- n. (q)-n
r*(q)
we
already
k)-f (k)
=0
r,(k)
)
n
at
coupled Boltzmann equation now becomes
28
Je-ph
f(k)
at
have
N
)e_,h
(1.1), the
"
=_f- "(k)-f:(k)+
at
d,,ft
r,(k)
Z [FkF,,(ka,qA)
- Afk,,]+ Z [F, 2(ka,k'p)-Anqj
k,',qA
ang(q)n,,
S -(q (q) -no () +
,ri
A(q)
at
k,8,qA
Z [Gk,(k'p,qA)-
Afka +G,,(ka,qA)- Afk,)o
ka,k7,(
(1.9)
where new relaxation times include part of the electron-phonon coupling (scattering of
electrons by equilibrium phonons as well as scattering of phonons by equilibrium
electrons)
=
-k)
+
GqA(ka,k'p8)
+F,(k'O,
r()r(k)
'Oq
'-a
qA)
(k)
(1.10)
=
A)
r,(q)
kak
and the remaining terms describe the coupling between non-equilibrium states in the
electron and phonon systems.
1.2. Phonon drag effect
We have above derived the coupled electron-phonon Boltzmann equation in a compact
form
as
at
",=
ift
an-(q)
at
k - O"k +Z [Fk,,(ka,qA)- Afk,,]+ Z [F,,(ka,k'p)- Anqj
r,(k)
k'/,qA
n, (q)-no(q) +
),if,
shown
Tr(q)
I
kak'p
in
Eq.
k,,,q2
[Gk.(k'pqA)- Afk, +G, (ka,qA) - Afkj
(1.9). The electron relaxation time ra (k) incorporates electron-impurity scattering and
electron scattering by equilibrium phonons, while the phonon relaxation time T. (q)
contains phonon-phonon interaction, phonon-impurity scattering and phonon scattering
by equilibrium electrons. The coupling through the non-equilibrium distribution is
manifested by the collision terms that are not described by the relaxation times in Eq.
29
t
at
=dft
"(k f(k
-r,(k)
n. 2(q)
at
)dri
-
+ IfF,( [ (ka,qA)-Afk,,]+ F [, (ka,k'8) -LAnq,,
kflqA
no(q) + Z [Gka(k'fl,qA)- Afka +Gk (ka,qA)- Afkf]
r(q)
(1.9), which are responsible for the phonon drag effect. In this section we will derive the
phonon drag Seebeck coefficient based on these equations.
We will adopt the Peltier picture, where an isothermal electric field is applied to drive
the coupled electron-phonon system. We choose this approach because it directly
provides the phonon drag contribution from each phonon mode. The derivation based on
fundamental relation between the Seebeck coeffient
S and Peltier coefficient
I
-
the Seebeck effect will be discussed in the next section, which clearly shows that the
Kelvin relation - still applies to the phonon drag effect.
For the Peltier effect, a non-equilibrium distribution of electrons Afk,
is generated
first by the electric field, which will then drive the phonons away from the equilibrium.
However, the determination of Afk,
requires the knowledge of Anq. , which also
appears in the electron Boltzmann equation and makes solving fully-coupled Boltzmann
equations
a
"
at
raf
an(q)J
at
)dri
formidable
task.
One
further
"= ,f (k), - + 1 [F1,,(ka,q2) -Afk,,]+ E [F 2(ka,k'p)- Anq,
T,(k)
k,',qA
-
step
towards
solving
Eq.
k,,,A
(q) -ni(q) +
[Gka(k'f,q2) Afka +Gk,,(ka,qA)- Afk,]
rA(q)
ka,k'#
(1.9) is to realize that the influence of the current-induced non-equilibrium phonons on
the phonon drag effect, indirectly through affecting the non-equilibrium electrons, is a
higher-order effect, which can be justified by the fact that the phonon drag phenomenon
is found to have a small influence on the electrical conductivity. Therefore for the
electron system, we can then assume that phonons are at equilibrium (An,, =0 can be
assumed
in
the
first
30
line
of
Eq.
=)t,(k - " ) "(k) + Z [Fk, (ka,qA)- Afk,,]+ I [F,,(ka,k')#)-An ]
[
(k)
,ra
rft
2 (q)
k,6,,A
k,',q
n2 (q)-n"(q)
An + Z [Gk,(k'#,q)- AfG
+Gk,,(ka, q)- Af],,)
ka.k,/
,,rr(q)
&t
(1.9) and note that for the Peltier effect the phonon drag comes in through the last term in
"a =
at
k) - ) "(k) + I [F,,(ka,qA)-Afk,,]+
n,,(q) n
at
drif, S(
I [F2(ka,k'i)-Anq 2]
k/J,qA
k,',qA
ra(k)
)dft
Eq.
of
line
second
the
q
+ ka,k',i[Gka(k'p,q) - Afka +G,,(ka,q)- Afk,)]
n 2 (q)-n(q)
-rA(q)
(1.9) instead of the last term in the first line). As a result, the electron-phonon Boltzmann
equation can be partially decoupled, which leads to a feasible computational approach for
calculating
(phonon drag contribution to the Seebeck coefficient). Now the electron
Sph
distribution function can be directly written down using the relaxation time model
V,(k)
Eq.
f,(k)
"(k)
E
(VT-ev(k)
aT
a
r(k)
T(qk) -a
01
(q)
A)-
VT=-n
VA (q)IT
+
-r(q)
we also assume that the term
_f_(k)
(
(see
Afea eavk
=-
+
at
eno(1.1); here
)e_,h
n~q
at
)e-ph
F,(ka,qA)-Afk,,]
will vanish, which is a
k',f,qA
commonly used approximation
as discussed in section 1.1.3). The normal electrical
conductivity and Peltier coefficient (related to the normal Seebeck coefficient via the
Kelvin relation II= TS) can be obtained by looking at the charge current and energy
current induced by such a non-equilibrium electron distribution, respectively:
31
3
KNk
<
kA
"."
3o7Nk
3Nk I
(E -pfka
k(-lVxafc;a/
e
=
2
-af=V(E ka kI
)/(V)
3Nk
(E f1)Va
DE)
aka
ka
J(1)
"
Or13Nk
ka
&E)
k
where o- is the electrical conductivity, H is the Peltier coefficient due to diffusion of
electrons and Nk is the number of points in the discrete reciprocal space mesh for
electrons. The Peltier coefficient shown here, only includes the energy current of the
electrons, and will be denoted as "diffusive" Peltier coefficient Haif in the later part
(and corresponding diffusive Seebeck coefficient Sdaff). Besides this, the energy current
also has its origin in the phonon heat flow. As we have discussed, in the isothermal
condition the phonons acquire the momentum via the electron-phonon coupling shown by
I(n,(q)
the
last
"
at
term
=-
,drift
the
+ Z
,(
-r (k)
)dri,
_ _n
at
in
Z
line
[F,,(ka,qA)-Afkfl]+ I
k,,,,
(q) - n (q) +
-r(q)
second
of
Eq.
[F,(ka,k'#f)-An]
k',f.qA
[Gka, (k'6,q)- Afka + Gk,, (ka, q2)-
Afk,,)
ka,kfl
(1.9) and lead to an extra heat flow, which manifests the "phonon drag" effect (In this
case "electron drag" might be a more suitable word). Given the electron distribution
Afka,
the phonon Boltzmann equation now becomes
+eV
(q)n,(q)-no(q)
+erpr= - kaG ,(k'fJ, q2)re Vka +Gk,, (ka,q),rkv, E
kaf(q)
Lak,
af[G#
aE
aE
(1.12)
where the drift term vanishes because there is no temperature gradient. It can be readily
solved to obtain the phonon distribution function
32
An,(q) = r(
-
G
(k'V,qk)va
" +Gk,,(ka,qA)rkv,, V W
l
ka,k',f
(1.13)
Considering the heat flow described by q=
hoqvq,,AnA
and the Kelvin relation, we
finally arrive at the formula describing the phonon drag Seebeck coefficient:
Sph=
2
2
3onNkNq kT
Jhq,~qAq~.*
q
L-
with HI= r gafl(k,k',q)
.(Ek,
- Eka - hoq)-
5(k'-k
where e is the electron charge, u is the electrical conductivity,
volume,
Nk
and
Nq
h
(TkaVka _'kfiVkflk41kfllq
,kakf,
fa(1.14)
-q)
0 is the unit cell
are the number of points in the discrete reciprocal space mesh for
electrons and phonons, and the term inside the bracket of Eq. (1.14) is the phonon drag
contribution (neglecting the common prefactor) from each phonon mode. Equation (1.14)
essentially describes the momentum delivery from non-equilibrium phonons to the
electron system via EPI. The total Seebeck coefficient is obtained by summing the
diffusion contribution and the phonon drag contribution, i.e. S = Sdif +
.ph*
Here we briefly comment on the various models used by previous work to study the
10
phonon drag effect. In Herring's original paper , an intuitive formula was used to
estimate the phonon drag magnitude, which is proportional to the relaxation time of
phonons that can deliver the momenta to the electron system and inversely proportional
to the relaxation time of the electrons. A more rigorous analysis based on solving coupled
electron-phonon Boltzmann equation was also discussed, with the results mostly focused
on the scaling analysis'0 . Following this, Ziman introduced a variational method to solve
the coupled electron-phonon Boltzmann equations5 , by first recasting the Boltzmann
equation into a variational form describing the entropy generation rate in the system.
Therefore solving the Boltzmann equation, in Ziman's approach, becomes a functional
minimization problem. Using this method, Ziman showed the order of magnitude of the
33
phonon drag Seebeck coefficient in metals is around a few kBT . Bailyn further extended
this variational method to study the phonon drag with more details 29. There are some
major assumptions used by Bailyn: (1) For phonons, dominant phonon-phonon scattering
is represented by relaxation time approximation;
(and
possibly
approximation;
electron-electron
(3)
scattering)
The electron-phonon
(2) For electrons, impurity scattering
is
represented
interaction is
by
relaxation
represented
by
time
transition
probabilities, with assumed perturbation potential. The first two assumptions also serve as
the basis in our derivation, but the model of deformation potential is no longer required
because the information of perturbed potential can be obtained from first principles
calculations.
Our formula presented above is identical to the results given by Cantrell and Butcher 32
who assumed weak electron-phonon coupling to ignore the effects of non-equilibrium
phonons back on the phonon system (Seebeck picture). It was argued for GaAs that the
electron-phonon scattering indeed can be neglected when considering the phonon
transport. However, our previous derivation has shown that this assumption can be
relaxed and in later chapters we will demonstrate that a wider agreement with experiment
can be achieved within this formalism. Particularly, we will study the whole doping
concentration range with the saturation effect taken into account, which is crucial for
evaluating the thermoelectric property (typical thermoelectric materials have high doping
concentrations) but not examined by previous work.
