A deeper insight into the fundamental issue concerning the possible existence of a Z T upper limit value (see Section 2.1) can only be gained by explicitly considering TE effects at a microscopic level, as we did in Section 1.2.2. Indeed, the transport properties of a given material greatly depend on the scattering processes determined by the mutual interactions between the carriers and the lattice, as well as by the presence of impurities and other structural defects, which often vary themselves with both temperature and external force fields. In order to properly include the physical processes determining the temperature dependence of the transport coef- ficients appearing in Eq. (2.4), it is convenient to introduce a characteristic function, which entails detailed information on the electronic structure of the material. This function is referred to as thespectral conductivity function,σ(E)>0 (measured in−1cm−1 units), which is defined as theT → 0 conductivity with the Fermi level at energyE.
For systems which can be described within the Boltzmann approach, one has
σ(E)=e2τ(E)n(E)v2(E), (2.8) whereτ(E) is the relaxation time,n(E) measures the charge carriers density (in cm−3), andv(E) is the group velocity of the carriers. On the other hand, in the case of systems for which the applicability
of the Boltzmann approach is not guaranteed, one can consider the more general relationship
σ(E)=e2
V N(E)D(E), (2.9)
whereN(E) is the DOS (measured in states (eV)−1per atom),D(E) measures the diffusivity of the states (in cm2s−1), and V is the volume of the system.
From the knowledge of the spectral conductivity function, one can derive the kinetic coefficients,a
Li j(T)=(−1)i+j +∞
−∞ σ(E) (E −μ)i+j−2
−∂f
∂E
d E, (2.10) wherei, j=1, 2,μ(T) is the chemical potential (which equates the Fermi energy,EF, atT =0), and
f(E,T)= 1
1+e(E−μ)β, (2.11)
is the Fermi–Dirac distribution function, whereβ≡(kBT)−1. In this formulation, all the microscopic details of the system are included in the spectral conductivity functionσ(E). Therefore, the temperature dependence of the transport coefficients appears in the Fermi–Dirac distribution, whereas all peculiarities of the scattering processes are incorporated in σ(E). The kinetic coefficients derived from Eq. (2.10) are valid for both extended and localized states. In fact, as we will see below, this formalism has been applied to study the TE properties of disordered systems undergoing a disorder-driven metal–insulator transition [97], as well as to study transport phe- nomena in quasicrystals and related approximants characterized by critical wavefunctions exhibiting spatial fluctuations at all scales (see Chapter 5).
The macroscopic transport coefficients introduced in Section 1.1 are related to the kinetic coefficients given by Eq. (2.10) through the
aThis approach is known as the Chester–Thellung–Kubo–Greenwood formulation.
[14, 16] This description is valid provided that the charge carriers are noninteracting among them and the scattering with impurities and lattice phonons is elastic. No assumption is made about the strength of disorder and the nature of the states.
Figure 2.3 (a) Diagram showing the spectral conductivity function for the Mott model, (b) temperature dependence of the electrical conductivity for a system undergoing a metal–insulator transition. More details in the text
expressions [16],
σ(T)=L11(T), (2.12)
S(T)= 1
|e|T
L12(T)
L11(T), (2.13)
κe(T)= 1 e2T
L22(T)−L212(T) L11(T)
. (2.14)
For the sake of illustration, let us consider the spectral conduc- tivity function describing an insulating-metal transition which takes place in certain disordered materials whose electronic structure can be described in terms of the so-called Mott spectral model. The corresponding spectral conductivity function is sketched in Fig. 2.3a and consists of two separate regions given by the step function
σ(E)=
0, if E <EC σM, ifE ≥ EC
(2.15) where σM is a constant and the energy EC is referred to as the mobility edgeseparating the region where all wave functions are localized (i.e.,E <EC) from that having extended wave functions in the energy spectrum (i.e.,E ≥ EC). Accordingly, an insulator–metal transition is expected to occur when the Fermi level crosses from the insulating to the conducting region.
