Physics of Semiconductors

Shingo Katsumoto Department of Physics and Institute for Solid State Physics

University of Tokyo

9th 2016.6.13

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Outline today

Answer to the question paused in the last week Heterojunction and quantum confinement to 2-dimensional systems Heterojunction connection rule Quantum well Quantum barrier Double barrier Resonant diode Superlattice Modulation doping

My question in the last week

0

0 V

J

Consider an ideal light emitting diode, which has no non-radiative recombination. Every injected carrier emits a photon with the energy 𝐸g. Now apply a voltage 𝑉1 < 𝐸g/𝑒 and a current 𝐽1 flows. The power of light emission is 𝑃L = 𝐸g𝐽1/𝑒 . 𝐸g

𝑒 𝑉1

𝐽1

On the other hand, the electric power source gives the power 𝑃S = 𝐽1𝑉1, which is smaller than 𝑃𝐿! Does the LED create energy? Or what is happening inside the LED?

An experiment

2.4 V: 0.517 µm Green!

Blue: 0.45 µm -> 2.76eV

pn junction as a heat pump E E

fc(E)

D(E)

Only carriers with high kinetic energies can diffuse into the other layer

Evaporation cooling occurs Environment heat bath

Electric power source pn junction Photon

Evaporation cooling of atoms

4 cm

Courtesy: Prof. Torii

Atoms in MOT

Magnetic trap

Zeemann splitting

rf

E

f

hν

Ch.3 Heterojunctions and quantum confinement to two-dimensional systems

Nobel prize for semiconductor heterostructure

Envelope function

Heterojunction and envelope function

Bloch type wavefuntion:

Lattice periodic function band structure

Plane wave Envelope function

Lattice Hamiltonian: Perturbation potential:

Bloch functions

Heterojunction and envelope function

Inverse Fourier transformation

Schrödinger equation with effective mass: Effective mass approximation

Heterojunction: difference in and normalize into step potential at the interface:

Anderson’s rule

R. L. Anderson, IBM J. Res. Dev. 4, 283 (1960).

II-VI, III-V, VI combinations

Lattice constant (Å)

Ener

gy g

ap (e

V)

GaN ZnO Graphene

Molecular beam epitaxy (MBE)

RHEED Substrate

Ga Al In

As Si

van del Waals heterostructure

A. K. Geim and I. V. Grigorieva Nature 499, 419 (2013).

Quantum well 𝑉0

−𝐿/2 𝐿/2 𝑥

𝑉(𝑥)

States localized inside the well: 𝐸 < 𝑉0

Quantum well

Continuous:

Differentiable:

Quantum well

Optical absorption in quantum well

Envelope function Lattice periodic function

𝐸g

Two dimensional density of states:

hh

lh

Optical absorption in quantum well

Quantum barrier 𝐴1(𝑘) 𝐴2(𝑘)

𝐵1(𝑘) 𝐵2(𝑘) 1 2

𝑄

𝑀𝑇

Transfer matrix: 𝑀𝑇

𝑀𝑇 for a barrier width 𝐿 height 𝑉0

Inside the barrier

Boundary condition:

Transfer matrix for a square barrier

t, r : complex transmission and reflection coefficients

Double barrier transmission

∵

Double barrier transmission

Resonant transmission

Double barrier conduction

Drain

Source

𝑒𝑉𝑠𝑠

heavy hole

light hole

𝐸/𝑉0

Tran

smis

sion

coe

ffici

ent

Double barrier conduction

𝑉𝑠𝑠

𝐼𝑠𝑠

Drain

Source

𝑒𝑉𝑠𝑠 z

𝑘𝑥

𝑘𝑦

𝑘𝑧

Double barrier and wave packet Resonant T =1

?

1. Immediately go through 2. Take some time and go through 3. Mostly be reflected by the potential 4. Others

Double barrier and wave packet

qu Quasi stationary

incoming

reflected

Semiconductor Superlattice

Raphael Tsu Leo Esaki

Bloch theorem

Eigenvalue 𝑒±𝑖𝑖𝑠

d

Kronig-Penny potential

: δ -function series potential

Bloch oscillation in solids

Cosine band:

Bloch oscillation

Formation of mini-bands

Experiment on Bloch oscillation

A

near infrared

THz

Y. Shimada et al. Phys. Rev. Lett. 90, 046806 (2003). N. Sekine et al. Phys. Rev. Lett. 94, 057408 (2005).

Stark ladder state

Experiment on Bloch oscillation

Modulation doping and 2-dimensional electrons

Electric field of sheet charge

Hartree potential

Modulation doping and 2-dimensional electrons

Step function

Schrödinger equation

Solve self-consistently

Approximations

Triangular potential

Airy function

Fang-Howard (variational approximation)

Electron mobility in MODFET

Exercise B-6-13

here is a GaAs (dielectric constant 13) 𝑝+𝑛 diode grown with molecular beam epitaxy. Doping is abrupt and uniform for both p and n layers. We have cut the grown film to a 1 mm2 area and measured the differential capacitance with applying the (negative) bias voltage 𝑉𝑏and obtained the results summarized in the table on the left. Obtain the built-in potential in unit of V. The measured 𝐶 contains some experimental errors. Assume that the capacitance is dominated by the doping in the n layer and obtain the donor concentration in the n layer in the unit of cm−3.

Submission deadline: 6/27