Physics of Semiconductorskats.issp.u-tokyo.ac.jp/kats/semicon3/ppt/semicon-3.pdf · Exercise B-6-13...
Transcript of Physics of Semiconductorskats.issp.u-tokyo.ac.jp/kats/semicon3/ppt/semicon-3.pdf · Exercise B-6-13...
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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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Site for uploading answer sheet
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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
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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?
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An experiment
2.4 V: 0.517 µm Green!
Blue: 0.45 µm -> 2.76eV
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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
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Evaporation cooling of atoms
4 cm
Courtesy: Prof. Torii
Atoms in MOT
Magnetic trap
Zeemann splitting
rf
E
f
hν
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Ch.3 Heterojunctions and quantum confinement to two-dimensional systems
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Nobel prize for semiconductor heterostructure
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Envelope function
Heterojunction and envelope function
Bloch type wavefuntion:
Lattice periodic function band structure
Plane wave Envelope function
Lattice Hamiltonian: Perturbation potential:
Bloch functions
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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:
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Anderson’s rule
R. L. Anderson, IBM J. Res. Dev. 4, 283 (1960).
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II-VI, III-V, VI combinations
Lattice constant (Å)
Ener
gy g
ap (e
V)
GaN ZnO Graphene
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Molecular beam epitaxy (MBE)
RHEED Substrate
Ga Al In
As Si
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van del Waals heterostructure
A. K. Geim and I. V. Grigorieva Nature 499, 419 (2013).
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Quantum well 𝑉0
−𝐿/2 𝐿/2 𝑥
𝑉(𝑥)
States localized inside the well: 𝐸 < 𝑉0
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Quantum well
Continuous:
Differentiable:
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Quantum well
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Optical absorption in quantum well
Envelope function Lattice periodic function
𝐸g
Two dimensional density of states:
hh
lh
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Optical absorption in quantum well
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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:
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Transfer matrix for a square barrier
t, r : complex transmission and reflection coefficients
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Double barrier transmission
∵
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Double barrier transmission
Resonant transmission
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Double barrier conduction
Drain
Source
𝑒𝑉𝑠𝑠
heavy hole
light hole
𝐸/𝑉0
Tran
smis
sion
coe
ffici
ent
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Double barrier conduction
𝑉𝑠𝑠
𝐼𝑠𝑠
Drain
Source
𝑒𝑉𝑠𝑠 z
𝑘𝑥
𝑘𝑦
𝑘𝑧
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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
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Double barrier and wave packet
qu Quasi stationary
incoming
reflected
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Semiconductor Superlattice
Raphael Tsu Leo Esaki
Bloch theorem
Eigenvalue 𝑒±𝑖𝑖𝑠
d
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Kronig-Penny potential
: δ -function series potential
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Bloch oscillation in solids
Cosine band:
Bloch oscillation
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Formation of mini-bands
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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
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Experiment on Bloch oscillation
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Modulation doping and 2-dimensional electrons
Electric field of sheet charge
Hartree potential
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Modulation doping and 2-dimensional electrons
Step function
Schrödinger equation
Solve self-consistently
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Approximations
Triangular potential
Airy function
Fang-Howard (variational approximation)
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Electron mobility in MODFET
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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