Electronic Band Structures for Dr. HoSung Lee April 2, 2015

1 Car et al. (1978) – Istituto di Fisica del Politecnico, Milano

Calculated bandgap: 2.1 eV Experimental value (Albers et al. (1962)): 0.9 eV

2 Soliman et al. (1995) – Dept. of Physics, Ain Shams University, Cairo

3 Lefebvre et al. (1998) – IEMN and LPMC, France

4 Makinistian and Albanesi (2009) – Universidad Nacional de Entre Rios, Argentina

Indirect bandgap, C1-V1: 1.05 eV

5 Chen et al. (2012) – Tongji University, China Polycrystalline SnSe

The can be adjusted by doping element Te from 0.643 (no doping) to 0.608 eV (doping).

Band gap: 0.643 eV 6 He et al. (2013) – Dept. of Material Science and Engineering, Nanjing Institute of Technology, China

Direct energy gap: 0.8 eV Debye temperature: 215 K Gruneisen parameter: 2.98

7 Sun et al. (2013) – Chinese Academy of Sciences, China

8 Zhao et al. (2014) – Dept. of Chemistry, Northwestern University

9 Zhao et al. (2014) – Dept. of Chemistry, Northwestern University

10 Shi and Kioupakis (2015) – Dept. of Material Science and Engineering, University of Michigan

11 Shi and Kioupakis (2015)

12 Park et al. (2010) – Dept. of Physics, Missouri University of Science and Technology

Scheidemantel et al. (2003) – Dept. of Physics, Pen State University

13 Shi and Kioupakis (2015)

14 Shi and Kioupakis (2015)

15 Experiments, Ab initio Experiments, Ab initio Semiclassical Soliman et al. Calculation, Zhao et al.(2014) Calculations, Shi and Nonparabolic Two-Band (1995) Chen et al. Kioupakis (2015) Kane Model (2012) (fit to measurements of Zhao et al. (2014)) Band edge LCB HVB LCB HVB LCB HVB LCB HVB LCB HVB

First band, - - - - 1 1 1 1 1 1 Second band 1 1 Degeneracy of first band, ------2 2 2 2 Degeneracy of second 2 2 (4) (4) band, DE (eV) = - - - - - First band – second band -4 Band gap, Eg (eV) 0.895 0.643 0.61-0.39 0.83-0.46 0.74 – 0.95x10 T

Single DOS effective - - 4.02mo - - - 2.4mo 0.74mo 5.35 mo 0.47 mo mass (md) 1.06mo 3.0mo 0.34mo (3.3mo) (0.3mo)

Integral DOS effective ------8.5 mo 0.75 mo mass (m*) LCB: Lowest conduction band HVB: Highest valence band Note: This work assumes that the multiple bands are equal to multiple valleys. The effective 1/3 2/3 masses are calculated using the relationship of md = (mxmymz) and m* = Nv md.

16 Nonparabolic two-band model for p-type SnSe by Dr. HoSung Lee on 7/26/2014 2 4  12 A s  19  23 J  31   8.854 10 e  1.602110 C k  1.380610 m  9.1093910 kg 0 3 c B K e m kg  34 6.6260810 Js 23 h  N  6.02213710 N  2 p 2 A v

o  2900 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_h0 0 Bejenari (2008) used exponent 0.2 for Bi2Te3 and exponent 0.2 for Si by Barber (1967) and m (T)  T eff_h 0 exponent of 0.8 for PbTe by Lyden (1964) (300K) 2  3 2 m (T)  N m (T)  d_h v eff_h 3 md_e  Nv meff_e

Lyden (1964) and Pei et al. (2012) mI_h(T)  md_h (T) mI_e  md_e

Nv m* h m*e Chen et al. (2014) 2 0.75 SPB model calculation

This work 2 0.75 Calculation Nv: multiplicity of valleys

17 Debye temperature θ =65K by Zhao et al. (2014), θ =215 K for SnSe by He et al. (2013) D  155K

gm gm d  5.76 d  4.81 Mass density= mass/volume = /(NA*a^3), Sn 3 Se 3 cm cm Goldsmid (1964) and Maldelung (1983)

Atomic (molecular) masses, Periodic table MSn  118.71gm MSe  78.96gm y  0.5

1 1 3 3 M M  Sn   10  Se   10 aSn    aSn  3.247 10 m aSe    aSe  3.01 10 m  NAdSn   NAdSe 

1 3  3 3   10 a  aSn (1  y)  aSe y a  3.133 10 m Atomic size, Vining (1991) Atomic size, 2.9x10^-8 cm used by Larson et al. (2000)

Mean atomic mass MSnSe  MSn(1  y)  MSey

MSnSe gm d  d  5.339 mass density, d = 8.219 gm/cm^3 by Malelung (1983) 3 3 NAa cm

 1 3 kB cm 2 5 Speed of sound, Zhao et al. (2014) gives 2.0 x 10^5 cm/s. vs  6  Da vs  1.631 10  hp s

18 20 110 Holes (This work) Electrons (This work)  2 Holes, Zhao et al. (2014)) 19 110 nhn1Ti Tin1  3  3 cm

n  n T T n 18 e  1 i i 1 110   3 cm  4 1 19 p_data 10

Carrier densityCarrier(cm^-3) 17 110

 5 3 200 400 600 800 110 16 110 3 Temperature (K) 200 400 600 800 110 0 n = 3.3 x10^17 cm^-3 TiTip_data T (K)

19 600

0.8 500

0.6  (  V/K) 400 This work Zhao et al. (2014) 0.4 300 (W/m*K) k 3 200 400 600 800 110

T (K) 0.2

0 o  4.5ecV 3 100 200 400 600 800 110 This work Z  0.1 80 Zhao et al. (2014) T(K) Ka  1 Total thermal conductivity 60 Electronic thermal conductivity  1  1 Lattice thermal conductivity  cm  40 Zhao et al. (2014)

20

0 3 200 400 600 800 110

20 800 300 K 900 K 800 niT1 T1ni 600 300 K niT1 T1ni V  1  1 K 900 K 600  cm 400 niT  T ni niT  T ni 3 3 400 3 3 V  1  1  cm K 200 200

0 0 16 17 18 19 20 110 110 110 110 110

ni  3 cm Shi and Kioupakis (2015) This work

21 3 This work 3 300 K Experiment, Zhao et al. (2014) 600 K ZT  n T T n 900 K 2   i 1 1 i2

ZT niT2 T2ni

ZT ZT  niT T ni   3 3 1 1

0 16 17 18 19 20 0 110 110 110 110 110 3 200 400 600 800 110 ni  3 Temperature (K) cm 0.3

1.5 Parabolic model Kane model g E SPM i  Ab Initio calc. He et al. (2013) 0.2  3  1 1 a ecV

gSKM Ei 

 3  1 (J/g.K) Cv a ecV 0.1 0.5 E_DOSti Prediction, this work Experiment, Zhao et al. (2014) 0 0 3  3  2  1 0 1 2 3 0 200 400 600 800 110 Ei Ei ti T (K)   22 ecV ecV ecV The End

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