Numerical Study of Bloch Electron Dynamics in Wide Band-Gap Semiconductors
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1 Department of Information Technology and Media, Mid-Sweden University, S-851 70 Sundsvall, Sweden, Hans-Erik.Nilsson@ite.mh.se
2 Department of Solid State Electronics, Kungl. Tekniska Högskolan (KTH), Electrum, S-164 40 Kista, Sweden
3 ITN/Campus Norrköping, Linköping University, S-601 74 Norrköping, Sweden
I NTRODUCTION
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A numerical study of the Bloch electron (hole) dynamics in 4H- SiC, 3C-SiC and wurtzite GaN is presented.
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The calculation is based on a 3-band k ⋅p model including spin orbit interaction, where the band structure parameters have been obtained from ab initio band structure calculations.
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The interband tunneling at different electric fields have been calculated around the maximum of the valence band structure where bands are close together in energy.
R ESULTS
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Significant tunneling between bands occur in 4H-SiC and wurtzite GaN at an electric field as low as 10 kV/cm.
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Interband tunneling is found to be present in 3C-SiC at an electric field of 40 kV/cm applied along the Γ-L segment.
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A smaller energy separation between bands provides stronger tunneling (see Fig. 4. and 5.).
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The tunneling is very weak for smooth bands while it becomes strong in regions where the curvature changes rapidly.
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The interband tunneling is expected to influence the transport properties at
⇒ low temperatures where the average time between scattering is large.
⇒ high fields where the hole may drift long distances in k-space between scattering events.
0.05 0 0.05
0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
Energy [eV]
4H-SiC
0.05 0 0.05
kvector [2 /a]
3C-SiC
0.05 0 0.05 Wurtzite GaN
Fig 1. K⋅p band structures. 4H-SiC and W-GaN: N⊥FD[LV is in the positive direction and NFD[LVis in the negative direction. 3C-SiC: NΓ/ is in the
negative direction and NΓ; is in the positive direction.
0 0.5 1 1.5 2 2.5 3
0 0.5 1
time [ps]
| |2
4H-SiC 1
2 3
0 0.5 1 1.5 2 2.5 3
0 0.5 1
time [ps]
| |2
W GaN 1
2 3
Fig 2. Time evolution of the probability to find the hole in different bands along the c-axis close to the Γ-point. The electric field is 10 kV/cm.
0 0.1 0.2 0.3 0.4 0.5
0 0.5 1
time [ps]
|α |2
3C-SiC
α
1α
2α
30 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 0.5 1
time [ps]
|α |2
4H-SiC
α
1α
2α
3Fig 3. Time evolution of the probability to find the hole in different bands for a drift in 3C-SiC (along the Γ-L segment) and 4H-SiC (along the c-axis). The
electric field is 40 kV/cm.
0.05 0 0.05
0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0
Normalized kvector [2 /a]
Energy [eV]
so
= 16 meV
so
= 8 meV
so
= 4 meV
so
= 0.1 meV
Fig 4. Energy dispersion along the c-axis in 4H-SiC for different spin orbit split ∆so.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
time [ps]
| |2
so
= 16 meV
so
= 8 meV
so
= 4 meV
so
= 0.1 meV
Fig 5. Time evolution of the probability find the hole in band 1 during a drift along the c-axis in 4H-SiC for different spin orbit split ∆so.
Γ Γ Γ