The existence of this phase transition can be readily confirmed by plugging Eq. (2.15) into Eqs. (2.10) and (2.12) to obtain
σ(T)=σM
+∞
EC
−∂f
∂E
d E =σM
f(EC)− lim
E→+∞
1 1+e(E−μ)β
= σM
1+e(EC−μ)β. (2.16)
Thus, depending on the relative positions of the chemical potential and the mobility edge, three possibleσ(T) curves can be observed in the system, as it is illustrated inFig. 2.3b. In the case μ <EC, the electrical conductivity vanishes in the limit T → 0 (insulator) and it progressively increases as the temperature increases due to a thermally activated transport involving hopping of charge carriers among localized states. On the contrary, in the caseμ > EC one getsσ(T → 0)=σM(residual conductivity) and the electrical conductivity progressively decreases asT increases (metallic behavior). Both the thermally activated and the metallic transport curves asymptotically approach the valueσ(T → ∞) = σM/2 in the high temperature limit, which corresponds to the particular case where the chemical potential is exactly located at the mobility edge (i.e.,μ=EC).
Let us now consider the Seebeck coefficient temperature dependence for this spectral conductivity model. According to Eq.
(2.13), we must calculate the kinetic coefficientL12(T) making use of Eqs. (2.10) and (2.15). In doing so, after an integration by parts, we geta
L12(T)= −σM(EC−μ)f(EC)−σM
+∞
EC
f(E)d E. (2.17) To perform the integral appearing in Eq. (2.17), it is convenient to introduce the dimensionless scaled energy variablex ≡(E−μ)β and to express the Fermi–Dirac distribution function in the form
f(x)= 1 1+ex = 1
2 1−tanhx 2
, (2.18)
aMaking use of the L’H ˆopital rule we have lim
E→∞
E−μ
1+e(E−μ)β = lim
E→∞
1 βe(E−μ)β =0.
so that,
+∞
EC
f(E)d E = β−1 2
+∞
xC
1−tanhx 2
dx
=β−1
ln 2 coshxC 2
−xC 2
, (2.19) where, in order to evaluate the improper integral limit, we take into account the relation limx→+∞ln cosh(x/2) x/2−ln 2. Making use of Eq. (2.19) in Eq. (2.17) and plugging the resultingL12(T) coefficient along with Eq. (2.16) into Eq. (2.13), we finally obtain
S(T)= −kB
|e|
xC+(1+exC)
ln 2 coshxC 2
− xC 2
. (2.20) As it occurred for the electrical conductivity, the temperature dependence of the Seebeck coefficient strongly depends on the relative positions of the chemical potential and the mobility edge.
In the particular caseμ= EC(i.e.,xC =0), we have a temperature independent Seebeck coefficient valueS(T) ≡ SM = −kBln 4/|e|
−121μVK−1, which defines a model reference value. Note that this value is close to the proposed lower limit value|S∗| 130 μVK−1yielding an optimal power factor (see Section 2.2.2). On the other hand, in the case μ <EC (thermally activated regime) the Seebeck coefficient diverges to−∞in the limitT → 0 (i.e.,xC → +∞), whereas it asymptotically approaches the value S = SM as the temperature is progressively increased. In a similar way, in the caseμ > EC (metallic regime) one gets S(T → 0) = 0 and S(T → ∞)=SM(Exercise 2.5).
In the description of realistic systems, one generally finds rather involved spectral conductivity functions, naturally leading to difficult to solve integrals when obtaining the transport coefficients.
