Physics of Semiconductors

Transcription

1 Physics of Semiconductors 9 th Shingo Katsumoto Department of Physics and Institute for Solid State Physics University of Tokyo

2 Site for uploading answer sheet

3 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

4 My question in the last week J Consider an ideal light emitting diode, which has no non-radiative recombination. Every injected carrier emits a photon with the energy E g. Now apply a voltage V 1 < E g /e and a current J 1 flows. The power of light emission is P L = E g J 1 /e. On the other hand, the electric power source gives the power P S = J 1 V 1, which is smaller than P L! Does the LED create energy? Or what is happening inside the LED? J V 1 E g e V

5 An experiment 2.4 V: µm Green! Blue: 0.45 µm -> 2.76eV

6 pn junction as a heat pump E E D(E) f c (E) Only carriers with high kinetic energies can diffuse into the other layer Environment heat bath Evaporation cooling occurs Electric power source pn junction Photon

7 Evaporation cooling of atoms Magnetic trap Atoms in MOT Zeemann splitting 4 cm f rf Courtesy: Prof. Torii hν E

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

9 Nobel prize for semiconductor heterostructure

10 Heterojunction and envelope function Bloch type wavefuntion: Lattice periodic function band structure Plane wave Envelope function Lattice Hamiltonian: Perturbation potential: Bloch functions Envelope function

11 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:

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

13 II-VI, III-V, VI combinations GaN ZnO Energy gap (ev) Graphene Lattice constant (Å)

14 Molecular beam epitaxy (MBE) RHEED Substrate Ga Al In As Si

15 van del Waals heterostructure A. K. Geim and I. V. Grigorieva Nature 499, 419 (2013).

16 Quantum well V 0 V(x) States localized inside the well: E < V 0 L/2 L/2 x

17 Quantum well Continuous: Differentiable:

18 Quantum well

19 Optical absorption in quantum well lh Envelope function Lattice periodic function E g Two dimensional density of states: hh

20 Optical absorption in quantum well

21 Quantum barrier A 1 (k) B 1 (k) 1 Q 2 M T A 2 (k) B 2 (k) Transfer matrix: M T M T for a barrier width L height V 0 Inside the barrier Boundary condition:

22 Transfer matrix for a square barrier t, r : complex transmission and reflection coefficients

23 Double barrier transmission

24 Double barrier transmission Resonant transmission

25 Double barrier conduction Source ev ss Drain Transmission coefficient light hole heavy hole E/V 0

26 Double barrier conduction Source ev ss Drain z I ss k z V ss k y k x

27 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

28 Double barrier and wave packet Quasi qu stationary reflected incoming

29 Semiconductor Superlattice d Raphael Tsu Leo Esaki Bloch theorem Eigenvalue e ±iii

30 Kronig-Penny potential : δ -function series potential

31 Bloch oscillation in solids Cosine band: Bloch oscillation

32 Formation of mini-bands

33 Experiment on Bloch oscillation Stark ladder state near infrared THz A Y. Shimada et al. Phys. Rev. Lett. 90, (2003). N. Sekine et al. Phys. Rev. Lett. 94, (2005).

34 Experiment on Bloch oscillation

35 Modulation doping and 2-dimensional electrons Hartree potential Electric field of sheet charge

36 Modulation doping and 2-dimensional electrons Step function Solve self-consistently Schrödinger equation

37 Approximations Airy function Triangular potential Fang-Howard (variational approximation)

38 Electron mobility in MODFET

39 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 mm 2 area and measured the differential capacitance with applying the (negative) bias voltage V b and obtained the results summarized in the table on the left. Submission deadline: 6/27 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.