Lattice Thermal Conductivity
for Tin Selenide
Dr. HoSung Lee
April 25,, 2015
Zhao et al. (2014) – Kanatzidis group
Carrete et al. (2014) – CEA – Grenoble, France
Carrete et al. (2014) – CEA – Grenoble, France
Carrete et al. (2014) – CEA – Grenoble, France
Serrano-Sanchez et al. (2015) – Institute of Ciencia of Materials of Madrid, Spain
Serrano-Sanchez et al. (2015) – Institute of Ciencia of Materials of Madrid, Spain
Serrano-Sanchez et al. (2015) – Institute of Ciencia of Materials of Madrid, Spain
Serrano-Sanchez et al. (2015)
Serrano-Sanchez et al. (2015) – Institute of Ciencia of Materials of Madrid, Spain
Nonparabolic two-band model for p-type SnSe by Dr. HoSung Lee on 7/26/2014
 23 J
 19
ec  1.6021 10
kB  1.3806 10
C
 34
h p 
6.6260810

J s
 31
me  9.1093910

K
23
 o  290  0
kg
Maldelung (1983)
Thomas (1991) used 90 for Bi2Te3
meff_h0  0.75 me
density-of-state effective mass of hole for multiple valleys
meff_e  8.5 me
density-of-state effective mass of electron for multiple valleys
meff_h ( T) 
meff_h0
( 300K )

md_h ( T)  Nv
0
T
Bejenari (2008) used exponent 0.2 for Bi2Te3 and exponent 0.2 for Si by Barber (1967) and
exponent of 0.8 for PbTe by Lyden (1964)
0
2
3
 meff_h ( T)
mI_h ( T)  md_h ( T)
3
m  kg
Nv  2
NA  6.02213710

2 
2 4
 12 A  s
 0  8.854  10

md_e  Nv
2
3
 meff_e
Lyden (1964) and Pei et al. (2012)
mI_e  md_e
Chen et al. (2014)
Nv
2
m* h
0.75
This work
2
0.75
Nv: multiplicity of valleys
m*e
SPB model calculation
Calculation
gm
cm
M
D
0.5


78.96gm
118.71
M
(110
M155K
1 y
gm
gm
gm
1 y )  MSe
1

10
y
510
Se
SnSe
Sn
SnSe
ad
1.631
3.01
3.247
Sn
10
10 mmm
vSn
d
3.133

5.339
4.81
5.76
10
Se
s
3
3 33 3s33
kB 3M
3cm
3 
cm
2 
aNA
 acm
Se
Sn
aa s
(
1

y
)

a

y
v

6





a
Se 
p
 D
Se
Sn  hSn
Sn 
 NA  d Se
 
Debye temperature θ =65K by Zhao et al. (2014), θ =215 K for SnSe by He et al. (2013)
D  155K
gm
d Sn  5.76
dSe  4.81
3
cm
MSn  118.71gm
gm
Mass density= mass/volume = molar mass/(NA*a^3),
Goldsmid (1964) and Maldelung (1983)
3
cm
MSe  78.96gm
Atomic (molecular) masses, Periodic table
y  0.5
1
 MSn 
aSn  

 NA  d Sn 
1
3
 10
aSn  3.247  10
 MSe 
aSe  

 NA  d Se 
m
3
 10
aSe  3.01  10
m
1
a  aSn  ( 1  y )  aSe  y


3
3
3
 10
a  3.133  10
m
Atomic size, Vining (1991)
Atomic size, 2.9x10^-8 cm used by Larson et al. (2000)
Mean atomic mass
M SnSe  M Sn  ( 1  y )  M Se y
d 
M SnSe
NA  a
d  5.339
3
gm
mass density, d = 8.219 gm/cm^3 by Malelung (1983)
3
cm
1
vs 
kB
hp
 2 3 Da
 6 
5 cm
v s  1.631  10 
s
Speed of sound, Zhao et al. (2014) gives 2.0 x 10^5 cm/s.
Electronic Thermal Conductivity

 kB 
Le(  T n )   
 ec 

2 F 
e 2  1 T n
  2


  3

 Fe 0 2 1 T n
 


 kB 
Lh (  T n)   
 ec 
3

2 F 
h  2  1  T n
3
  2


  3

 Fh  0 2 1  T n
 

 F  1 3 1 T n 

 e 2



3
 Fe 0  1 T n 
  2

2





 F  1 3 1 T n 

 h  2



3

 Fh  0  1 T n 
  2

ke(  T n )  Le(  T n)  e(  T n)  T  Lh (  T n )  h (  T n )  T 
2





e(  T n )  h (  T n )
(  T n )
Lattice Thermal Conductivity
 s  39
Strain parameter for point defects, Abeles (1963) used 39 for SiGe.
M  MSn  MSe
Vo  a
3
Mean atomic volume

