Supporting information:
High thermoelectric performance of oxyselenides:
Intrinsically low thermal conductivity of Ca-doped
BiCuSeO
Yan-Ling Pei,1 Jiaqing He,2,* Jing-Feng Li,3 Fu Li,3 Qijun Liu,4, 5 Wei Pan,3 Celine
Barreteau,6 David Berardan,6 Nita Dragoe,6 Li-Dong Zhao,6, ‡,*
1
School of Materials Science and Engineering, Beihang University, Beijing 100191,
China
2
Frontier Institute of Science and Technology (FIST), Xi’an Jiaotong University,
Xi’an 710054, China
3
State Key Laboratory of New Ceramics and Fine Processing, School of Materials
Science and Engineering, Tsinghua University, Beijing 100084, China
4
Institute of High Temperature and High Pressure Physics, School of Physical Science
and Technology, Southwest Jiaotong University, Sichuan 610031, China
5
State Key Lab of Solidification Processing, School of Materials Science and
Engineering, Northwestern Polytechnical University, Xi’an 710072, China
6
LEMHE, ICMMO, University Paris-Sud & CNRS, Orsay F-91405, France
*Corresponding Author:
hejiaqing@mail.xjtu.edu.cn (J. He) & lidong-zhao@northwestern.edu (L.-D. Zhao)
Author information:
‡L.-D. Zhao is presently in Department of Chemistry, Northwestern University,
Evanston, Illinois 60208, USA.
S1
1) Lorenz number calculation:
The total thermal conductivity (κtot) includes a sum of the electronic (κele) and
lattice thermal conductivity (κlat). The electronic part κele is directly proportional to the
electrical conductivity σ through the Wiedemann-Franz relation, κe = LσT, where L is
the Lorenz number.
[1]
Here, it would be worth discussing in detail the two
contributing parts of the total thermal conductivity. Generally, the lattice thermal
conductivity κlat can be estimated by directly subtracting κele from κtot with using:
L0 
 2  kB 
2
8
2
   2.45 10 W K
3  e 
(1)
Although this may be a good estimation for the lattice thermal conductivity кlat at
room temperature, it does not hold true for heavily doped semiconductors where a
strong change with temperature is observed in the chemical potential. [2] For most
thermoelectric materials, the true Lorenz number is in fact lower than L0 (2.45 × 10-8
W Ω K-2) especially at high temperature. The Lorenz number depends on the
scattering parameter r and will decrease as the reduced Fermi energy η decreases with
increasing temperature. The Lorenz number can be given as: [3]
2
 k B   (r  7 2F)r 5 2  ( ) r (
L 

 e   (r  3 2F)r 1 2  ( ) r (


F
5r 2 3) 
2

F
3r 21)2 
2
( )
 (2)
( )

For the Lorenz number calculation, we should get reduced Fermi energy η firstly; the
calculation of η can be derived from the measured Seebeck coefficients by using the
following relationship:
S 

k B  (r  5 2) Fr 3 2 ( )
 


e  (r  3 2) Fr 1 2 ( )