1.3.
Kelvin relation for phonon drag
The above derivation is based on the Peltier picture, for which an isothermal electric
field drives a charge current, which transfers the momenta to the phonon system and
gives rise to an extra heat flow. Here we want to further clarify the phonon drag effect in
the context of the Seebeck effect, which completes the picture and provides a clear proof
for the Kelvin relation applied to the phonon drag effect.
34
For the Seebeck effect, a temperature gradient drives the phonon flow, which then
transfer their momenta to the electron system via the non-zero Anfl
appearing in the
electron
line
Boltzmann
equation
= - fa(k)f(k)
at
2 (q)
-
at
)drif,
in
k,(k)
- nG(q) + I
the
first
of
Eq.
[q,(ka,k'6).Anq,
k',,2
[G, (k'fl,qI) Afk, +G,,,(ka,qA)- Afkf)
fk,
k,k',
TA(q)
-
term
+ Z[F,(ka,q)-Afk, ]+
-a
drift
(last
(1.9)). Similar as before, we realize that the influence of non-equilibrium electrons on the
phonon drag effect, indirectly through affecting the non-equilibrium phonons, is a
higher-order effect (i.e. Afk,
0 can be assumed in the phonon Boltzmann equation).
Accordingly the non-equilibrium phonon distribution can be directly expressed as
AnqA2 = rqAv,2-VT
at
n0
and
q,
aT
the
electron
a=ak - kL(k) + Z [F,(ka,qA)- Afk,,]+
drft
r,t (k)
k,',q
a Bngn
) ()n(q)
=at ,,,
- n"(q)
)
) + I
(q)
ka,kf,
Boltzmann
equation
in
Eq.
[,(ka,k'6)- Anq,]
W,6,
[Gk,(k',q2)- Afk, +Gkfl(ka,qZ)- Afk,,]
(1.9) can then be readily solved given the phonon distribution function (for the electron
Boltzmann equation we again assume that the term
[,(ka,qA)F Afk, ] will
vanish). Considering the electrical current density described by
j= e
Vkagfka, afr
ka
some rearrangements we will find that the same equation (Eq. (1.14)) is derived,
describing the phonon drag contribution to the Seebeck coefficient.
We note that the two methods' merging into the same result is not a coincidence. In fact
the Kelvin relation can be derived from a more fundamental law - the Onsager's
reciprocity theorem. This theorem states that linear reciprocal processes (examples
include thermoelectrics, piezoelectrics, pyroelectrics, etc.) are not independent of each
other, and simple relationships can be established essentially originating from the detailed
35
balance under the time-reversal symmetry. In our problem, since the time-reversal
symmetry is not broken, the Kelvin relation should still applies as for the normal
diffusive Seebeck coefficient (and Peltier coefficient).
1.4
Experimental investigation into phonon drag
Apart from the experimental work mentioned in the beginning of the introduction in
which the phonon drag is discovered in various materials, here we discuss some of other
experimental work that dived into the details of the phonon drag, which potentially lead
to further manipulation of the phonon drag effect.
One difficulty associated with phonon drag is that simple thermopower measurement
cannot distinguish between the contribution from the diffusion of electrons and the
phonon drag effect. At low temperatures, it is believed the phonon drag dominates the
measured Seebeck coefficient since theoretical models predict a low value for the
diffusive Seebeck coefficient.
At higher temperatures (for example around room
temperature), however, the phonon drag effect becomes weaker, and as a result there is
ambiguity in how much the phonon drag contributes to the total Seebeck coefficient. The
experimental work by Trzcinski et. al. seeks to provide such information by "quenching"
the phonon drag contribution at room temperature 60. They designed a point contact device
where two tips make contact only at a small region, where the temperature gradient is
applied across the point contact and the voltage bewteen the two ends is measured.
Because phonons that contribute to phonon drag typically have long mean free path (a
qualitative understanding since Herring's work'0 ), it is expected that the Seebeck
coefficient will decrease as the phonon drag makes less contribution if the contact area is
reduced. In this way, they found that the phonon drag still plays a role at room
temperature. However, because it is not clear how much the point contact geometry
destroys the phonon drag, it is hard to reach quantitative conclusion on the contribution of
the phonon drag.
36
The other approach to examine the phonon drag is to study the effect of nanocomposite
on modifying the phonon drag magnitude. It has been known for a long time that sample
size will affect the phonon drag at very low temperatures due to boundary scattering of
phonons9"0 . Recently, making nanocomposite materials has become a standard way to
.
reduce the thermal conductivity, thus improving the thermoelectric performance3 8
However, it is not clear how this approach will be applied to materials with large phonon
drag effect for cryogenic cooling applications. Experiments on FeSb 2 have been
conducted 27 , which showed greatly-reduced thermal conductivity, but also with a
largely-compromised
Seebeck
compared to the peak of
-
coefficient
-45000pV/K
(hundreds
of
pV/K in nanocomposites
in bulk crystal). Although debates exist for
whether the large Seebeck coefficient in FeSb 2 is due to phonon drag or Kondo effect, the
experiments in the first place provides a clue into the mechanism of the Seebeck
coefficient, indicating that FeSb 2 might have significant phonon drag, which is largely
reduced due to the scattering of long wavelength phonons at the grain boundaries.
Furthermore, if this argument is true, it also poses the question as how one should reduce
the thermal conductivity with minimum detrimental influence on the phonon drag effect.
As we have mentioned in the beginning, phonon drag will also be weakened if the
carrier concentration is increased (so-called saturation effect1 0).
Although we will
uncover that even in heavily-doped samples the phonon drag still makes sizeable
contribution to the total Seebeck coefficient and thus contradicts previous beliefs that
phonon drag almost vanishes at high doping concentrations, it is still desirable to
minimize the effect of the high-level doping on the phonon drag. One idea is to separate
the
doping region
where
charges
flow
and
the
phonon active
region
where
non-equilibrium phonons are generated. A simple geometry could be a heavily-doped
conductive layer deposited onto an insulating substrate. Under the in-plane (along the
interface) temperature gradient, non-equilibrium phonons will be generated in the
substrate, which experience less scatterings (because these phonons mostly see a
37
low-doping level environment) but couple to the phonons and also electrons in the
conductive
layer
and
therefore
produce
extra
electrical
current
through
the
electron-phonon coupling. It is then expected that the peak of the phonon drag will follow
the phonon properties of the substrate instead of the conductive layer, which has been
61
shown by the experiment conducted by Wang et. al. , who have claimed that such
thin-film-on-substrate geometry could be a way to tune the phonon drag property (in this
case the phonon drag of the thin film is tuned by the phonon properties of the substrate).
However, this simple structure is not very useful for thermoelectric applications because
the substrate is thick compared to the conductive layer, and therefore the large thermal
conductivity through the substrate compromises the thermoelectric efficiency. One way to
circumvent this problem is to make stacked structure of conductive layer and insulating
layer, or superlattices, which is examined in the work of Ohta et. al.2 3 , where superlattices
of alternating SrTiO 3 and SrTio.8No.20 3 are used and conduction electrons are strongly
confined in SrTio.8Nbo.20 3 layer, essentially forming a 2D electron gas. It is argued that
the phonon drag leads to the large Seebeck coefficient observed in such superlattice
structure and gives rise to the high thermoelectric efficiency, which could benefit from
the separation of the doping region and the phonon active region.
1.5
Organization of thesis
Despite of the efforts that have been devoted to manipulate the phonon drag, currently
it is still hard to rationally design the materials and also structures due to a lack of the
quantitative knowledge of how much each phonon mode contributes to the phonon drag
effect in different temperatures and doping concentrations. In the following we will use
an ab initio approach to study the phonon drag effect, providing such detailed
information of the role of each phonon, thus bringing the optimization of the phonon drag
into material designs for better thermoelectric performance. The organization of this
thesis is as follows: Chapter 2 provides the details of the ab initio computational
38
approach we used and developed to calculate the phonon drag effect. Basic concepts of
the first principles method (density functional theory) and an interpolation scheme for the
fast determination of electron-phonon coupling matrix (Wannier interpolation) are
presented, with focuses on the various scattering mechanisms. Chapter 3 compares the
results on silicon with the experimental data and justifies our computation framework.
Key features of the phonon drag are revealed, which uncover the fact that phonon drag is
not negligible
in heavily-doped
samples as generally believed. Furthermore, the
quantitative contribution of each phonon mode to the phonon drag Seebeck coefficient is
obtained, which shows distinct difference from the contribution to the heat conduction.
These findings are explored in Chapter 4, where we demonstrate an ideal phonon filter
that can largely
enhance the thermoelectric
performance
of silicon from room
temperature down to lOOK. A practical phonon filter based on nanocluster scattering is
also discussed to reach the limit set by the ideal phonon filter. Chapter 5 provides a
summary and future works on the direction of optimizing phonon drag in a wider range of
material systems.
39
Chapter 2. Ab initio approach for transport
property calculations
The calculation of the phonon drag Seebeck coefficient relies on the evaluation of various
variables appearing in Eq. (1.14). These include the electronic band structure Ek,
associated electron group velocity va(k)), phonon dispersion coq
(with
(with associated
phonon group velocity vA(q)), electron relaxation time rT,(k), phonon relaxation time
r, (q) as well as the electron-phonon interaction matrix element g,, (k, k', q). The first
two are equilibrium properties describing the eigenstates of electrons and phonons, which
can be obtained from density functional theory (DFT). The remaining ones require
consideration of various scattering processes. In this chapter we will first describe how
these variables can be obtained using an ab initio approach based on DFT. For faster
convergence of the problem, we also made tricks when integrating over all the available
scattering processes, which will also be discussed later in this chapter. We note that, from
here on, all the calculations will be carried out on silicon. The possible extension to other
materials will be discussed in the last chapter as we look forward to more material
systems.
2.1.
Density functional theory
In obtaining the band structure and phonon dispersion of periodic solids, quantum
mechanics-based density functional theory (DFT) has become a widely accepted tool,
which gives information on various physical properties (electrical, thermal, optical,
mechanical, etc.) of the material. This so-called first principle method originates from the
famous Hohenberg-Kohn theorems for quantum mechanics 62 :
Theorem 1
"For any system of interacting particles in an external potential
40
V, (r), the potential V,,(r) is determined uniquely, except for a
constant by the ground state particle density no (r)"
Theorem 2
"A universal functional for the energy E[n] in terms of the density
n(r) can be defined, valid for any external potential V, (r). For any
particular V, (r), the exact ground state energy of the system is the
global minimum value of this functional, and the density n(r) that
minimizes the functional is the exact ground state density no (r)"
The proof of these theorems can be seen in Hohenberg and Kohn's original paper63
Here we want to discuss the consequence of these theorems. For interacting particle
system with a great amount of particles, there are many degrees of freedom if one tries to
solve the Schrodinger equation, namely the wavefunction will have many independent
variables, each of them characterizing one electron. It therefore seems that to know the
information of the whole system one must solve the wavefunction which considers every
electron. However, the first theorem of Hohenberg and Kohn essentially states that only
the ground state particle density is enough to know the information of the whole system
(because as long as the density is known, according to the first theorem the external
potential is determined, which uniquely defines the total Hamiltonian and therefore the
whole system). This is a surprising result since the particle density at one point includes
the contribution of all the electrons, but the theorem states that restricting our knowledge
to only the particle density will not let us lose any information. This appears even more
non-trivial if we notice that this theorem is exact, in the sense that it is fundamentally
lying in the many-particle Schrodinger equation. Although the theorem reduces the
problem of solving many-particle wavefunctions to solving particle density that only
depends on r, it will not be useful until relation between the particle density and the
total energy can be established, which is essentially the task of the second theorem.