In order to simplify the mathematical treatment, it is convenient to use Eq. (2.18) in order to express the Fermi distribution derivative in terms of hyperbolic functions as
−∂f
∂E = β
4 sech2 x 2
. (2.21)
This is an even function with respect to the variablexexhibiting a narrow, pronounced peak around the chemical potential value (x= 0). Making use of Eq. (2.21), one can then rewrite Eq. (2.10) in terms of the scaled variablexto obtainLi j = (−1)nβ−nJn/4, wheren ≡
i + j −2, and we have introduced thereducedkinetic coefficients (measured in−1cm−1units),
Jn(β)= ∞
−∞xnσ(x,β) sech2(x/2)dx. (2.22) According to Eq. (2.22), J0 ≥ 0 andJ2 ≥0, whereas the sign of J1will depend on the chemical potential relative position, and allows for both n-type (μ < EF) and p-type (μ > EF) behaviors. Making use of Eq. (2.22), one can express Eqs. (2.12)–(2.14) in the form
σ(T)= J0
4, (2.23)
S(T)= −kB
|e| J1
J0, (2.24)
κe(T)= kB
2e 2
T J0
J0 J1
J1 J2
, (2.25)
and, for the sake of completeness, one can also express the Lorenz function in the form
L(T)≡ κe
Tσ = kB
e 2
J0−2 J0 J1
J1 J2
. (2.26)
Some general conclusions about the importance of the spectral conductivity functionparity on the resulting TE performance can be drawn from Eqs. (2.22)–(2.24). To this end, without loss of generality, let us expressσ(x)≡ σ+(x)+σ−(x), whereσ+(x) is an even function in the variablex(i.e.,σ+(−x) = σ+(x)) andσ−(x) is an odd one (i.e.,σ−(−x)= −σ−(x)). By pluggingσ(x) into Eq. (2.22) and keeping into account that sech2(x/2) is an even function, we realize that the electrical conductivity (see Eq. (2.23)) is completely determined by the even component of the spectral conductivity, whereas the Seebeck coefficient (see Eq. (2.24)) depends on both the even (denominator) and the odd (numerator) components. Thus, if σ(x) is an odd function (i.e., σ+(x) = 0) we have J0 = 0, therebyσ(T)= 0, and Z T =0. On the other hand, if the spectral conductivity is an even function (i.e.,σ−(x) = 0), we have J1 = 0 (with J0 = 0), thereby S(T) = 0 and the FOM vanishes as well.
Therefore, we conclude that the spectral conductivity function of promising TEMs must lack any definite parity. Thus, we realize that
materials exhibiting asymmetrically shaped electronic structures must be the preference choice in the search for novel TEMs [98–
100]. This fundamental conclusion nicely illustrates the useful guide provided by the use of the mathematically inspired spectral conductivity function. We also see that the Mott’s spectral model can be regarded as a suitable instance of a spectral conductivity function having no definite parity and exhibiting good TE properties.
As another illustrative example, we can consider the model describing the metal–insulator transition referred to asAnderson transition. This transition is related to the wave functions localiza- tion due to the presence of disorder in the sample and typically leads to significantly large Seebeck coefficient values once the transition takes place. On this basis, the idea of exploiting this effect for TE cooling has been discussed in the recent literature [97]. In analogy with the previously considered Mott model, the Anderson transition is also described in terms of a spectral conductivity function given by the step function
σ(E)=
0, if E <EC
σA|E −EC|ν, if E ≥ EC (2.27) whereσAis a constant,ECgives the mobility edge location, andυ >
0 is a universal critical exponent [101]. We see that forν = 0 this step function is now continuous atE = EC, whereas forυ =0 Eq.
(2.27) reduces to the Mott’s spectral conductivity model given by Eq.
(2.15). Expressing Eq. (2.27) in terms of the scaled variablex, and plugging it into Eq. (2.22) we obtain
Jn(β)=σAβ−υ ∞
xC
xn|x−xC|ν sech2(x/2)dx. (2.28) Experimentally one can usually approach the Anderson transi- tion from the metallic side.aThus, we will consider the limitμ→EC, withμ > EC (i.e.,xC → 0−) in Eq. (2.28), which takes the form Jn(β)=σAβ−υJ˜m, where we have introduced the integrals
J˜m≡ ∞
0
xmsech2(x/2)dx, (2.29) withm≡n+ν >0,n=0, 1, 2. These integrals can be analytically calculated. To this end, we express them in the form
J˜m≡ ∞
0
xmsech2(x/2)dx=2 ∞
0
xm d
dx tanhx 2
dx, (2.30)
aFor instance, by systematically increasing the disorder amount in the sample.
