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Simulating the Energy Spectrum of Quantum Dots

Simulating the Energy Spectrum of Quantum Dots. J. Planelles. PROBLEM 1 . Calculate the electron energy spectrum of a 1D GaAs/AlGaAs QD as a function of the size. Hint: consider GaAs effective mass all over the structure. L. AlGaAs. AlGaAs. 0.25 eV. GaAs. m* GaAs = 0.05 m 0.

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Simulating the Energy Spectrum of Quantum Dots

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  1. Simulating the Energy Spectrum of Quantum Dots J. Planelles

  2. PROBLEM 1. Calculate the electron energy spectrum of a 1D GaAs/AlGaAs QD as a function of the size. Hint: consider GaAs effective mass all over the structure. L AlGaAs AlGaAs 0.25 eV GaAs m*GaAs = 0.05 m0

  3. Let us use atomic units (ħ=m0=e=1) The single-band effective mass equation: Lb Lb L 0.25 eV V(x) BC: f(0)=0 f(Lt)=0 m*GaAs = 0.05 m0

  4. fi+1 fi fi-1 i-1 i i+1 h h Numerical integration of the differential equation: finite differences Discretization grid f1 f2 ... fn ... xn x1 x2 How do we approximate the derivatives at each point? Step of the grid

  5.  h 0 Lt i=n i=1 i=2 FINITE DIFFERENCES METHOD 1. Define discretization grid 2. Discretize the equation: 3. Group coefficients of fwd/center/bwd points

  6. We now have a standard diagonalization problem (dim n-2): 0 0 i=n i=1 i=2 Trivial eqs: f1 = 0, fn = 0. Extreme eqs: Matriz (n-2) x (n-2) - sparse

  7. The result should look like this: finite wall infinite wall n=4 n=3 n=2 n=1

  8. PROBLEM 1 – Additional questions • Compare the converged energies with those of the particle-in-the-box with infinite walls for the n=1,2,3 states. • b) Use the routine plotwf.m to visualize the 3 lowest eigenstates for L=15 nm, Lb=10 nm. What is different from the infinite wall eigenstates?

  9. PROBLEM 2. Calculate the electron energy spectrum of two coupled QDs as a function of their separation S. Plot the two lowest states for S=1 nm and S=10 nm. L=10 nm L=10 nm AlGaAs AlGaAs S GaAs GaAs x

  10. The result should look like this: bonding antibonding bonding antibonding

  11. PROBLEM 3. Calculate the electron energy spectrum of N=20 coupled QDs as a function of their separation S. Plot the charge density of the n=1,2 and n=21,22 states for S=1 nm and L=5 nm.

  12. The result should look like this (numerical instabilities aside):

  13. n=1 n=21 n=2 n=22

  14. ρ  PROBLEM 4. Write a code to calculate the energies of an electron in a 2D cylindrical quantum ring with inner radius Rin and outer radius Rout, subject to an axial magnetic field B. Calculate the energies as a function of B=0-20 T for a structure with (Rin,Rout)=(0,30) nm –i.e. a quantum disk- and for (3,30) nm –a quantum ring-. Lb=10 nm. Discuss the role of the linear and quadratic magnetic terms in each case. B V(ρ,Φ) GaAs AlGaAs

  15. Hint 1: after integrating Φ, the Hamiltonian reads (atomic units): with Mz =0, ±1, ±2...the angular momentum z-projection. 1 atomic unit of magnetic field = 235054 Tesla. Hint 2: describe the radial potential as AlGaAs Rout Lb where ρ=0 is the center of the ring Rin GaAs ρ f(Lt)=0 (i.e. fn=0) If Mz=0, then f’(0)=0 (i.e. f1=f2) If Mz≠0, then f(0)=0 (i.e. f1=0) Hint 3: use the following BC

  16. The results should look like this: Disk (Rin=0) Ring (Rin=3 nm)

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