According to the second theorem of Hohenberg and Kohn, a universal functional can be
established that relates the total energy and the particle density, and therefore finding the
ground state energy becomes a minimization problem of the density functional. This is
41
why the method is called density functional theory. There are more steps and
considerations towards establishing this functional, but the key idea behind DFT is
already illustrated by these theorems.
Because no major approximations come into DFT except the atom types and structure
information, it can be regarded as an ab initio approach. We should note that there are
assumptions made to speed up the calculation (so-called pseudo-potential concept, among
others)62, but these do not change the quantum mechanics basis of this method and are
usually well justified. We also should mention that, as described by Hohenberg-Kohn
theorem, what DFT solves is the ground state of the system, which essentially assumes
the temperature of the system to be zero. For equilibrium properties like band structure,
DFT can usually give very accurate results compared to the experiment (except the band
gap). However, the calculation of electrical transport properties like mobility and Seebeck
coefficient has received less investigation, mainly because of the increasing difficulty in
extracting higher-order information (coupling matrix between eigenstates compared to
eigenstates themselves) from the density functional theory and also the dense mesh
entailed to reach convergence. In the later chapters we will mainly focus on the problems
of electrical transport property calculations and try to extend the current methodology to
include the phonon drag effect.
For the equilibrium properties of electrons and phonons, we use the QUANTUM
ESPRESSO package 64, with the local density approximation (LDA) of Perdew and
Zunger 6 and a cutoff energy of 60 Ryd. A 12x12x12 k-mesh is used for calculating
the band structure and a 6 x 6 x 6 q-mesh is used for the phonon dispersion. We note that
ground-state DFT calculations solve the wavefunctions
Ika) with corresponding band
energies, while the density functional perturbation theory (DFPT)66 can provide the
phonon dispersion as well as the mode-specific perturbed potential aqV , which will be
used in calculating the electron-phonon matrix element. DFPT is an efficient way to
calculate the response of the system to changes of variables other than the electronic
42
degree of freedom. Examples include response to electric field (dielectric constant),
response to atomic displacement (phonon), second-order response to both electric field
and atomic displacement (effective charge) and so on. It is a perturbed version of DFT
and therefore directly characterizes the changes due to the perturbation. More information
on DFPT can be found in the literature6.
2.2. Electron-phonon interaction
Key to the calculation of the phonon drag effect is the electron-phonon interaction matrix
elements, which explicitly appears in Eq. (1.14). This information is also implicitly
included in the electron and phonon relaxation times showing in Eq. (1.14). Here we first
describe how the electron-phonon matrix elements are obtained on a dense mesh, which
is required for convergence, and their effects on the relaxation times will be discussed in
later sections.
2.2.1. Wannier function-based interpolation scheme
The
calculation
of
(
|ka)
electron-phonon
interaction
VIka)
requires
element
the
knowledge
of
the
and the perturbed potential aqAV. These can be easily calculated
on coarse meshes (for example we use a 12x12xl2
6x6x6
matrix
1/2
-(k'l,
gap(k,k',q)
wavefunctions
the
q-mesh for phonons).
k-mesh for electrons and a
However, the stringent convergence demands the
knowledge of both wavefunctions and perturbing potentials on an ultra-dense mesh (as
we find, the q-mesh needs to be as dense as lOOx1 OOx 100), which is an formidable task,
and only became accessible recently due to a Wannier function-based interpolation
scheme 4 3,67 , allowing us to interpolate between the matrix elements from the coarse
meshes to produce finer meshes. Here we illustrate the basic concept of the Wannier
interpolation scheme and more information can be found in related references43,44
43
Given the information on a coarse mesh, there are various ways to perform the
interpolation. A simple method is to use linear interpolation, which averages the
contribution from the discrete sampling points on the mesh, but in general is only a
mathematical manipulation and does not provide physical insights. Therefore the linear
interpolation has accuracy limited by the mesh density used. A more accurate way should
utilize the features of the physical system. In systems which only have short-ranged
forces, the extent of single particle wavefunction is very localized to certain atomic site.
For such systems, tight-binding model is often used to describe the electron states and
calculate the band structure (here we assume only one atom in the unit cell, but it can be
easily generalized) 46
Vf,k(r)
- Ie
'c,,,
(2.1)
(r - R)
m,R
where $,
is single particle wavefunctions for an isolated atom and R
describes
different unit cells. The tight-binding model describes the wavefunction as a linear
superposition of the wavefunctions for isolated atoms on different sites, with coefficients
incorporating the overlap. In the reciprocal space when plotting the band structure, it may
seem that the system has different information at different wave vector k , which
changes across the first Brillouin Zone. Indeed Eq.
eikR c,.,q,(r-R)
'fnk(r)rn.R
(2.1) gives the wavefunction and associated band energy for each given k . However, we
see that the band structure is actually only governed by the superposition coefficients
C..,, essentially because of the short-range character of the interaction in the system. This
indicates that, only a coarse mesh in the reciprocal space is enough to obtain such
coefficients, which then fully determine the wavefunction and thus band energy on every
k-point (from coarse mesh to dense mesh). This is the key idea behind the Wannier
interpolation scheme. For real materials, however, the tight-binding model is not always
valid because the wavefunctions are not necessarily localized to certain atomic sites.
Nevertheless, functions (denoted as
f
) playing similar roles as the atomic wave
44
functions
#,,
in the tight-binding model can still be found46.
Ak
where
'nk
(r) =
R
e
Rf(r,
(2.2)
R)
(r) is the true Bloch wavefunction. This is because we can always define
them through the Fourier transform of the true Bloch wavefunctions
f,(r, R) =
-
f dk
(r)
eik-
(2.3)
VO
which also shows that
f
must has the form f(r, R) = f(r - R) because as the both r
and R is displaced by a Bravais lattice vector
f(r - R) into Eq. VA (r) =
f
is not changed. The insertion of
ek-Rf(r, R)
(2.2) makes
R
direct analogues to the tight-binding model shown in Eq.
V,
eik-R cnn, (r - R)
(r) =
mR
(2.1) and these functions (f(r - R)) are known as Wannier functions6 8 . The Wannier
interpolation scheme essentially is trying to set up a tight-binding model from the
information (in our case electron wavefunctions and perturbed potentials) on a coarse
mesh 43, so that information on any point in the reciprocal space can be obtained similar as
fl(r) -le
what Eq.
ikR
nin
(2.1) describes.
(r-R)
rn.R
However,
we
need
to
mention
that
states
defined
by
Eq.
(2.3) are not necessarily localized
dk -e-*k-R Vk (r)
(R)=
Wannier
VO
in the real space. To achieve a meaningful tight-binding model, another key concept is to
combine different bands, whose superpositions may lead to more localized states. There
has been development on how to achieve the maximal localization of such states via the
Fourier transform of the Bloch wavefunctions to the real space and the result also
depends on the definition of the extent of the localization
. Here we will adopt the
concept known as the maximally-localized Wannier functions6 9 , which have been shown
45
to connect with local orbitals in the material. By transforming the Bloch wavefunctions to
the maximally-localized Wannier functions in the real space, a tight-binding model can be
established, which can then be mapped back to any point in the reciprocal space.
Above we have only explicitly discussed the electron states. For obtaining the
electron-phonon matrix element, the perturbed potential is also required on a dense mesh.
These can be accomplished by the same idea of the Wannier interpolation scheme. For
certain phonon mode (q, A), the perturbed potential is due to a collective motion of
many atoms. It has contributions from different atoms sitting on different atomic sites. Its
Fourier transform gives rise to the perturbed potential due to the movement of one atom,
which is intuitively seen as localized in the real space. Similarly, the dynamical matrix of
certain phonon mode (q, A) is determined when knowledge of force constants (the force
acting on one atom due to the movement of another atom; two atoms can be the same one)
is known. The Fourier transform of them therefore gives the force constant. These are
meaningful physical variables in the real space and clearly will be enough to know the
phonon perturbed potentials as well as the phonon dispersion as long as the forces
between atoms are short-ranged. It has been known that in polar materials, long range
electric field may arise due to the zone center longitudinal optical phonon , which makes
such interpolation harder and requires taking extra care. In the material we will explore in
this thesis (silicon), this long range force does not occur and therefore we can safely use
the Wannier
interpolation
scheme. We note here that the interpolation
of the
electron-phonon matrix element involves the Fourier transform for both electrons and
phonons. The Wannier states (tight-binding model) in the real space are first established
based on information on coarse meshes, and then interpolated to a finer reciprocal mesh
to obtain the electron-phonon matrix element, as well as band structure and phonon
dispersion.
46
2.2.2. Electron scattering by phonons
We
have
interaction
that electron-phonon
mentioned
leads
to the
so-called
electron-phonon relaxation time (for electrons), described by the second term in the first
E+ Fk1(k'pf,q2)
=
line of Eq.
a)
(k)
___
1
TA (q)
=
1 +
r* (q)
(1.10). Here we want
kflqA
Gq (ka, k'p)
to rewrite its expression in a more explicit way (after some algebraic manipulations):
r'-Ph(k)
1P
ITr
2
h 1
dq Ig,6z(,q
azZBBZ
(nq,
2,
XL
+
ffl
n,+
q
)5(E.k+ CoqA-Epk,)
fl
)d
tgfl(kq) %A
)
I
f
+q
,k
(2.4)
where the wave vector k' of the final electron state is changed according to the crystal
momentum conservation. After obtaining the electron-phonon matrix element as well as
other information on a dense mesh, the major problem is how to perform the integration
while incorporating the energy conservation. Here we use the Gaussian smearing method
for calculating the electron relaxation time. The smearing method represents the delta
function as a Gaussian function with the width (in unit of energy) determined by the
Gaussian broadening parameter. As the Gaussian broadening parameter approaches zero,
the Gaussian function better describes the energy conservation imposed in the scattering
processes. However, it should be emphasized that the mesh density must increase along
with the decreasing Gaussian broadening parameter. Otherwise, the scattering rate will be
underestimated because as the energy conservation condition described by the Gaussian
function becomes more stringent, the sampling points that can satisfy become less, which
leads to lower scattering rates. To avoid such spurious result, for given Gaussian
broadening parameter the mesh density has to be large enough so that increasing the
mesh density will not lead to apparent changes of the scattering rates. Considering these,
we have checked the convergence with respect to the Gaussian broadening parameter,
which is necessary when using the smearing-based method for the integration.