 T h (  T n)  e(  T n)
2
Drabble and Goldsmid (1961)
a  aSn  aSe
s  y  ( 1  y ) 
  2.1
2
 M  2
 a  



s a 
 MSnSe 
 


Disorder parameter or mass-fluctuation-scattering parameter
Abeles (1963)
s  0.096
Gruneisen parameter, Zhao et al. (2014)
γ = 2 is often used to describe the anharmonisity effects on the thermal conductivity.
  2
The ratio of Normal to Umklapp processes, β = 2 is used for SiGe by Steigmeier and Abeles (1964)
1




5


3 1  
 20 
2
2
3

 6   
9 

T  2

 
 U( xT)  
 NA  h p  


x

  
2 D
 4   1   M
 3
 
SnSe a 

T1  300K
1
Phonon relaxation time for Umklapp process, Klemens
(1958), Steigmeier and Abeles (1964), and Vining (1991)
 N( xT)     U( xT)


11
 U x1 T1  10
4

 kB T  4

 Vo  s   h   x 
 p  
 PD( xT)  
3


4   v s


1 md_h ( T)  v s
Erc ( T)  
2
kB T
x1  2
3
 3.3  10
s
1
Point defects ,Klemens (1955), Vining (1991)
and Klemens (1958)
2
Reduced carrier energy


11
 PD x1 T1  10
5
 4.837  10
s
2




x
x   
 2


1

exp
E
(
T
)




3
rc

16 Erc ( T)
2   
 o  md_h ( T)  vs 



 EP(  xT)  
  x  ln
 
4
2

 4   h p  d  Erc ( T) 

x
x   


 1  exp Erc ( T)    16 E ( T)  2   
rc




  
 latt (  xT)    N( xT)

1
  U( xT)
3
 kB T  
klatt (  T) 

 
2
hp
 
2   v s 
1
  PD( xT)
1
  EP(  xT)
 1
1
1
Electron-phonon scattering,
Zieman (1956) and Vining (1991)
Combined phonon relaxation time

D
kB
T


0
k(  T n )  klatt (  T)  ke(  T n )
4 x
 latt (  xT) 
x e
ex  1
2
dx
x
h 
kB T
Debye-Callaway formula
This work
3
110


 U xi T 2  f  xi
100
ps


 PD xi T 2  f  xi
ps
0.8
k (W/m*K)
10
 


1
 latt  n1 T 2 xi T 2  f  xi

0.1
ps
0.01
 EP  n1 T 2 xi T 2  f  xi
ps
 
0.6

0.4
3
110
0.01
0.1
1
10
xi
0.2
2
0
200
400
600
800
3
110
 

W
m K
T(K)
Total thermal conductivity
Electronic thermal conductivity
Lattice thermal conductivity
Zhao et al. (2014)

k  ni T 1 T 1 ni
1.5
300 K
900 K
 

k  ni T 3 T 3 ni
W
m K

1
0.5
0
16
110
17
110
18
110
ni
3
cm
19
110
20
110
Specific Heat
Prediction, this work
Debye cutoff freq.
0.8
 
 1012  Hz   1 0.6
1
 max
 1012  Hz   1
d  5.339 
cm
Dulong-Petit limit
D  155K
gm
3
Cv_DP  3
kB
da
3
 0.252
J
Cv
gm K
D

9 kB T 3  T


Cv( T) 


3  D  

d a 


0.4
0.2
0
4 x
x e
e
x

2
dx
1
0
0
1
2
 i 10
3
 12
Hz

4
0.3
5
0
 12
 max  10
Hz
0.2
Cv (J/g.K)
gph  i
density
This work
0.1
Prediction, this work
Experiment, Zhao et al. (2014)
0
0
200
400
600
T (K)
800
110
3
3
Na
V
 kB
The End