and integrate by parts to obtain J˜m =2 lim
x→∞ xmtanhx 2
−2m ∞
0
xm−1tanhx
2dx. (2.31) Making use of Eq. (2.18) in the second term of Eq. (2.31), one can express this equation as
J˜m= lim
x→∞(xm−xm)+4m ∞
0
xm−1dx 1+ex =
4 ln 2, ifm=1 4mIm, ifm=1 ,
(2.32) whereIm ≡(1−21−m)(m)ζ(m),(m) is the Gamma function and ζ(m) is the Riemann zeta function [101]. By plugging Eq. (2.32) into the expression Jn(β) = σAβ−υJ˜m, the reduced kinetic coefficients can be explicitly obtained in terms of the model parameters σA
andυ, and substituting the obtained values in Eqs. (2.23)–(2.25) one finally derives the temperature dependence of the transport coefficients as follows,
σ(T)=
σAln 2β−1, ifυ=1
υσAIνβ−υ, ifυ=1, (2.33)
S(T)= −kB
|e|
⎧⎪
⎪⎨
⎪⎪
⎩ π2
6 ln 2, ifυ=1
υ+1 υ
Iν+1
Iυ , ifυ=1
, (2.34)
κe(T)= σAkB
e2
⎧⎪
⎪⎨
⎪⎪
⎩
9 2ζ(3)−
π2 3
2
1 ln 16
β−2, ifυ =1 F(υ)
υIυ β−(υ+1), ifυ =1
, (2.35)
where F(υ) ≡ υ(ν +2)IνIν+2 −(ν +1)2Iν+12 . We note that the physical conditionκe >0 imposes the restrictionF(υ) >0 which, in turn, imposes certain constrains on the possible values of the critical exponentυ. On the other hand, making use of Eqs. (2.33) and (2.35) into Eq. (2.26) we get the following expression for the Lorenz number
L(T)=L0
⎧⎪
⎪⎨
⎪⎪
⎩ 27
π2ζ(3)− π2 3 ln 4
1
ln 4, if υ=1 3
π2 F(υ)
(υIυ)2, ifυ=1.
(2.36) By inspecting Eqs. (2.33)–(2.36), the following conclusions regarding the temperature dependence of the transport coefficients at the Anderson transition edge value can be drawn:
• Both the electrical and thermal conductivities obey a power law temperature dependence of the formσ(T)∼Tυandκe(T)
∼ Tυ+1, respectively. Therefore, both transport coefficients vanish in theT →0 limit.
• The Seebeck coefficient is temperature independent and it takes on remarkably large values. For instance, in the case of a linear spectral conductivity function (i.e., υ = 1) one has S(T)= −k|eB|6 ln 2π2 −206μVK−1.
• The value of the Lorenz number is temperature independent as well, that is, the WFL holds in systems undergoing the Anderson transition, and its value depends on the adopted critical exponent value, though it is generally smaller than the Sommerfeld’s reference value L0 = 2.44×10−8 V2K−2. For example, forυ =1 we haveL=1.61×10−8V2K−2, whereas forυ=2 (parabolic spectral function) one hasL=2.24×10−8 V2K−2.
The relatively large Seebeck coefficient values, along with the increase of the electrical conductivity with the temperature suggest that large power factors may be obtained for materials undergoing an Anderson transition, as far as the chemical potential is properly located close to the mobility edge. Nevertheless, the power law increase of the electronic contribution to the thermal conductivity will reduce the expected FOM as well. In order to estimate the potential of these systems as promising TEMs, we will determine the FOM expression by plugging Eqs. (2.33)–(2.36) into Eq. (2.4) to obtain
Z T(T)=
⎧⎪
⎨
⎪⎩
3
π
4
ζ(3) ln 4−1+L06 ln 4 π2 κl
σAkBT2
−1
, ifυ=1
υIν (ν+1)2Iν+12
(ν+2)Iν+2+ keB2σκAlTβυ
−1 −1
, ifυ=1 . (2.37) As we see the temperature dependence is related to the lattice contribution only. Since κl(T → 0) ∼ T3 for most materials (see Section 1.2.2.4), we conclude that the FOM value becomes temperature independent in the low temperature regime, and its value is mainly determined by the adopted critical exponent value as far asν ≤ 2. For instance, in the caseν = 1 after Eq. (2.37) we haveZ T(T →0)2.6, which is a large value indeed.