47
For the electronic transport properties of semiconductors, the major contributions come
from the electron states lying close to the band edge (usually within 0.5eV from the band
edge). This
is because
the electron
distribution
function generally
follows the
Fermi-Dirac distribution function (with only small deviations; this is why we can assume
linearized Boltzmann equation before), which exponentially decays as the energy is
above the Fermi level and therefore electrons with energy far above the Fermi level will
only have negligible contribution to the transport properties. For this reason, a band
energy cutoff (measured from the band edge) is used to select only electron states near
the band edge, which also speeds up the calculation and saves the memory required to
store the wavefunctions.
The electron-phonon scattering is responsible for the intrinsic mobility of materials
with low defect density. In Figure 2-1 we show the resulting intrinsic mobility with
respect to temperature as a test to the electron relaxation times we obtain. We note here
that previous
first principles
calculation
have obtained
similar
agreement
with
experiments for the electron mobility in silicon 7' 58' 70. As is seen in Figure 2-1, overall the
results agree well with the experimental data. There is some discrepancy for the hole
mobility in p-type silicon near room temperature. This shows that the experimental
samples experience more scattering and therefore bear a lower mobility. We speculate this
to be a result of the split valence bands due to the spin-orbit coupling at valence band
edge. We cannot confirm this point yet because the spin-orbit coupling is not included in
our calculation. However, this discrepancy should not affect the Seebeck coefficient
calculation much, because we know from the Boltzmann description of the diffusive
=3N
Seebeck
coefficient
(Eq.
-I=
k
(E -p)VxkaAfk
T1
=__N
(
(afk
k
e
(E- P)Vrk
3o4Nk ka8E
48
OE
3Nk ka
l
_
xka fkj
k
(1.11)) and phonon drag Seebeck coefficient (Eq. (1.14)) that both of them will not be
changed if the electron scattering time is just changed by a single constant factor.
Therefore the results on p-type silicon can provide insights into the phonon drag effect in
p-type materials.
10
o
9
EPO
>
2.10
E
O
Experiment, n-type
Experiment, p-type
--
DFT, n-type
DFT, p-type
El
1013
102
10
102
Temperature (K)
Figure 2-1. Temperature dependence of the calculated intrinsic mobility in n-type and p-type silicon
compared with that of sufficiently pure samples from the experiment 1 . The intrinsic mobility is
calculated assuming a carrier concentration of 101 cm.
2.2.3. Phonon scattering by electrons
For heavily-doped samples, electron-phonon scattering of phonons also needs to be
considered for calculating the phonon relaxation time, described by the second term in
1
-
the second line of Eq.
Zra(k)
1
= ,+
r (k)
1
1
1
ra,(q)
4 (q)
2Fak,
fqA)
k'fi,qA
(1.10). Here
+ 1 Gq,,(ka,k'/#)
ka,k-6
we also want to rewrite its expression in a more explicit way:
1
1 rPh (q)
22;r
h
j
dkX~
dk Ig a(k,q)
2
f_
X (fak
25
fk+q )()qA
+Eak -Eg.)
(2.5)
a# Bz fBz
where the crystal momentum conservation has been taken into account by changing k'
49
to k + q and rearranging the terms. This scattering has been found to account for some
fraction of the reduction of the thermal conductivity in heavily-doped silicon 9 . We will
show later that it is also the major cause of the reduction of the phonon drag at higher
doping concentrations.
Different from calculating the electron relaxation time due to EPI, here we will use the
tetrahedral integration method to deal with the energy conservation for the scattering of
phonons by electrons. The tetrahedral method linearly interpolates the numbers on the
discrete mesh to the continuous reciprocal space, allowing analytic evaluation of the
integration involving delta functions (which describes a surface in a 3D space). It has
advantage in that the only parameter is the mesh density and the convergence is easily
checked, while for the smearing method one must be careful to simultaneously increase
the mesh density and decrease the broadening parameter. However, the tetrahedral
method can be hard to achieve the convergence when the integrand has large variation in
the space. Numerically we find that even for a lOOx100x100 q-mesh the electron
relaxation times are still not converged using the tetrahedral method. Therefore we
choose the smearing method for the calculation of the electron relaxation time. However,
for the calculation of the phonon relaxation time due to EPI, although Ek still changes
dramatically near the band edge (note that the conduction band of silicon has six pockets,
which are very anisotropic), the variable that actually plays the role in the energy
conservation is the different between two electrons states Ek - E1k
q changes more smoothly (for an anisotropic band, Ek -
Eaq
, which for a given
for a given q turns
out to be a linear function). Therefore the tetrahedral function leads to better convergence
in this case. We use the tetrahedral functions whenever the convergence can be achieved
within the accessible range of the mesh density, because it has only one tuning parameter
and the convergence is more easily checked. The integration to obtain the phonon drag
Seebeck coefficient also uses the tetrahedral method.
50
2.3.
Phonon-phonon interaction
In most range of the temperature, the dominant scatterings for phonons (and thus the
thermal conduction) are determined by the phonon-phonon interaction. The phonon
eigenstates are obtained by diagonalizing the dynamical matrix, which is derived based
on (harmonic) force constants. This implies that the total energy of the system can be
expressed as a quadratic function of the atom displacements. However, because materials
are never perfect harmonic crystals, there are also higher order force constants. For
example, third order force constant describes the force acting on one atom due to the
movement of two other atoms (these atoms can be the same one or different ones). In
general the higher order force constants are referred as anharmonic force constants,
because they give rise to the correction to the total energy which is non-quadratic with
respect to the atom displacements. The anharmonicity therefore serves a perturbation to
the harmonic system, allowing transitions of phonons between different states. Similar as
before, phonon phonon interaction matrix element can be defined, which can be
calculated given the anharmonic force constants. The Fermi's golden rule then establishes
the phonon-phonon relaxation time based on these interaction matrix elements. The key
component in calculating the phonon phonon relaxation time is then the anharmonic force
constants.
Here only third order force constants will be discussed, which have been found to be
enough for achieving good comparison of the thermal conductivity with the experiment.
The method we use is based on a real-space fitting approach and more information can be
found in the literature 4,4.
Firstly, the symmetry of the material (silicon in our case) is
used to determine a minimal set of third-order force constants, which only include the
interactions between atoms within certain neighbor shell of the origin. Then we use DFT
to calculate forces acting on different atoms with different atom
displacement
configurations in a large supercell (2 x 2 x 2 conventional unit cells, 64 atoms). For
harmonic force constants, each atom along each direction is moved from the equilibrium
51
by small amounts. For anharmonic
force constants,
we consider random atom
displacements in the supercell. Combining all the force-displacement data, the harmonic
and third-order anharmonic force constants are fitted together, with imposed translational
and rotational invariances7.
Figure 2-2 shows the resulting thermal conductivity with respect to temperature as a
test to the phonon relaxation times we obtain. Excellent agreement has been achieved
between our simulation and the experimental data. We note here that previous DFT
calculations
have obtained similar agreement
with experiments
for the thermal
conductivity4042 in silicon.
3500
0
-
-3000
Experiment
DFT
-
2500
'5 20001500
0
in 1000
E
-
jE5W
0
40
80
120
160
200
240
280
320
Temperature (K)
Figure 2-2. Temperature dependence of the thermal conductivity of pure silicon compared with the
experiment. The calculation is performed on a 70x70x70q-mesh.
2.4.
Impurity scattering
For heavily-doped samples, impurity scattering must be taken into account. Due to the
lack of accurate and computationally feasible methods for calculating the impurity
scattering, the effects from impurities are described using empirical models - the
Brooks-Herring model for electron-impurity scattering '74 and the Tamura model for
phonon-impurity scattering75.
We limit the temperature range to 100K-300K for
52
heavily-doped silicon, where the electron-impurity scattering has a small influence on the
phonon drag effect. This small influence lies in the fact that the Seebeck coefficient
essentially represents the ratio of the temperature-gradient
induced current to the
electric-field driven current. When the electron relaxation time is reduced, both of them
are weakened and therefore the ratio between them is less affected.
The Tamura model7 5 is used to examine the effect of the ionized impurity scattering on
the phonon drag effect, where the mass difference ratio AM /
is chosen to be 1 to
represent both the mass disorder and strain effect. We found from our calculation that
phonon-ionized impurity scattering has small influences on the phonon drag effect. As a
consequence, the use of empirical models in our calculation will not significantly affect
the evaluation of the phonon drag.
2.5. Phonon drag modeling
Knowing all the quantities in Eq. (1.14), phonon drag Seebeck coefficient can be
calculated on an equal electron and phonon Brillouin zone mesh, for which the
tetrahedral integration method is implemented7 6 . We also note that the final electron
relaxation
time
rka
combines
both electron-phonon
scattering
as
well
as
the
electron-impurity scattering, according to Matthiessen's rule. Similarly, the phonon
relaxation
time
scattering
and
-rq,,
combines
phonon-impurity
both
phonon-phonon
scattering.
Table
2-1
scattering,
phonon-electron
summarizes
the
scattering
mechanisms considered in our work and how they are treated.
Table 2-1. Scattering mechanisms for electrons and phonons
Carrier
Scattering
Method
Electron-phonon
DFT
Electron-impurity
Brooks-Herring model74
Electron
53
Phonon
Phonon-phonon
DFT
Phonon-electron
DFT
Phonon-impurity
Tamura mode 75
To further speed up the calculation, based on the knowledge that only phonons that are
close to the zone center contribute to the phonon drag effect (which becomes clear in the
next chapter), we define a phonon wave vector cutoff, above which phonons will not be
considered for Eq. (1.14). This cutoff has been checked and the change of the result is
within 1% of the original value. Table 2-2 summarizes the parameters we used for all the
transport property calculations in this work.
Table 2-2. Parameters used in determining the electron and phonon relaxation times as well as in the
calculation of the phonon drag effect. The parameter " a " in the last column is the lattice constant of
silicon.
