Physics of Semiconductors 7th 2016.5.30

Shingo Katsumoto Institute for Solid State Physics,

University of Tokyo

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

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

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

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)

Classical transport: Boltzmann equation (1)

𝒓

𝒑

(𝒓,𝒑) 6-dimensional phase space

Distribution function 𝑓(𝒓,𝒑, 𝑡)

𝑑𝒓

𝑑𝒑

Classical transport: Boltzmann equation (2)

Boltzmann equation:

Collision term

Relaxation time approximation:

≈ 0 (stable state) around thermal equilibrium

Expansion to the first order of dt

Relaxation time

Currents: Particle flows(fluxes)

𝒑: Anisotropic distribution = Current

Diffusion current Drift current

Drift current:

Drift current for Fermi-degenerated system

𝑓(𝑘)

𝑘

𝑘𝑥

𝑘𝑦

−𝑘𝐹

Drude formula

Maxwell distribution: 𝑓0 ≈ 𝐴exp −𝐸/𝑘𝐵𝑇

: Drude formula for metals

: Drude formula

Diffusion current, Einstein relation

Relaxation time approximation:

: Diffusion constant

Einstein relation

Heat transport, thermoelectric effect Heat flux density:

Thermal conductivity:

Seebeck effect:

A B B

A B B

: Seebeck coefficient

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

Kelvin (Thomson) relations

A B B

𝑇 𝑇 + Δ𝑇

Quasi-static

𝑇𝑚 𝑇𝑚

𝑉𝐴𝐵

:First law

:Second law

: Kelvin relations

Unit charge

Seebeck coefficient as material constant

Material specific

Δ𝑇 A

B

Thermocouple

Boltzmann equation and thermoelectric constants

Replace with 𝑓0

Boltzmann equation and thermoelectric constants (2)

𝑗𝑥 = 0 Drift current Diffusion current :balance

Peltier device

Ch.2 Transport in pn junctions

Transport in pn junctions

Equilibrium

pn junction : spatially non-uniform

Diffusion current: Entropy increase

Drift current: Internal energy decrease

Balance: Minimize Free energy

pn junction thermodynamics

Consider electrons

+ +

+ + +

donors

e−

e−

e− e−

e−

Vacuum for electrons

diffusion

− − − − −

+ + + + +

voltage (polarization) → energy cost

𝐹 = 𝑈 − 𝑇𝑇

Voltage (internal energy cost) Diffusion (entropy)

Minimization of 𝐹 → Built-in (diffusion) voltage 𝑉𝑏𝑏

Built-in potential

Einstein relation

mobility

Rigid band model:

Current-Voltage characteristics equilibrium

External voltage V

Current-Voltage characteristics (2)

𝑛 𝑥 = 𝑁𝑐exp −𝐸𝑐 𝑥 − 𝜇𝑒(𝑥)

𝑘B𝑇

𝜇𝑒 𝑥 = 𝐸𝑐 𝑥 + 𝑘B𝑇ln𝑛(𝑥)𝑁𝑐

quasi-Fermi level

Diffusion equation

Minority carrier diffusion length

𝐿𝑒 = 𝐷𝑒𝜏𝑒 , 𝐿ℎ = 𝐷ℎ𝜏𝑒

generation

𝑛𝑝 𝑥 = 𝛿𝑛0exp𝑥 + 𝑤𝑝𝐿𝑒

+ 𝑛𝑝0

𝐸F𝑛 − 𝐸F𝑝 = 𝑒𝑉

𝑗 𝑉 ≈ 𝑒𝑛𝑏2𝐷𝑒𝐿𝑒𝑁𝐴

+𝐷ℎ𝐿ℎ𝑁𝐷

exp𝑒𝑉𝑘B𝑇

− 1

Response to illumination

G(x) =G constant

𝑛𝑝 𝑥 = 𝛿𝑛0exp𝑥 + 𝑤𝑝𝐿𝑒

+ 𝑛𝑝0 + 𝐺𝜏𝑒

𝛿𝑛0 = 𝑛𝑝0 exp𝑒𝑉𝑘𝐵𝑇

− 1 − 𝐺𝜏𝑒

𝑗 = 𝑗0 exp𝑒𝑉𝑘𝐵𝑇

− 1 − 𝑒𝐺 𝐿𝑒 + 𝐿ℎ

jm Short circuit current

Vm

fill factor 𝐹𝐹 = 𝑉𝑚 𝑗𝑚𝑉oc 𝑗sc

− +

Majority carrier → ignore increase in density Minority carrier → huge increase in density

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).

Spin Seebeck effect

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