Electronic (Band) Structures
for Lead Telluride
Dr. HoSung Lee
March 23, 2015
1
Allgaier (1961) – US. Naval Ordnance Laboratory, Mayland
2
Lyden (1964) - MIT
3
Lyden (1964)- MIT
4
Rogers (1967)-Zenith Radio
Research Corp., UK
5
Ravich et al. (1971)
Figure. Brillouin zone and symmetry
points for an fcc lattice.
6
Tsang and Cohen (1971) – Dept. of Physics UC Berkeley
Band gab of SnTe resembles that of PbTe
throughout the zone except at near L point.
7
Harris and Ridley (1972) - Zenith Radio Research Corp., UK
The polar optical phonon scattering is being reduced by screening at these
high carrier concentrations.
The effect of the higher-lying conduction bands at the L points is almost completely compensated by
that of the lower-lying valence bands, thereby making the two-band model a good approximation.
It is assumed that in low carrier density PbTe at 300 K, only optimal phonon and acoustic phonon
scattering need be considered.
The ionized impurity scattering is negligible at room temperature; however, it does become
significant at very low temperatures
8
Martinez et al. (1975) – Dept. of physics, UC Berkeley
Cuff et al. (1964)
Empirical pseudopotential method (EPM) was
used to calculate the band structure.
9
Bilc et al (2006) – Kanatzidis group
n-type PbTe
The longitudinal effective mass is defined along the LG direction in the Brillouin zone, whereas the transverse
effective mass is defined along the perpendicular direction to LG.
A two-band model (conduction and valence) has to be employed to calculate the transport coefficient correctly.
A nonparabolic (fourfold degenerate) conduction band is seen rather than parabolic one in the figure.
10
Ahmad et al. (2006) – Kanatzidis group
11
Hummer et al. (2007) – Dept. of Computational Materials Physics, Univ. Wien, Austria
12
Vineis et al. (2008)-Dr. Shakouri group
13
Vineis et al. (2008)-Dr. Shakouri group
14
Singh (2010) – Oak Ridge National Laboratory
It is important to note that both the valence and
conduction bands are highly nonparabolic.
15
Pei et al. (2011)-Snyder group
16
Degeneracy
Band gap (eV)
Integral DOS
Effective mass
Experiments,
Ravich (1971)
Ab initio
Calculations, Vineis
et al. (2008)
Semiclassical
Nonparabolic Two-Band
Kane Model
(fit to measurements of
Pei et al. (2012))
n-type
n-type
n-type
LCB
HVB
LCB
HVB
LCB
HVB
4
4
4
4
4
4
0.19+4.2x10-4T
0.2
0.22 mo
0.28 mo
0.23 mo
0.36 mo
0.18+4x10-4T
0.22 mo
0.22 mo
Note: While two bands (one conduction and one valence) are capable to estimate the transport
coefficients in n-type material, three bands (one conduction and two valence bands) should be
deployed in p-type material as mentioned in the previous slides.
17
18
Debye temperature, Maldelung (1983)
D  136K
dTe  6.25
dPb  11.34
gm
3
cm
M Pb  207.2gm
gm
Mass density= mass/volume = molar mass/(NA*a^3), Maldelung (1983)
3
cm
M Te  127.6gm
Atomic (molecular) masses, Periodic table
y  0.5
1
 MPb 
aPb  

 NA  d Pb 
1
3
 10
aPb  3.119  10
 MTe 
aTe  

 NA  d Te 
m
3
 10
aTe  3.236  10
m
1
a  aPb  ( 1  y )  aTe  y


3
3
3
 10
a  3.179  10
Mean atomic mass
M PbTe  M Pb  ( 1  y )  M Te y
d 
M PbTe
NA  a
d  8.654
3
Atomic size, Vining (1991)
m
gm
mass density, d = 8.219 gm/cm^3 by Malelung (1983)
3
cm
1
vs 
kB
hp
 2 3 Da
5 cm
 6 
v s  1.452  10 
s
Speed of sound, Eq. (142) of the class note
2.94*10^5 cm/s by Pei et al. (2012)
longitudinal sound velocity=3.59*10^5 cm/s by Maldelung (1983)
e V
4 c
Eg ( T)  0.18ec V  4.0 10

K
T
Reduced band gap, Ravich (1970), Maldelung (1983)
19
3
2

1
0
1
300
400
500
600
700
800
Temperature (K)
n = 1.9 x10^19 cm^-3
n = 2.9 x 10^19 cm^-3
0.7
 
0.6
Eg T i 0.5
ec V 0.4
0.3
0.2
200
400
600
800
110
3
3
1.210
3
1.410
Ti
20
21
22
The End
23