Quantities
required
k-mesh (electron)
q-mesh
(phonon)
Integration method
C)
(broadening
Electron
relaxation
.
time
Phonon-phonon
relaxation time
70' ~-M3
803
Gaussian
Gaussian
703 _100 3
Gaussian
(1cm')
Electron-phonon
scattering of
phonons
Phonon drag
effect
70'
703 _1001
703
703 _1003
Tetrahedra
Tetrahedra
parameter if any)
energy: 0.5eV
Energy / wavevector
(eycutoff
q) cutoff
0.5 eV
2
q:(0.2 -2)-aq:(0.2
54
1.0 eV
~ 2)
Nearest neighbor
2 nd
force constant
considered in the
force constant
(harmonic): 7 3 rd
force constant
fitting 41
(anharmonic): 1
55
Chapter 3. Simulation of phonon drag
To justify the formalism and our numerical implementation clarified in previous chapters,
we will first examine the temperature and carrier concentration dependence of the phonon
drag effect and compare with the experimental results. The phonon drag Seebeck
coefficient in a lightly-doped silicon is studied first, which we will refer as intrinsic
phonon drag (no effect of doping concentration comes in). Then we will discuss how the
increased carrier concentration reduces the magnitude of phonon drag. With the
information revealed by the calculation, we will provide detailed mode contribution to the
phonon drag and uncover several key features that will lead to the optimization of phonon
drag in heavily-doped samples in the next chapter.
3.1.
Intrinsic phonon drag
Here we first examine the intrinsic phonon drag contribution to the Seebeck coefficient
for electrons and holes in silicon. Here "intrinsic" means that the boundary scattering is
omitted and low doping levels are examined, implying that impurity scattering 6 as well as
the phonon scattering by electrons 59' 77 is negligible. Correspondingly, experimental data
with larger sample size and lower net doping concentration is chosen for comparison. As
shown in Figure 3-1, good agreement is obtained between the calculation results and the
experimental data' 2 from 300K down to 60K for electrons and to 80K for holes,
validating the applicability of Eq. (1.14). An extremely dense sampling mesh of
100x100x100 q-points in the phonon Brillouin zone is made, which is necessary for the
convergence at very low temperatures. It is seen that the diffusion contribution to the
Seebeck coefficient changes slowly with decreasing temperature. On the other hand, the
phonon drag part increases dramatically at low temperatures and is at least one order of
magnitude larger than the diffusion part below 80K. At room temperature, the phonon
drag still contributes to a sizable fraction of the total Seebeck coefficient, 30% for
56
electrons and 40% for holes. This has only been inferred previously based on theoretical
models 78 and quenched thermopower experiments60 . From the first-principles approach,
we clearly verify that for lightly-doped silicon, even though the phonon drag effect is
only dominant at very low temperature, it has influences across a wide range of
temperatures, extending to above room temperature.
0 Experiment
----Total Seebeck
.
Diffusion part
(a)
0
--
104
Phonon drag part.
1 Experiment
---- Total Seebeck
.--.. Diffusion part
-- "Phonon drag part
5
>
'.
.............................
3
50
(b)
100
250
200
150
Temperature (K)
50
300
100
250
200
150
Temperature (K)
300
Figure 3-1. Intrinsic phonon drag effect for (a) electrons and (b) holes in lightly-doped silicon. The open
circles and squares are taken from the experiment , with the corresponding net doping concentration of
2.8x1014cm- for electrons and 8.xO14cm- for holes, respectively. Lines are first principles results
assuming the same doping concentrations. Dotted lines represent the diffusion contribution to the
Seebeck coefficient while dash-and-dot lines represent the phonon drag contribution on a semilog plot
of the Seebeck coefficient with respect to the temperature. The phonon drag contribution increases
dramatically as the temperature decreases and converges to the total Seebeck coefficient, shown by the
dashed lines.
3.2.
Saturation effect
3.2.1. Reduction of phonon drag at high doping concentrations
To optimize zT, the carrier concentration is another common experimental variable and
usually sits around 1019 to 1021 cm-3 , for achieving higher electrical conductivity
9
. It was
experimentally observed that, higher carrier concentration causes a reduction of the
phonon drag effect - called the "saturation" effect by Herringio - which is beyond the
intrinsic phonon drag regime. In Figure 3-2(a), the Seebeck coefficient as a function of
57
carrier concentration in n-type silicon is shown. The intrinsic phonon drag Seebeck
coefficient is independent of the carrier concentration. This can be understood because
for lightly-doped samples the carrier concentration plays a role in Eq. (1.14) only via the
position of the Fermi level Ef , which affects the occupation numbers of available
electrons fA. and the electrical conductivity u through the same exponential term
exp(Ef / kBT) and therefore these factors cancel each other. As is clear in Figure 3-2(a),
the total Seebeck coefficient combining (black dotted curves in Figure 3-2) the diffusion
contribution and the intrinsic phonon drag agrees well with experiment, up to 1017 cm
3
doping concentration.
As the carrier concentration increases above 1017 cm-3 , an apparent discrepancy occurs
between the dotted curves in Figure 3-2(a) and the experiment. At higher carrier
concentrations, several scattering processes start to change the transport properties.
Impurity scattering of electrons can largely decrease the mobility but is found from our
calculation to have only a small influence on the phonon drag effect (see section 2.4). It is
the scattering of phonons that reduces the non-equilibrium degree of phonons, therefore
leading to a significant reduction of the phonon drag effect, as shown in Figure 3-2(a).
Our calculation suggests that phonon scattering by electrons 59' 77 is the major reason for
this saturation effect (discussion in the next section), which leads to a further decrease of
the total Seebeck coefficient at high carrier concentrations. After taking into account the
scattering of phonons by electrons, the reduction of phonon drag at high carrier
concentrations is well captured and agrees well with the experiment.
In Figure 3-2(b) we also show the doping concentration dependence of the Seebeck
coefficient for p-type silicon. Similarly, the diffusion part alone cannot explain the
experimentally observed values. With the phonon drag effect considered, the total
Seebeck coefficient agrees reasonably well with the experiments (take into account the
scatterings of phonons by electrons). The small discrepancies which increase at lower
58
temperature in the low doping concentration range could be a result of the spin-orbit split
bands at the valence band edge. This effect is not included in this calculation because the
3000
1
..
----------
-
150-
....
4.
*%
4000
----------
0
44
1000
500.
500
C 14
10
1
2000
drag part
1500-Phonon
1500
0
2500
%
2000
1000
Experiment 200K
Experiment 300K
Total Seebeck
--Diffusion
part
Phonon drag part
13
Experiment 200K
0 Experiment 300K
-- Total Seebeck
Diffusion part
2500
-
spin-orbit coupling has been omitted.
- - 1
10
- -
6
- - 1
-
-
6
- - -
10
10
10
10
Doping concentration (cm-)
0
- - 1014 15i 1617
18192
10
10
-
-
10
- --
10
-.
10
-
-
10
- ~
10
10
Doping concentration (cm-)
Figure 3-2. Calculated Seebeck coefficient with respect to doping concentrations for (a) n-type silicon
and (b) p-type silicon at 300K and 200K on a semilog plot. The solid lines describe the calculated results
12
at 300K while dashed lines represent 200K. Circles and squares are taken from the experiment . At
each temperature the total Seebeck coefficient (black) as well as the decomposition into the phonon drag
part (red) and diffusion part (green) is shown. Dotted lines are the total Seebeck coefficient calculated
using the low doping level value of the phonon drag contribution and assuming this value will not be
reduced as the doping concentration increases, therefore neglecting the "saturation" effect. After taking
into account electron scattering of phonons, the total Seebeck coefficient (black solid lines for 300K and
black dashed lines for 200K) agrees with experiments across the full range of the doping concentration.
In spite of the saturation effect, phonon drag at high carrier concentrations cannot be
3
simply ignored. As is seen in Figure 3-2, at 1019 cm- doping concentration, the phonon
drag contribution to the Seebeck coefficient is comparable to the diffusion contribution
for both n-type and p-type silicon. This indicates that, in heavily-doped silicon samples,
phonon drag can still play an important role in the Seebeck coefficient, which will be
underestimated if phonon drag is totally neglected. This is contrary to what is generally
believed (high doping concentration destroys the phonon drag) and crucial to our
utilization of the phonon drag for improving thermoelectric performance.
We also want to note that, for the saturation effect, only temperatures down to 200K
59
are examined. Results for 100K will be provided when analyzing the optimized doping
concentration but not here for the saturation effect. This is because at low doping
concentrations when the temperature is below a certain value (-100K for silicon from a
simplified dopant model), carriers will freeze out, which means that the carrier
concentration will drop below the doping concentration. In this case, most of the
electrons are trapped in the dopant positions and are no longer "free electrons" that will
contribute to the conduction. As a result, the mechanism of the scattering of these
electrons with phonons will therefore change. This is usually called phonon-bound
electron scattering and has been studied using theoretical models80 . However, due to a
lack of the suitable first-principle method, it is not yet possible to accurately capture this
phenomenon which becomes dominant below 100K. Therefore, results for the saturation
effect below 100K for low doping levels are not discussed in this paper. However, at
sufficiently high doping levels (normally above 2x 1019 cm-3 ) the Fermi level will
penetrate into the conduction band and carriers will not freeze out as the temperature is
reduced. Therefore for optimization, for which the doping concentration will be high, we
will still use our numerical framework to predict possible enhancement of thermoelectric
performance by carefully optimizing the phonon drag effect.
3.2.2. Cause of saturation effect
As we mentioned above, it is the scattering of phonons that reduces the phonon drag
effect. There are two sources of this scattering as doping concentration increases. One is
the impurity scattering of phonons and the other is the electron scattering of phonons. We
found from our calculation that the phonon-impurity scattering is not strong enough to
provide the observed reduction of phonon drag above 10" cm-3 doping concentration. In
comparison, the electron scattering of phonons leads to a significant reduction of the
phonon drag effect. Here we want to understand why it is the electron scattering instead
of the impurity scattering that plays the major role in reducing phonon drag. This
60
difference stems from the scaling behavior of their scattering rates. As we will show
below (section 3.3.1), phonons that contribute to the phonon drag effect mostly have long
wavelengths (low frequencies). For long-wavelength phonons, the phonon-impurity
scattering"l (scattering rate ~
C4)
drops much faster with phonon frequency than the
59
phonon-electron scattering (scattering rate
-
c) and therefore the phonon-electron
scattering is much stronger in decreasing the non-equilibrium degree of phonons that are
significant for phonon drag. At sufficiently high doping concentration, we see from
Figure 3-3 that the phonon-electron scattering will eventually dominate over the
phonon-phonon scattering (scattering rate
(02) and
-
as a result, the phonon drag effect is
reduced as doping concentration increases (we note that the phonon-impurity scattering is
small compared to the other two at low frequency range and therefore is not shown in the
plot).
