Thermoelectric Merit of Two-band Systems

Chapter 4 P-type PbSe with Na doping

4.4 Thermoelectric Merit of Two-band Systems

83 and p-type. It also shows the zT calculated for the L valence band only, assuming there is no Σ band. Due to its larger deformation potential coefficient, the maximum zT if there is no Σ band would be much smaller around 0.6.

(S₁!1+S₂!2)²

!1+!2

= 9 e

k_B

!

"

# $

%&

2

"L

( )

T ^{2 B1!}"#^{1F1'21'!0F1'21}$

%&+B2 1F'2

1 2

'(!' ()⁰F'2 1

! 2

"

# $

%&

)

*+ ,

-.

2

3 e k_B

!

"

# $

%&

2

"L

( )

T ^!"#^{B10F1'21+B20F2'21}$

%&

= 3!L

TB₁ ¹F_!2¹

1

!"⁰F_!2¹

" 1

#$ %

&

'+B₂ B₁

1F_!2¹

2

!("! ()⁰F_!2¹

" 2

#$ %

&

' )

*+

, -.

2

0F_!2¹

1

+B₂ B₁

0F_!2¹

" 2

#$ %

&

'

=A

Equation 4.15

L1+S1

(

2

)

^!1+

(

^L2+S22

)

^!2=3B1!LT

2F!2 1 1

!2!¹F!2 1 1

+!^{2 0}F!2 1

" 1

#$ %

&

'+B2

B1 2F!2

1 2

!2(!! ")¹F!2 1 2

+(!! ")^{2 0}F!2 1

# 2

$% &

'( )

*+ ,

-.=C

Equation 4.16 The energy integrals are defined as:

nF_l^m

1

=ⁿF_l^m(!),ⁿF_l^m

2

=ⁿF_l^m(!! ") Equation 4.17

Thus zT is expressed as:

zT (B₁,B₂

B₁,!,!)= A C!A+!L

T

= A^* C^*!A^*+1

Equation 4.18 Where

A^*=A !_L

( )

T _, ^C*=C

( )

^!LT Equation 4.19

It is now easy to visualize the relation between zT and different combination of B₁, B₂/B₁, Δ and η.

For simplicity, we will assume both bands are parabolic so α1 = α2 = 0. Two examples are given in Figure 4.12 and Figure 4.13. Assuming the quality factor of the first band is 0.35, which is about the value for the L valence band in PbSe, and 0.7, which is about that for the L conduction band in PbSe. The value of B1 and B2 are allowed to change only in a reasonable range based on known quality factors for real compounds. Specifically, for the first band, 0.35 correspond to a mediocre thermoelectric compound that most researches start with, while 0.7 is close to the value for a state- of-the-art thermoelectric material. For the second band, 0.17 is about the minimum quality factor for it to be of any value to contribute for zT, whereas 1.4 is almost impossibly high based on the knowledge on various compounds.

There are several interesting implications from these two figures:

85 1. Once the quality factor of the first band is fixed, how much improvement of zT the system could achieve with the second band is determined by its quality factor, instead of any individual parameter. More generally it also matters how much the non-parabolicity is for the second band.

2. The quality factor of the first band determines the basis of the zT of the system. The second band adds its contribution depending on the offset between them, when they are aligned (Δ = 0), the zT reaches its maximum, which is about the sum of zT expected for each band using their own B factor.

3. Depending on their ratio the second band only starts to benefit zT when it is closer than 2 to 4 kBT from the first band edge, Being on the large side when the ratio of B₂/B₁ is large, and around 3 k_BT when the ratio is about 1.

4. There is always a single maximum zT as function of η, given realistic combinations of quality factors, the optimum chemical potential is always close to the edge of the first band, regardless of the quality factor of the second band. There is a shoulder in zT for some B₂/B₁ ratios as the chemical potential moves away from the edge of the first band towards the second band. As seen in Figure 4.12 when B1 is 0.35, even when B2 is ten times as good the maximum zT when two bands are not aligned is still found around the edge of the first band. One should try to move the second band closer towards the first band while maintaining the chemical potential near the band edge for better zT.

5. As an extreme exception, when B1 is very small while B2 is large, then the maximum zT is found when η moves towards the second band (Figure 4.14). More generally, the optimum η to get maximum zT is always found below or close to the edge of the first band unless the first band has a very small quality factor B1 close to zero, while the second band is many times as good. For B1 ≥ 0.3 the optimum η is always around the edge of first band, regardless of the quality of the second band (Figure 4.15).

6. The optimum chemical potential slightly shifts toward the second band as it comes close to around 1 kBT. This shift is less than half a kBT, even when the second band has a B factor twice as high. As the second band comes even closer η begin to shift back to the band gap again.

7. This relation is between zT and chemical potential η. When the more directly observable nH is considered, one need to increase nH in order to keep η constant when there is a second band being brought closer.

Figure 4.12. zT of a two-band system for different combination of parameters for the second band and chemical potential. Quality factor for the first band is 0.35. A secondary band with quality

factor greater than 1, as suggested in d) and e) is not very likely in reality.