(3)
where Fn(η) is the n−th order Fermi integral,
Fn ( )  
0
=
n

1  e  
Ef
k BT
S2
d
(4)
(5)
In the above equations, kB is the Boltzmann constant, e the electron charge and Ef the
Fermi energy. Meanwhile, acoustic phonon scattering (r = −1/2) has been assumed as
the main carrier scattering mechanism.
[3]
The Lorenz number can be obtained by
applying the calculated reduced Fermi energy η and scattering parameter r into Eq. (2),
following Table lists the calculated results for BiCuSeO.
η
T
Sexpt
Sfit
(K)
(μVK-1)
(μVK-1)
303
323
373
423
473
523
573
623
673
723
773
823
873
931
898
848
798
748
698
648
598
548
498
448
398
348
329.01635
331.79449
339.32302
347.49017
356.05282
364.76786
373.39219
381.68268
389.39623
396.28973
402.12006
406.64412
409.61878
410.8075
410.44912
408.34032
404.56057
399.35299
392.96068
385.62677
377.59436
369.10656
360.40649
351.73725
343.34196
335.46373
329.016
331.7945707
339.3230888
347.4902319
356.0529017
364.7678858
373.3923049
381.682795
389.3963155
396.2897761
402.120
406.6442079
409.6187705
410.8074137
410.4489983
408.3401874
404.5605005
399.352905
392.960669
385.6269054
377.594
369.1066194
360.4065046
351.7372759
343.3420266
335.4636993
L
(WΩK-2)
-1.732159371
-1.767043322
-1.861140818
-1.962553796
-2.068203667
-2.175093786
-2.280301747
-2.380953072
-2.474214536
-2.557274401
-2.627330893
-2.681575676
-2.717190033
-2.731410533
-2.727123237
-2.701886501
-2.656603281
-2.594101444
-2.517193983
-2.428684056
-2.428683424
-2.331373136
-2.228088732
-2.121676646
-2.015036803
-1.911125041
1.527345892
1.526402699
1.523994516
1.52160943
1.519340151
1.517250825
1.515381132
1.513751884
1.512370493
1.51123633
1.510345379
1.509694401
1.509284621
1.509124785
1.509172749
1.509459029
1.509989993
1.510760742
1.511772742
1.513030109
1.513030118
1.514535768
1.516286846
1.518270026
1.520455706
1.522792641
[1] Fitsul, V. I. Heavily Doped Semiconductors; Plenum Press: New York, 1969.
[2] Kumar, G. S.; Prasad, G.; Pohl, R. O.; J. Mater. Sci. 1993, 8, 4261.
[3] Zhao, L.-D.; Lo, S.-H.; He, J. Q.; Li, H.; Biswas, K.; Androulakis, J.; Wu, C.-I.;
Hogan, T. P.; Chung, D.-Y.; Dravid, V. P.; Kanatzidis, M. G. J. Am. Chem. Soc. 2011,
133, 20476.
S3
2) Callaway model calculation:
In a solid solution system, point defects scattering originates from both the mass
difference (mass fluctuations) and the size and the interatomic coupling force
differences (strain field fluctuations) between the impurity atom and the host lattice.
Callaway model [1−3] has been applied to describe the influence of point defects on the
lattice thermal conductivity. Here we present a phonon scattering model on the basis
of the above theory and try to describe the effect of Ca doping on the lattice thermal
conductivity of BiCuSeO system.
At temperature above the Debye temperature, the ratio of the lattice thermal
conductivities of a material containing defects to that of the parent material can be
written as: [1−3]
 lat
tan 1 (u )

 lat , p
u
(1)
Here κlat and κlat,p are the lattice thermal conductivities of the defected and parent
materials, respectively, and the parameter u is defined by: [1−3]
12
  2 D 

u 
 L, p  
2
 hva

(2)
where h and Ω stand for the Planck constant and average volume per atom,
respectively, average sound velocity va can be extracted from:[4−7]
1  1 2 
va    3  3  
 3  vl vs  
1
3
(3)
Here, the longitudinal (vl, 3290m/s) and transverse (vs,1900m/s) sound velocities have
been obtained as described in the experimental details section, Eq. (3) gives va about
2107 m/s. Debye temperature θD is defined by: [4]
13
h  3N 
D = 
va
kB  4V 
(4)
where the V is the unit−cell volume, N is the number of atoms in a unit cell, kB is
Boltzmann parameter, and h presents the Planck constant. Eq. (4) gives θD about 243
S4
K. The imperfection scaling parameter Γ in Eq. (2) represents the strength of point
defects phonon scattering, which includes two components, the scattering parameter
due to mass fluctuations ΓM and the scattering parameter due to strain field
fluctuations ΓS. A phenomenological adjustable parameter ε is always included
because of uncertainty of ΓS, one writes Γ = ΓM + εΓS. In this paper, ε is directly
estimated by following relationship: [5]
2  6.4   (1   p ) 