This theoretically derived high FOM value naturally reminds us the question regarding the possible existence of a physical upper bond for Z T, an issue that was previously discussed in Section 2.1 without any explicit consideration to the sample’s electronic structure. In order to further ascertain this appealing issue, we will make use of the general expressions given by Eqs. (2.23)–(2.25) into Eq. (2.4). After some rearrangement, one obtains (we will explicitly assumeJ0=0 andJ2=0, henceforth),
Z T =
1−+, (2.38)
where
≡ J12
J0J2 ≥0, (2.39)
is completely determined by the electronic structure, whereas ≡
2e kB
2κl(T)
T J2−1>0 (2.40) depends on both the lattice dynamics (κl) and the electronic structure (J2). Several important conclusions regarding FOM optimization can be drawn by inspecting Eqs. (2.38)–(2.40):
(1) The case=0 (i.e.,J1=0 andS(T)=0) corresponds to the worstTEM withZ T =0 at any temperature
(2) The physical requirement Z T ≥ 0, leads to ≤ 1 +. The upper bond = 1+, leading to Z T → ∞(thereby η → ηC after Eq. (1.73)), will correspond to the best possible TEM. Nevertheless, since > 0, this will imply > 1, therebyJ12> J0J2. According to Eq. (2.25), this condition leads tonegative values for the charge carrier contribution to the thermal conductivity, against any physical evidence. Therefore, the condition ≤ 1 must be fulfilled instead and Z T will always remain bounded above.
(3) The upper limit = 1 defines the FOM upper bond value Z T = −1(T). However, in this case we have J12 = J0J2, implyingκe(T) = 0 and L(T) = 0 at any temperature (see Eqs. (2.25) and (2.26)). In that case, electrical conductivity decouples from the (vanishing) charge carrier contribution to the thermal conductivity, and the WFL is no longer obeyed.
As we will see in the following section, the condition = 1 can only be attained when singular electronic structures, described in terms of δ-Dirac distribution functions, are considered. Accordingly, we conclude that real materials will always satisfy the condition < 1. Notwithstanding this, the search for materials approaching the → 1 limit provides a simple and powerful guide in TEMs theoretical research, as will be further discussed in the next section (Exercise 2.6).
(4) Making use of Eqs. (2.39) and (2.40), one can express Eq. (2.25) in the formκe(T) = κl(T)(1−)−1. Sinceκe(T) = 0 for realistic systems, we can replacein Eq. (2.38) to obtain the factorized expression [98, 102, 103],
Z T = 1−
1+κl
κe
−1
, (2.41)
where 0 < < 1. Since the second factor in Eq. (2.41) is always less than unity, one realizes that any significant breakthrough in the quest for new TEMs must come from suitable electronic structure based approaches, aimed at attaining the → 1 optimal condition, which ideally defines the best TEM (Exercise 2.7).
From the general results presented in this section, three main criteria orienting the fundamental research in TEMs can be outlined:
• In order to optimize the Z T expression given by Eq. (2.41), one must consider materials exhibiting a very low lattice contribution to the thermal conductivity (i.e.,κl(T)κe(T)) over a wide temperature range.
• According to Eq. (2.4), the charge carriers contribution to the thermal conductivityκe(T) must be also small. However, in order to guarantee that the small value ofκe(T) does not lead to a very smallσ(T) value, the materials of interest shouldnot obeythe WFL within the previously determined temperature range.
• For those materials satisfying the two previous requirements, one should be able to properly control their electronic structure close to the Fermi level in order to approach the
optimal limit→1. To this end, a suitable electronic structure engineering of the sample, based on both chemical structure refinement and material processing should be undertaken.
This topic we will consider in the next section.