10i
-
10
-
-
by phonons
10~
D-by elecrons
01
1
Phonon frequency (THz)
10
Figure 3-3. Phonon scattering rates due to phonon-phonon interaction (red points) and electron-phonon
3
interaction (blue points). The calculation is carried out for the n-type silicon with 1019 cm- doping
concentration at 300K. The dashed curves show the theoretical scaling behavior of the scattering rates at
long wavelength (low frequency) limit. We note that the phonon-electron scattering rate does not
clearly follow the linear frequency trend. One possible reason is that the phonon frequency we
examine here is still large for the frequency scaling to be well satisfied. However, the crossover
between phonon-phonon scattering and phonon-electron scattering can be seen, which explains why
low frequency phonons get more scatterings due to increased number of electrons.
61
3.3.
Mode contribution to phonon drag
Having discussed the dependence of the phonon drag effect on the temperature and the
carrier concentration and justified our calculation by comparing with experiments, we
proceed to quantify the mode contribution to the phonon drag effect, which is the key
information for advancing our understanding of phonon drag. We will reveal that
phonons that contribute to phonon drag and to the thermal conduction do not spectrally
overlap, which leads to designs of phonon filters that can maintain phonon drag while
reducing the thermal conductivity. Combined with significant phonon drag at high doping
concentration discovered in the last section, this "decoupling" idea will show an
enhanced thermoelectric efficiency for lower temperatures in the next chapter.
3.3.1. Phonon mode contribution
In Figure 3-4(a) we show the accumulated percentage contributions to the phonon drag
Seebeck coefficient and to the thermal conductivity from each phonon mode with respect
to their frequencies from lOOK to 300K. It can be seen that compared with the modes
contributing to the thermal conductivity, the specific phonons that are significant in the
phonon drag effect have lower frequencies, indicating that they are closer to the zone
center and also possess longer wavelengths (Figure 3-4(b)). This is further confirmed in
Figure 3-4(c), which shows that phonons involved in the phonon drag processes have
significantly longer mean free path than those that carry heat.
These features were understood previously from simplified theoretical models'
82 83
,
Here with the full knowledge of the spectral contribution, we can take advantage of this
information to quantitatively determine how important each phonon mode is in
contributing to phonon drag. In order to enhance zT, the factor S 2 / /
needs to be
maximized. Provided that the phonon drag contribution is non-negligible in the total
Seebeck coefficient, one can ask whether we can reduce the thermal conductivity without
sacrificing the Seebeck coefficient much. According to Figure 3-4(c), one can achieve
62
this by designing a mean free path selective phonon filter. For example, at 300K phonons
with mean free paths smaller than 1 gm contribute around 70% to the total thermal
conductivity while contributing negligibly to the phonon drag effect, implying that the
thermal conductivity can be reduced by 70% without changing the Seebeck coefficient
much by "filtering out" these phonon modes. At lower temperatures, the accumulated
contribution to the phonon drag effect has a larger shift towards the long mean free path
region compared with the contribution to the thermal conductivity. Therefore this
"decoupling" strategy becomes even more effective at lower temperatures.
(b)
(a)
0.8 -
Thermal conductivity
Phonon drag (electron$
-
Phonon drag (hole)
-- -----
..
..
0.
.0
.
0
-
tX0.8
0
0.2
0.
C
&0.4
~0.2
-J
Thermal conductivity
Phonon drag (electron)
-
30.2
(C 04
E
Phonon drag (hole)
2
1
0.5
0.2
Phonon frequency (THz)
1
5
-----.
--
conductivity
-Thermal
0.8
--
0
10
,-
0.1
20
15
10
Wavelength (nm)
5
25
--
,---
Phonon drag (electron)
Phonon drag (hole)
I0.6
j
30.2-
~0I
4
-
'
le
1014 10
0
1 10"
le
10j
10
Mean free path (pn)
102
10
10 3e
Figure 3-4. Phonon mode-specific accumulated contributions to the phonon drag Seebeck coefficient
and the thermal conductivity with respect to (a) phonon frequency, (b) phonon wavelength and (c)
phonon mean free path. Solid lines show the contribution at 300K, while dashed lines are used at 200K
and dotted lines at 100K. Green curves show results for the thermal conductivity. Red and blue curves
represent n-type and p-type silicon, respectively. These results are obtained for lightly-doped silicon.
The small differences between electrons and holes lie in the detailed band structures near the band edge.
63
3.3.2. Electron mode contribution
Here we show the mode-specific contributions to the phonon drag Seebeck coefficient
as well as the electrical conductivity and diffusive Seebeck coefficient from the electron
side. This provides us the knowledge of what portion of electrons contributes to these
transport properties and especially the phonon drag effect most notably. Eq. (1.14)
directly presents the phonon mode-specific contribution to the phonon drag (term inside
the bracket). For the electrons, we can similarly combine all the terms that are labeled
with the same electron wave vector k and band number a in the summation. The result
is given below for the lightly-doped n-type silicon with a doping concentration of 1014
cm-3
In general, we see in Figure 3-5 that at the same temperature the accumulated
contribution curves for the three physical quantities (electrical conductivity, diffusive
Seebeck coefficient and phonon drag Seebeck coefficient) almost overlap. This is because
the Fermi-Dirac distribution, which modifies the population of the electrons, changes
more strongly with the electron states compared to other properties such as scattering
rates, and essentially confines the electron states that are important for transport
properties to a small region near the band edge. Therefore
we see a general
monotonically-increasing accumulated contribution curve. To be more specific, we see in
Figure 3-5 that, when temperature decreases, the curves move towards the band edge.
This is a result of the temperature characteristics of Fermi-Dirac distribution function,
which decreases more rapidly with energy as temperature decreases. As a result, the
electrons that participate in the transport are more confined to the band edge at lower
temperatures. At 300K, most electron states that are important for the transport properties
are located within 0.2eV from the band edge, as shown in Figure 3-5(a). Because
electrons are confined to the band edge, which is made up of six equivalent electron
pockets, and the conduction band minimum corresponds to a wavelength of 0.67nm,
significant contributions to the transport properties should come from electrons with
64
wavelength around 0.67nm. In Figure 3-5(b), indeed we see that most of the contributions
come from electrons with wavelength between 0.6nm and 0.7nm. At lower temperatures,
the curve becomes slightly more narrow, and the reason is the same as before: electrons
become more confined to the band edge and the reciprocal space they occupy then
shrinks. In terms of the electron mean free path, Figure 3-5(c) indicates that, electrons at
300K have mean free paths between 20nm and 80nm. The mean free path increases as
temperature decreases, and at lOOK the majority of electrons have mean free paths
around 100nm-300nm. We should note that these plots are obtained for a lightly-doped
silicon. For the heavily-doped silicon, the characteristics of the energy-dependence 57 and
wavelength-dependence will remain the same, meaning that electrons involved in the
transport process are still confined within -0.2eV
from the band edge and have
wavelengths around 0.6nm-0.7nm, because the qualitative argument given above does
not change. In comparison, mean free paths of electrons will decrease due to the impurity
scattering, and therefore the mean free path accumulated curve will move towards the left.
For example, it was known that for the n-type silicon with 1019 cm-3 doping concentration,
the electrons have mean free paths below 20nm within the temperature range of
.
100K-300K5 7
3.3.3. Effect of normal scattering and Umklapp scattering
We have shown that phonons that participate in the phonon drag effect are closer to the
zone center and therefore are characterized by small wave vectors. Besides, the electrons
that contribute to the transport are located near the band edge (because the Fermi-Dirac
distribution function drops fast as a function of electron energy when the electron energy
is far from the band edge). These features can be clearly seen in Figure 3-6. For p-type
silicon, only normal scattering will be important because the hole pockets are located at
zone center and to fulfill the momentum conservation the phonon will only have a very
small wave vector. For n-type silicon, we have six equivalent electron pockets (valleys)
65
but located in different directions from the zone center. The intravalley scattering, which
(b)
(a)
0
0
0.8-
.0
C
0
0. 6-
0.8
0.6
0)0
C0.
4Fa 0.
CO.
1-
0.
E
2
.
7.3
7.35
Electrical conductivity
Diffusive Seebeck
Phonon drag Seebeck
-
7.4
Band energy (eV)
0
.0
7.45
E
a
4
7.5
0.2 6
-
0.5
Electrical onductivityDiffusive S sebeck
Phonondnig Seebeck
0.6
0.7
Electron wavelength (nm)
0.8
Electrical conductivity
-Diffusive
Seebeck
0.8 -
~0.
Phonon drag Seebeck
(c)
CL0.4
ES0.2
a
10
20
50
100
200
Electron mean free path (nm)
Figure 3-5. Accumulated contribution to electrical conductivity, diffusive Seebeck coefficient and
phonon drag Seebeck coefficient, with respect to (a) electron band energy, (b) electron wavelength and
(c) electron mean free path. Physical quantities are labeled with different colors (blue for electrical
conductivity, green for diffusive Seebeck coefficient and red for phonon drag Seebeck coefficient). In all
three plots, solid curves describe results at 300K, while dashed curves represent 200K and dotted lines
100K. The results are calculated assuming a doping concentration of 1014 cm 3 . In part (a), the band
energy is chosen with some arbitrary reference and the conduction band edge is denoted by 7.3eV.
occurs only between electrons that are in the same pocket, involves phonons with small
wave vectors. Umklapp scattering only occurs for intervalley scattering, where one
electron is scattered from one valley to another. However, in such cases, phonons will
have large wave vectors and therefore higher frequency, which leads to lower mean free
path and smaller occupation number (see Eq. (1.14)). Therefore, the contributions from
the intervalley scattering will be small and the major electron-phonon coupling involved
66
in the phonon drag effect is of normal type (intravalley scattering).
(b)
(a)
00/
\ -V/
8
0
25
0
15
5
50
0
tX .11 ,7N\ .77
5
50
5
5
5
55
L
G
X
G
A
60
L
G
X
G
(c)
16
-15
14
912
-20
10
......
56
.
C
0
2
9%
IG
G
x
L
Figure 3-6. Electron mode contribution to the phonon drag effect for (a) n-type silicon and (b) p-type
silicon, as well as (c) the phonon mode contribution to the phonon drag effect for n-type silicon. The
phonon mode contribution to the phonon drag effect for p-type silicon is similar to that of n-type silicon
and therefore is not shown here. The color bars show the log scale of the contribution to the phonon drag
Seebeck coefficient. For electrons it is log (IS,, (k, a) [V/K]) where SP,, (k, a) is obtained from
Eq. (2) by grouping the terms which correspond to the same electron state Yka , while for phonons it is
log (IS, (q,2) [V/K]I) where
SPh (q, 2)
is obtained by choosing one phonon mode in Eq. (2).
Red colors show larger contributions compared to yellow and green colors, while blue colors indicate
negligible contributions. All of the plots are calculated on a 70 x 70 x 70 mesh with 1014 cm-3 carrier
concentration at 300K.
