Heterostructures Handout

Heterostructures Handout - University of California,...

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University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor Heterostructures Handout.fm Heterostructures and sub-bands (Read Datta 6.1, 6.2; Davies 4.1-4.5) Quantum Wells In a quantum well, electrons are confined in one of three dimensions to exist within a region of length L z . If the barriers at have infinite height and the electrons are free in the x, y directions then We still have freely propagating plane waves in the x-y directions with the energies given by: This gives us a series of 1D sub-bands (indexed by p ), each with a 2D dispersion relation zL z ± = x y z L z k z p π L z ---- = p 123 ,,, = φ xy , A , ik x xk y y + () [] exp = k , 2 π n , L , --------------- = E p h _ 2 2 m ------- k x 2 k y 2 k z 2 ++ π 2 h _ 2 p 2 2 mL z 2 ---------------- h _ 2 2 m k x 2 k y 2 + + == k x,y E p=1 p=2 p=3
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- 69 - University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor The sub-band energies are each shifted up by with (with effective mass m * ). 2D density of states Where the sub-bands overlap, the density of states add. Quantum wires In this case we have 2D confinement and free propagation in 1D Now we have 2D sub-bands with a 1D dispersion. The 1D density of states is: ε z p 2 ε z h _ 2 π 2 2 mL z 2 ------------- m 0 m ------ 10 nm L z ⎝⎠ ⎛⎞ 3.8meV == 14 9 1 2 3 D(E)/D 0 E ε z L z x y z L y E np , n 2 ε y p 2 ε z + () h _ 2 2 m ------- k x 2 + =
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- 70 - University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor Quantum dots 3D confinement Density of states is a delta function Other confining potentials Square well, finite depth. This is the actual situation for heterostructured quantum wells (more on this later). In this case, we have a finite number of sub-bands. States above the top of the barrier merge into 3D states. The solution for the wavefunctions and energy states is given in many stan- dard textbooks. 14 9 1 2 3 D(E)/D 0 E ε z L x x y z L y E mnp ,, m 2 ε x n + 2 ε y p 2 ε z + =
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- 71 - University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor Infinite triangular well. This case is approximated in doped quantum wells and in a MOSFET channel. Here, the potential is given by: The Schrodinger equation for the z-dependent part of the wavefunction for this potential is:
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- 72 - University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor Define and Then, with , “Airy equation” The solutions are “Airy” functions, a special function related to Bessel functions of order 1/3. See Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, (Dover, 1972). The
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Heterostructures Handout - University of California,...

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