Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2....

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Physics of Semiconductors 7 th 2016.5.30 Shingo Katsumoto Institute for Solid State Physics, University of Tokyo

Transcript of Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2....

Page 1: Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum

Physics of Semiconductors 7th 2016.5.30

Shingo Katsumoto Institute for Solid State Physics,

University of Tokyo

Page 2: Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum

Syllabus

1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum structures Quantum wells, wires and dots 4. Coherent quantum transport Landauer-Buttiker formalism Interference devices 5. Single-electron effects Charges and spins in quantum dots 6. Quantum Hall effect 7. Spin physics, spintronics, topological insulators

Page 3: Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum

Lecture notes http://kats.issp.u-tokyo.ac.jp/kats/

http://kats.issp.u-tokyo.ac.jp/kats/semicon3/

Page 4: Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum

Outline today

Classical Transport Boltzmann equation Drift current, diffusion current Drude formula, Einstein relation Electromagnetic effect (Hall effect) Heat transport Thermal conductivity Thermoelectric effect Transport in pn junctions Thermal equilibrium Current-voltage characteristics Response to illumination (minority carrier injection)

Page 5: Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum

Classical transport: Boltzmann equation (1)

𝒓

𝒑

(𝒓,𝒑) 6-dimensional phase space

Distribution function 𝑓(𝒓,𝒑, 𝑑)

𝑑𝒓

𝑑𝒑

Page 6: Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum

Classical transport: Boltzmann equation (2)

Boltzmann equation:

Collision term

Relaxation time approximation:

β‰ˆ 0 (stable state) around thermal equilibrium

Expansion to the first order of dt

Relaxation time

Page 7: Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum

Currents: Particle flows(fluxes)

𝒑: Anisotropic distribution = Current

Diffusion current Drift current

Drift current:

Page 8: Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum

Drift current for Fermi-degenerated system

𝑓(π‘˜)

π‘˜

π‘˜π‘₯

π‘˜π‘¦

βˆ’π‘˜πΉ

Page 9: Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum

Drude formula

Maxwell distribution: 𝑓0 β‰ˆ 𝐴exp βˆ’πΈ/π‘˜π΅π‘‡

: Drude formula for metals

: Drude formula

Page 10: Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum

Diffusion current, Einstein relation

Relaxation time approximation:

: Diffusion constant

Einstein relation

Page 11: Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum

Heat transport, thermoelectric effect Heat flux density:

Thermal conductivity:

Seebeck effect:

A B B

A B B

: Seebeck coefficient

Page 12: Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum

Heat transport, thermoelectric effect (2)

A B

Peltier effect J J

Electric current J : continuous

Heating at A-B interface QAB

Heat flux Q : discontinuous

: Peltier coefficient

Thomson effect

:Thomson coefficient

A J J x

Material specific

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Kelvin (Thomson) relations

A B B

𝑇 𝑇 + Δ𝑇

Quasi-static

π‘‡π‘š π‘‡π‘š

𝑉𝐴𝐡

:First law

:Second law

: Kelvin relations

Unit charge

Page 14: Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum

Seebeck coefficient as material constant

Material specific

Δ𝑇 A

B

Thermocouple

Page 15: Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum

Boltzmann equation and thermoelectric constants

Replace with 𝑓0

Page 16: Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum

Boltzmann equation and thermoelectric constants (2)

𝑗π‘₯ = 0 Drift current Diffusion current :balance

Page 17: Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum

Peltier device

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Ch.2 Transport in pn junctions

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Transport in pn junctions

Equilibrium

pn junction : spatially non-uniform

Diffusion current: Entropy increase

Drift current: Internal energy decrease

Balance: Minimize Free energy

Page 20: Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum

pn junction thermodynamics

Consider electrons

+ +

+ + +

donors

eβˆ’

eβˆ’

eβˆ’ eβˆ’

eβˆ’

Vacuum for electrons

diffusion

βˆ’ βˆ’ βˆ’ βˆ’ βˆ’

+ + + + +

voltage (polarization) β†’ energy cost

𝐹 = π‘ˆ βˆ’ 𝑇𝑇

Voltage (internal energy cost) Diffusion (entropy)

Minimization of 𝐹 β†’ Built-in (diffusion) voltage 𝑉𝑏𝑏

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Built-in potential

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Einstein relation

mobility

Rigid band model:

Page 23: Physics of Semiconductors 7th 2016.5...Syllabus 1. Classical transport, Transport in pn junctions 2. Junction transistors, field effect transistors 3. Hetero-junctions and quantum

Current-Voltage characteristics equilibrium

External voltage V

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Current-Voltage characteristics (2)

𝑛 π‘₯ = 𝑁𝑐exp βˆ’πΈπ‘ π‘₯ βˆ’ πœ‡π‘’(π‘₯)

π‘˜B𝑇

πœ‡π‘’ π‘₯ = 𝐸𝑐 π‘₯ + π‘˜B𝑇ln𝑛(π‘₯)𝑁𝑐

quasi-Fermi level

Diffusion equation

Minority carrier diffusion length

𝐿𝑒 = π·π‘’πœπ‘’ , πΏβ„Ž = π·β„Žπœπ‘’

generation

𝑛𝑝 π‘₯ = 𝛿𝑛0expπ‘₯ + 𝑀𝑝𝐿𝑒

+ 𝑛𝑝0

𝐸F𝑛 βˆ’ 𝐸F𝑝 = 𝑒𝑉

𝑗 𝑉 β‰ˆ 𝑒𝑛𝑏2𝐷𝑒𝐿𝑒𝑁𝐴

+π·β„ŽπΏβ„Žπ‘π·

expπ‘’π‘‰π‘˜B𝑇

βˆ’ 1

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Response to illumination

G(x) =G constant

𝑛𝑝 π‘₯ = 𝛿𝑛0expπ‘₯ + 𝑀𝑝𝐿𝑒

+ 𝑛𝑝0 + πΊπœπ‘’

𝛿𝑛0 = 𝑛𝑝0 expπ‘’π‘‰π‘˜π΅π‘‡

βˆ’ 1 βˆ’ πΊπœπ‘’

𝑗 = 𝑗0 expπ‘’π‘‰π‘˜π΅π‘‡

βˆ’ 1 βˆ’ 𝑒𝐺 𝐿𝑒 + πΏβ„Ž

jm Short circuit current

Vm

fill factor 𝐹𝐹 = π‘‰π‘š π‘—π‘šπ‘‰oc 𝑗sc

βˆ’ +

Majority carrier β†’ ignore increase in density Minority carrier β†’ huge increase in density

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Spin Seebeck effect

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K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa and E. Saitoh, Nature 455, 778 (2008).

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Spin Seebeck effect

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