67
Chapter 4. Optimization of phonon drag
Having shown that phonons contributing most to the phonon drag have much longer
wavelengths than those that carry heat and that significant phonon drag contributes to the
Seebeck coefficient even at high doping concentration up to the room temperature (300K),
we now proceed to quantify how one can benefit from utilizing the phonon drag effect.
We have uncovered that, although phonons play roles in both the phonon drag and
thermal conduction, different phonons have different contributions. In order to utilize the
phonon drag, one seeks to reduce the thermal conductivity with minimum influences on
the phonon drag (thus Seebeck coefficient), by taking advantage of the difference in their
spectral contributions and selectively filtering out phonon modes. In this light, we will
first identify "preferable" phonon modes, defined as those contributing more to the
phonon drag than the thermal conductivity. Ideal phonon filters as well as more practical
filtering mechanisms will then be discussed, which show that the thermoelectric
performance in silicon at low temperatures can be greatly enhanced by reducing the
thermal conductivity while maintaining significant phonon drag.
4.1.
Preferable phonon modes for phonon drag
Each phonon mode labeled by wave vector q and branch number A
makes a
contribution to phonon drag given by Eq. (1.14) and to the thermal conductivity given by
K(q, 2) =
3NV
quality factor
(1fk )
'q2
as the ratio between the contributions:
Sh
!
(ka Vka ~Tk'6 Vk'fl fka
h q2(8lnq / aT) . We can define a mode-specific phonon drag
q
C;,A
=
(
q
ShqA)
(q, A)
=
2e
22(4.1)
-NkkT2
(
ka,k73
Vq(an., / aT)
The material will have little use for thermoelectrics if the thermal conductivity is too high.
68
achieve when reducing the thermal conductivity), described as
I
i(q, A) ic.
,
Therefore we consider an upper bound for the thermal conductivity (as a goal we want to
(q,A)EC
where set C denotes the phonon modes that are involved and kr
is the upper bound
of the thermal conductivity. Given this constraint, the largest phonon drag contribution
that one can achieve is obtained by selecting those modes that have phonon drag quality
factors as large as possible, noting that
max
(4.2)
ax,*, =i-(q, A)
The quality factor defined above distinguishes the "preferable" phonon modes that are
more significant in phonon drag from those that are less important, and serves as the
criterion to select phonons if one seeks to maximize the phonon drag contribution.
In Figure 4-1 we show the distribution of the preferable phonon modes as a function of
wavelength and frequency, where it is clearly seen that more preferable modes typically
have longer wavelengths and lower frequencies. In general, designs of phonon filters will
require mechanisms that can provide strong scatterings for high frequency and short
wavelength phonons, while have minimum influences on low frequency and long
wavelength phonons.
4.1.1. Optimization of n-type silicon with ideal phonon filters
First we will consider the enhancement of zT using an ideal phonon filter, which means
that the optimal zT will be achieved by selecting those phonons that have larger figures of
merit (Eq. C;,2 -
S,,
2e
(q,A)
K(q,
q
A)
-
0
VqA q
0
(_r
aq
<(8ng
aN k T 2
P
21l
,8-fA
kakp
(4.1)), and then "filtering out" all other phonons. Modes with
4
/qT) T
,A from the largest to the
smallest are selected to maximize the phonon drag Seebeck coefficient, until an upper
69
bound for the thermal conductivity is reached. Figure 4-2(a) shows the largest possible
phonon drag Seebeck coefficient with different upper bounds for the thermal conductivity
in n-type silicon. In heavily-doped silicon, it was previously believed that
3.5
3
-10
S2.5
-12
j.5
0
-16
2.5*
00
CA0
0
2
1.5
1
0.5
Phonon wave vector (1/nm)
Figure 4-1. Distribution of preferable phonon modes in wave vector and phonon frequency. The lower
branch describes the transverse acoustic modes while the higher branch shows the longitudinal
acoustic modes (Note that they are plotted against the length of the phonon wave vector, therefore
projected onto this 2D plot). Data are obtained on a 70x70x70 mesh and only long-wavelength phonons
4
1
are shown. The color bars show the log scale of the phonon drag quality factor - ln( 4q,) ( ,, in
unit of m -V/W ). Red colors represent higher phonon drag quality factors
(;2 than blue colors. The
gradually changing background represents the impurity scattering exploited as a phonon frequency
selective mechanism, where low frequency phonons are less scattered.
the phonon drag effect is completely suppressed, especially when one also tries to reduce
the thermal conductivity. State-of-the-art material synthesis techniques have shown the
capability of reducing the room temperature thermal conductivity of silicon to below 4
W/mK'&.Our results show, however, even at such a low value of the thermal
3
conductivity in a 1019 cm- doped n-type silicon sample, there still can be a phonon drag
contribution that is about 25% of the diffusion contribution, if preferable modes are
chosen carefully. This reduced thermal conductivity and maximized phonon drag should
2
lead to enhancement of the factor S I C compared to the original bulk material. In
Figure 4-2(b), it is clearly seen that the phonon drag effect can be utilized to boost zT by
70
a factor of 20-30 if preferable phonon modes are carefully selected. Without the phonon
drag effect, the enhancement of the factor
S2 / K
is around 10, which does not change
too much between 200K and 300K. The reason is that the diffusion contribution to the
Seebeck coefficient decreases while the thermal conductivity actually increases as the
temperature decreases. If we assume the same reduced thermal conductivity, the
reduction of the thermal conductivity at lower temperature is larger and benefits the
S2
/
K
factor more. However, the Seebeck coefficient also decreases so that we do not
gain much from lowering the temperature. The situation is different if the phonon drag
effect is included. First, because the phonon drag magnitude is comparable to the
diffusion contribution, the inclusion of phonon drag with selective phonon modes helps to
boost
S2
/
(for instance by a factor of 2 at 300K). Besides, when the temperature is
lowered, the phonon drag effect becomes even more pronounced and therefore makes the
enhancement of
S2
be utilized to boost
even larger. These results show that the phonon drag effect can
/
S2
/
K
and therefore to enhance zT at lower temperatures as well.
Figure 4-3(a) compares the zT when selecting preferable modes with that when
neglecting the phonon drag effect, assuming that the thermal conductivity is reduced to 4
W/mK. The optimized zT for normal bulk silicon is -0.01 at 300K around 4x101 9 cm-3
doping concentration1 4 , which can be boosted to -0.25 by combining the optimized
phonon drag effect and the reduced thermal conductivity. Omission of the phonon drag
effect will diminish such enhancement by half. As we mentioned, the benefit from the
phonon drag becomes more pronounced, which is clearly seen in Figure 4-3(b), where we
fix the doping concentration to be 4x101 9 cm- 3 and vary the temperature. The zT
enhancement increases as the temperature decreases, reaching a value of 30 at 200K and
can be as large as 70 times at 100K. This striking result clearly shows the importance of
recognizing the different spectral distributions of phonons contributing to the phonon
71
drag effect and to thermal conduction.
(b)
(a)
500-
-
400 -
0
with Sph, 300K
30 - ---- 300K
with Sph, 200K
E25 - --- '200K
~300--------------E
201
;_
10 14 cm-3, 300K
10
-10 19 cm-3, 300K
100
.
.--
:32.2222
-10 19 cm-3, 200K
0
10
20
30
410
5
1
Thermal conductivity(W/(mK))
10
10
Doping concentration (cm-)
Figure 4-2. (a) Contribution of the most preferable modes to the phonon drag Seebeck coefficient at
2
different reduced thermal conductivity values and (b) the enhancement of the factor S /
K
as a
function of doping concentration when phonon modes are selectively scattered. In part (a), for
lightly-doped silicon, we see that the thermal conductivity can be reduced to 30 W/(mK) before
observing significant diminishment of the phonon drag effect. Dashed lines represent the diffusive
Seebeck coefficient for heavily-doped silicon at different temperatures. For heavily-doped silicon, the
phonon drag part is still non-negligible and becomes larger compared with the diffusion part when the
temperature is decreased. In part (b), solid lines represent the results where phonon drag is included and
preferable phonon modes are chosen, while for the dashed lines it is assumed that phonon drag is
neglected. The results examined in (b) assume that the thermal conductivity is reduced to 4W/(m- K)
for all the curves.
4.1.2. Optimization of p-type silicon with ideal phonon filters
We have also analyzed the optimization of phonon drag effect in p-type silicon, as
shown in Figure 4-4. Comparing results from n-type and p-type, we see that p-type
silicon has a slightly larger phonon drag contribution to the Seebeck coefficient compared
3
to n-type silicon. For example for 1019 cm- doped silicon at 300K, the n-type silicon
shows a phonon drag contribution about 230pV/K while p-type silicon shows a number
72
around 350ptV/K. For p-type silicon, the largest phonon drag Seebeck coefficient that
can be achieved drops below the diffusion part when the thermal conductivity is reduced
(a)
0.25
Phonon drag S + reduced
--
E
-*"'4x10
7
Reduced
19
cm-
\
S0.2 --- None
60
E 50
0.15
U040-
_
S0.1-
(b)
so
oC
1;130
20
0.05
10
.
0
1018
.......................
20
19
10
Doping concentration (cm-)
100
10
200
300
Temperature (K)
Figure 4-3. (a) The enhancement of zT compared to bulk crystal achieved by selecting preferable modes
at 300K for n-type silicon with respect to the doping concentration and (b) the zT enhancement at a
3
doping concentration of 4 x 1019 cm- as a function of the temperature. In calculating zT, the
experimental data is used for the electrical conductivity as a function of doping concentration for n-type
85
silicon , and we assume that the electrical conductivity stays the same.
to 20W/(m-K)
at 300K (10W/(m-K)
at 200K). At lower temperatures the phonon
drag effect becomes more pronounced, which is the same for both n-type and p-type
2
silicon. By carefully selecting preferable phonon modes, the ratio S 1
in p-type
silicon can be enhanced by a factor of 10-20 between 200K and 300K, which is a factor
of 2-3 larger than the case when the phonon drag effect is neglected, as shown in Figure
4-4(b). This enhancement is slightly smaller than that in the case of n-type silicon (Figure
4-2(b)). This can be understood if we compare Figure 4-2(a) with Figure 4-4(a), where
we see that in n-type silicon the curve representing the largest phonon drag Seebeck
coefficient drops significantly only after the thermal conductivity is largely reduced while
in p-type silicon phonon drag reduction is seen when thermal conductivity is still large. It
indicates that phonons that are involved in the phonon drag effect in p-type silicon are not
as close to the zone center as those in n-type silicon, which is a manifestation of the
73
different band structures for n-type and p-type silicon near the band edge.