= 
9  1   p  


2
(5)
where υp the Poisson ratio, which can be derived from the longitudinal (vl) and
transverse (vs) sound velocities by the relationship as: [1-3, 5]
p 
1  2  vs vl 
2
2  2  vs vl 
2
(6)
Eq. (6) gives υp about 0.25.
Gruneisen parameter (γ) has been calculated using Poisson ratio (υp) as: [1-3, 5]
3  1p
2  2  3 p
= 




(7)
Gruneisen parameter (γ) was calculated to be ~1.5 for BiCuSeO. Combination of Eq.
(5), (6) and (7), gives phenomenological adjustable parameter ε about 57.
There is no change on the sites of Cu, Se and O when the substitution between Bi
and Ca occurs, ΓCu = ΓSe = ΓO = 0, which gives:
2
 Bi1 xCaxCuSeO
1  M ( Bi ,Ca ) 
 
  ( Bi ,Ca )
4 M 
(8)
  M  S
(9)
 ( Bi ,Ca )   M ,( Bi ,Ca )   S ,( Bi ,Ca )
(10)
 M ,( Bi ,Ca )
 M 
 x(1  x) 

M
 ( Bi ,Ca ) 
2
(11)
M ( Bi ,Ca )  (1  x) M Bi  xM Ca
where M  M Bi  M Ca , and
S5
2
 S ,( Bi ,Ca )
 r 
 x(1  x) 

r
(
Bi
,
Ca
)

 ,
(12)
r
 (1  x)rBi  xrCa
where r  rBi  rCa , and ( Bi ,Ca )
then,
 ( Bi ,Ca )
2
 M 2
 r  
 x(1  x) 
   
 
r( Bi ,Ca )  
 M ( Bi ,Ca ) 



(13)
and then,
 Bi1xCaxCuSeO
2
2
 M 2
 r  
1  M ( Bi ,Ca ) 
 
   
 
 x(1  x) 
4 M 
r
 M ( Bi ,Ca ) 
(
Bi
,
Ca
)

 

(14)
After the calculation of point defects scattering on phonon for Γ from those
relative physical properties, we obtained perfect agreement between the calculated
and measured values.
[1] Callayway, J.; Von Baeyer, H. C. Phys. Rev. 1960, 120, 1149;
[2] Dey, T. K.; Chaudhuri, K. D. J. Low. Temp. Phys. 1976, 23, 419;
[3] Yang, J.; Meisner, G. P.; Chen, L. Appl. Phys. Lett. 2004, 85, 1140.
[4] K. Kurosaki, A. Kosuga, H. Muta, M. Uno, S. Yamanaka, Appl. Phys. Lett. 2005,
87, 061919.
[5] C. L. Wan, W. Pan, Q. Xu, Y. X. Qin, J. D. Wang, Z. X. Qu, M. H. Fang, Phys. Rev.
B 2006, 74, 144109.
[6] C. L. Wan, Z. X. Qu, Y. He, D. Luan, W. Pan, Phys. Rev. Lett. 2008, 101, 085901.
[7] D. S. Sanditov, V. N. Belomestnykh, Technical Physics, 2011, 56, 1619.
S6
Fig. S1. Lattice parameters (a, circles and c, squares) as function
of Ca doping contents for Bi1-xCaxCuSeO samples;
S7
Fig.S2. Heat capacity as a function of temperature for
Bi1-xCaxCuSeO.
S8
Fig.S3. Thermal diffusivity as a function of temperature for
Bi1-xCaxCuSeO.
S9
Fig.S4. Lorenz number as a function of temperature for
Bi1-xCaxCuSeO.
S10
Fig.S5. Electronic thermal conductivity as a function of
temperature for Bi1-xCaxCuSeO.
S11