By calculating the maximum zT for a few B₁ values as function of the ratio B₂/B₁ and energy offset Δ, one can estimate the effective quality factor Beff for two-band systems that is equivalent to a single band system respect to maximum zT. The ratio of B_eff/B₁ is plotted in Figure 4.16. Generally when the two bands are aligned (Δ = 0), Beff is roughly the sum of B1 and B2 (the maximum zT is lower than the sum of two systems with each band along). For small offsets Δ < 0.5, the two bands can be approximately considered as aligned. With larger Δ, Beff will be smaller than the sum of B1

and B₂. Also as the quality factor of the first band B₁ increases, it requires the second band to have higher B2 or to be closer to the first band to result in a Beff greater than B1 (f > 1).

zT, B1 = 0.35, B2/B1 = 0.5

Δ

η

zT, B1 = 0.35, B2/B1 = 1

Δ

η zT, B₁ = 0.35, B₂/B₁ = 2

Δ

η

a b

c

η

Δ

d ^{zT, B1 = 0.35, B2/B1 = 4}

η

Δ

zT, B₁ = 0.35, B₂/B₁ = 10

e

87

Figure 4.13. zT of a two-band system for different combination of parameters for the second band and chemical potential. Quality factor for the first band is 0.7.

Figure 4.14. zT of a two-band system for different combination of parameters for the second band and chemical potential. When the first band has very low quality factor B1 = 0.05 while B2 = 0.5,

the chemical potential need to be moved to the edge of the second band for best zT.

zT, B1 = 0.7, B2/B1 = 0.5

Δ

η

zT, B1 = 0.7, B2/B1 = 1

Δ

zT, B1 = 0.7, B2/B1 = 2 η

Δ

η

a b

c

Δ

η

zT for two-band system, B1 = 0.01, B2/B1 = 10

Figure 4.15. The optimum reduced chemical potential η in two-band systems with different energy offset Δ (=ΔE/kBT) between two bands and the ratio of their quality factor B2/B1, for different B1 values of a) 0.01, b) 0.1, c) 0.3, and d) 1. Dashed lines in c) and d) are rough limits of

the ratio B2/B1 in each case, the region to their right is not likely achievable in real systems.

In the end, let’s come back to the case of p-type PbSe. Using the parameters determined from this study, the quality factor for L valence band is calculated to be 0.32 at 850 K. Compared with the known quality factors of different compounds, this is not quite promising for thermoelectrics.

Notice the quality factor for its counterpart the L conduction band, was found about twice as high at 0.67. The reason for such difference is the difference in the deformation potential coefficient or, the strength of electron-phonon interaction. The p-type PbSe would be only ordinary in performance at best if there were not the Σ valence band. For the Σ band we could also give an estimate of its quality factor, which is 0.37. This means the Σ band is better than the L valence band in thermoelectric performance, even though it is not superior (also keep in mind it is parabolic with α

= 0 while the L band is non-parabolic with α = 0.17). But as we learned from the relation between zT and quality factors in a two-band system, the two ordinary band when working together could add up to a remarkable zT. In binary PbSe, the two bands are on the right track when they move close to each other at high temperature: the Σ band is 1.8 kBT away from the L band edge at 850 K.

With this configuration the maximum zT is expected to be 1.0 at 850 K, which is very close to the more careful calculation shown in Figure 4.11, as well as experimental results. But obviously there is extra room for zT if one could make the two bands come even closer at this temperature. We later

Δ

B2/B1

optimum η, B₁ = 0.01

6.05.5 5.04.5 4.03.5 3.02.5 2.01.5 1.0 0.50

Δ

B2/B1

optimum η, B1 = 0.1

5.04.5 4.03.5 3.02.5 2.01.5 1.00.5 -0.50 -1.0-1.5

0.005 -0.495 -0.995 -1.495 -1.995

Δ

B2/B1

optimum η, B1 = 0.3

-1.0 -1.5 -2.0 -2.5 -3.0

Δ

B2/B1 optimum η, B1 = 1.0

a b

c d

89 demonstrate the implement of this in Chapter 8. But before that we shall draw a blueprint of zT to look for assuming this could be done. The result indicates a maximum zT of 1.5 when the two bands are aligned (Figure 4.17), a 50% increase of zT.

Figure 4.16. The ratio f between effective quality factor Beff and B1 as function of B2/B1 and Δ for systems with different B1 a) 0.1, b) 0.3, c) 0.5 and d) 1.

Figure 4.17. zT map of p-type PbSe assuming the energy level of second band can be adjusted freely.

Δ

B2/B1

f in Beff = fB1, B1 = 0.1 _1.0

1.5 2.02.5 3.03.5

4.5 5.5 6.5 7.5 8.5 1.25 1.75

9.5 10.5

Δ

B2/B1

f in Beff = fB1, B1 = 0.3 _1.0

1.5 2.02.5 3.03.5

4.5 5.5 6.5 7.5 8.5 1.25 1.75

9.5 10.5

Δ

B2/B1

f in Beff = fB1, B1 = 0.5 _1.0

1.5 2.02.5 3.03.5

4.5 5.5 6.5 7.5 8.5 1.25 1.75

9.5 10.5

Δ

B2/B1

f in Beff = fB1, B1 = 1.0

1.0 1.5 2.02.5 3.03.5

4.5 5.5 6.5 7.5 8.5 1.25 1.75

9.5 10.5

a b

c d

B1 = 0.32, B2/B1 = 1.15, parameter for p-PbSe

Δ

η

91