(a)
600
0
500
20
--
with Sp, 300K
---
300K
with S
200K
200K
-
-- ------ - - ----- - - -
E
-
.-- -- - -
2400
(b)
'"0
700
300
-
14
200
-10
100
-
----
cm-3
cm-3
10
a-10
300K
co
300K-
1019en-3, 200K
u0
20
10
40
30
1020
le
1019
10181
50
19
Thermal conductivity (W/(mK))
Doping concentration (cm-)
Figure 4-4. (a) Contribution of preferable modes to the phonon drag Seebeck coefficient at different
reduced values of the thermal conductivity and (b) the enhancement of the factor S2 /
K
as doping
concentration for p-type silicon. In part (a), solid lines represent the phonon drag part of the Seebeck
coefficient while dashed curves describe the diffusion part. In part (b), solid lines represent the results
where phonon drag is included and preferable phonon modes are chosen while for dashed lines it is
assumed that phonon drag is neglected. The results examined in -part (b) assume that the thermal
conductivity is reduced to 4W/(m.- K).
4.2.
Other Phonon filters
While the discussions above set the upper bound for the enhancement of S2 /
K
by
examining the effect of an ideal phonon filter, in practice it is not easy to arbitrarily select
phonon
modes
Sh(q, A)
K(qA)
solely
2e
o-NkT 2
based
on
their
quality
kf"6)
vq,,no,,
q
(rkaVkark'#Vk,6)-f a
VqA(anfl
/ T)
factors
21-1
h
(Eq.
(4.1)).
Nevertheless, simple selective mechanisms can be devised based on one single variable,
such as frequency or wavelength. For example, nanoclusters can be used as impurities to
selectively scatter phonons with different frequencies. Impurity scattering is generally
74
stronger for phonons with higher frequencies and therefore serves as a low-pass filter.
4.2.1. Nanocluster scattering for frequency selectivity
We propose to make use of nanocluster scattering to effectively scatter high-frequency
phonons. Nanoclusters are clusters that have impurity atoms different from the host atoms
with sizes ranging from sub-nanometer to a few nanometers. One extreme case of the
nanocluster is a single impurity atom embedded in the host, for which the theoretical
model developed by Tamura can be used to estimate the phonon-impurity scattering rate
75.
For clusters that contain more than one impurity atom, there has been development of
first principles approach based on Green's function calculation8 6'8 7 , which can provide
more accurate results. For simplicity, we will not use such a rigorous method to describe
the nanocluster scattering. Instead, we use an analytical formula86 generalized from the
Tamura model for the description of the nanocluster scattering effect. In the Born
approximation, it can be shown that the phonon-nanocluster scattering rate is
Ti. (q,) --
12N
~-D
_
M
(coq)
(4.3)
F
D* (o
2)
61
6
q'A'
e*.(q', ') E,(qA) .IS,15 -fq(2 -C
L
where N is the total number of unit cells,
f
is the volume fraction of the nanoclusters,
AM is the mass difference of the impurity atom and the host atom, M is the average
mass of all the atoms,
%Ax
describes the phonon frequency and E,(q,A) is the unit
vector along the polarization of the atom labeled by o in the unit cell.
SAq
is the
structure factor with the sum includes all the unit cells occupied by one nanocluster.
D*(oq2)
can be regarded as a generalized phonon density of state. It can be shown that
when the nanocluster contains only one impurity atom (the structure factor is one in this
75
case), D*(Cq2,)
reduces to the normal phonon density of state, and as a result, Eq. (4.3)
is essentially the same as the Tamura model. The unit vectors along the polarizations of
the atoms as well as the phonon frequencies are obtained from first principle calculations.
The mass fraction term
AM
_
M
is chosen to be 1 to represent both mass disorder and force
constant disorder. This is a typical number for alloys. For example, if the host is silicon,
then the addition of germanium atoms act as impurities with a mass fraction of around
1.6.
We test nanocluster size up to 1nm (this is the equivalent diameter defined through the
total volume of the unit cells contained in the nanocluster). For the value of Inm, electron
wavelengths are comparable to the nanoclusters size (see section 3.3.2). We can use the
geometric limit to estimate the upper bound for the electron-nanocluster scattering, which
gives a corresponding mean free path of Aw,,,,c,
-
3
3f
~ 330nm , where
r is the
characteristic radius of the nanoparticle and chosen to be 0.5nm for estimation (volume
fraction
f
here is chosen to be 0.2%, which is the maximum value we set). For
nanoclusters that are smaller, the geometric limit becomes smaller. However, the
wavelengths are now large compared to the nanocluster sizes and enter into the Rayleigh
scattering regime, where the scattering rate falls below the geometric limit. Therefore
Amvwpaice
should be on the order of 300nm or even larger. From previous work on first
principles calculation of silicon57 we know that for the doping concentration of 1019 cm 3
the electron mean free paths are less than 20nm. The comparison with A
indicates that the dominant scatterings for electrons still come from the phonons and
dopants. Therefore the nanocluster scattering for electrons can be neglected and the
electrical conductivity is barely affected.
Combining
the
phonon-nanocluster
scattering
76
and the
normal phonon-phonon
scattering rates using the Matthiessen's rule, we can calculate the modified phonon
relaxation times and their effects on phonon drag Seebeck coefficient as well as the
thermal conductivity. The enhancement ratio of S 2 / ic and thus zT can then be obtained.
Figure 4-5 shows the effect of the nanocluster scattering on the enhancement of zT at
300K and 200K. In the volume fraction range we explored, the electrical conductivity is
barely affected. A large enhancement of zT can be seen, which is due to the fact that the
thermal conductivity is largely reduced while the Seebeck coefficient is less affected. An
optimal nanocluster size which is Inm in diameter with a volume fraction of 0.2% (Note
that we have confined the volume fraction to be within this range because the scattering
model is no longer appropriate as the volume fraction becomes very large), is found to
enhance zT by a factor of 5 at 300K, for which a significant portion of the Seebeck
coefficient comes from the phonon drag, and the neglect of the phonon drag effect will
compromise the enhancement. At lower temperatures, this benefit becomes even larger,
with the enhancement reaching 7 times at 200K using Inm nanocluster with a volume
fraction of 0.2%. Generally, at the same volume fraction, nanoclusters with larger size
scatter phonons more strongly. This fact is known for decades5 and is utilized here to
select low-frequency phonons. However, nanoclusters that are much larger will also
significantly reduce the phonon drag effect. This is the reason why we will see an optimal
choice for the size of the nanoclusters, which turns out to be around Inm for silicon at a
doping concentration of 1019 cm-3. This example using phonon frequency filtering again
emphasizes that the phonon drag effect is not negligible and can be engineered to enhance
zT even for heavily-doped silicon around and below room temperature.
We also want to briefly comment on the use of frequency selectivity in p-type silicon.
Similar to n-type silicon, the thermoelectric figure of merit zT in p-type silicon can also
be enhanced using a low-pass phonon frequency filter, as shown in Figure 4-6. Here we
only calculate the single impurity case to illustrate the concept. Separated impurities can
be viewed as an extreme case of the nanoclusters (one nanocluster only contains one
77
impurity atom). As mentioned above, in this case the formula we used to describe the
nanocluster scattering (Eq. (4.3)) essentially reduces to the Tamura model. From Figure
4-6, we see that the enhancement of zT is slightly smaller than that in n-type silicon
(compared with the red solid lines in Figure 4-5). The reason for this is that at the same
doping concentration, the effects of electron-phonon scattering on phonons are different.
It turns out that in p-type silicon the phonons are more strongly scattered, which leads to
a lower thermal conductivity in p-type than that in n-type at the same doping
concentration5 9. Therefore assuming the same reduced thermal conductivity implies that
the p-type silicon gains a smaller benefit. We note that this gain in zT is guaranteed
because the phonon drag Seebeck coefficient is less affected while the thermal
conductivity is largely reduced.
(b)
(a)
8
6
single impurity + phonon drag
7 -O-- 0.77nm + phonon drag
Inm + phonon drag
6 ,. single Impurity
-Ar- 0.77nm
-- A--1nm
0
-4- single Impurity + phonon drag
-0-- 0.77nm + phonon drag
5 -- O"1nm + phonon drag
-"-single impurity
* 4 -Ar0.77nm
L",-1nm
E
S2.0
0
2-#
0
0.00 .5002 0.0005
2
0.005%
0.002
0.02% 0.05%
0.2%
Volume fraction of nanoparticles
Volume fraction of nanoclusters
Figure 4-5. The enhancement of the thermoelectric figure of merit zT with respect to the volume
fraction of nanoclusters using phonon frequency selectivity for n-type silicon with a doping
concentration of 1019 cm-3 at (a) 300K and (b) 200K. Curves labeled with phonon drag include the
phonon drag contribution to the Seebeck coefficient, while others negelct the phonon drag effect. The
single impurity case is an extreme case of nanocluster scattering, where one nanocluster only contains
one impurity atom. For nanoclusters with more than one impurity atom, we calculate the equivalent
diameter corresponding to the total volume of unit cells contained in that nanocluster. We note that the
0.77nm size contains 6 unit cells, while the lnm size contains 14 unit cells.
78
19 cm,
-3'300K,
101
phonon drag
3
-"'1019cm , 200K, phonon drag
1.5 - -A. 19 cm-, 300K
-Ar 1 19 cm3, 200K
E
0.5-
-- - ~
-- - -- --0.2%
0.05%
0.02%
0.002%0.005%
Volume fraction of impurities
Figure 4-6. The enhancement of the thermoelectric figure of merit zT if single impurities are used to
scatter short-wavelength phonons preferentially. Dashed curves describe the results for which the
phonon drag effect is neglected.
79
Chapter 5. Conclusion
5.1.
Summary
In summary, the phonon drag effect is investigated in detail in this work to study the
possibility of enhancing the thermoelectric performance around and below 300K via
first-principles simulation. Although it is widely believed that the phonon drag effect is
only dominant at very low temperature and will be largely suppressed when the carrier
concentration becomes higher and especially after the thermal conductivity is reduced,
we clearly show that even for silicon with a 1019 cm-3 carrier concentration at room
temperature, the phonon drag contribution to the Seebeck coefficient is still appreciable,
the neglect of which leads to a significant drop in the enhancement of zT. The benefit of
this aspect of the phonon drag effect becomes greater when the temperature is decreased.
Moreover, we have proposed filtering mechanisms based on the identification of
preferable phonon modes. An ideal upper bound as well as more feasible phonon filtering
approaches (including nanocluster scattering as a frequency selective mechanism) are
discussed, which show the enhancement of zT coming from the resulting reduced thermal
conductivity and optimized phonon drag effect.
5.2.
Future work
The computational approach we developed to solve the electron-phonon coupled
transport for the phonon drag effect should be generally applicable beyond silicon. We
envisage that along this path more material systems can be systematically studied with
quantitative understanding of their coupled electron-phonon transport. Furthermore, other
filtering mechanisms can be designed and engineered to better select the preferable
phonon modes, thereby bringing benefit to thermoelectric applications, particularly at
lower temperatures.
80
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