PHYSICS OF LOW-DIMENSIONAL SYSTEMS Problems e-e interactions-
2. Consider a 2D parabolic quantum dot with single particle Energy levels of 2 HO E(n,m)=w0 (n+m+1)
a)Draw energy spectrum, including s and p-shell, and add (1,1) state from the d-shell
b)Assign indices 1 to (0,0), 2 to (1,0) , 3 to (0,1), 4 to (1,1) levels
The Hamiltonian for N electrons in second quantization is written as
+
1
å< ij| Vee + + H =å Eiciscis + |kl >cisc js’cks’cls
is
2 ijklss’
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c) Write the Hartree-Fock state for
N=2 electrons with Sz=0 (one up one down)
Matrix elements in units of <00,00|V|00,00> are <00,11|V|11,00>=0.6875 <00,11|V|00,11>=0.1875
PHYSICS OF LOW-DIMENSIONAL SYSTEMS Problems excitons-Hubbard model topology Nov28-2014
1.Excitons – consider a 1D system with single particle
energy levels E(m)=w0*m where m=1,2,3,4 is angular momentum.
Hamiltonian is given as
+
1
å< ij| Vee + + H =å Emcmscms + |kl >cisc js’cks’cls
ms
2 ijklss’
U
With direct elements
1+ | m1- m2 |
exchange elements
U
1
+ 4 | m1- m2 |^2
a) Draw the single particle energy spectrum as a function of m.
PHYSICS OF LOW-DIMENSIONAL SYSTEMS Problems excitons-Hubbard model topology Nov28-2014
b) put 2 spin up and two spin down electrons into lowest energy configuration |GS>.Write explicitly this state in terms of creation operators. Calculate total energy and express it as a sum of
energy of quasiparticles. Calculate selfenergies and plot quasiparticle Energy spectrum for U=1, w0=1.
c) Consider singlet and triplet excitations from |GS> with delta m=2 and delta Sz=0 (no change of spin). Picture them as electrons and holes using your single particle energy spectrum. Demonstrate explicitly that they are indeed spin singlet and triplet states by using first quantization representation.
d)How many singlet and triplet excitons with delta m=2 do you find? e) Is there a bi-exciton possible?
PHYSICS OF LOW-DIMENSIONAL SYSTEMS Problems excitons-Hubbard model topology Nov28-2014
f) write the Hamiltonian in the space of singlet/triplet excitons only diagonalise it and determine your excitation spectrum.
g) Plot your excitation spectrum with increasing degree of accuracy
i) noninteracting,
ii) including self-energy,
iii) adding direct e-hole attraction and exchange interaction
iv) including interaction of several e-h pairs, i.e., correlations?
PHYSICS OF LOW-DIMENSIONAL SYSTEMS Problems excitons-Hubbard model topology Nov28-2014
2. Hubbardc model, e-e and topology
consider three quantum dots with one orbital per dot ,
energies of these orbitals Ei=0 and tunneling matrix elements tij=-t as in first problem. The triple quantum dot molecule is described by Hubbard Hamiltonian:
3
3
3
+
+
ni
H=åEiciscis +
åtijcisc js +åUini¯
s ,i =1
s ,i , j=1
i=1
j¹i
There are two electrons in this triple quantum dot molecule.
PHYSICS OF LOW-DIMENSIONAL SYSTEMS Problems excitons-Hubbard model topology Nov28-2014
2. Hubbardc model, e-e and topology
2a) write all possible 2 electron configurations with total spin Sz=+1 (both electrons have spin up). How many configurations do you have?
2b) evaluate Hubbard Hamiltonian in the space of these configurations. Write the Hamiltonian matrix explicitly and compare with one electron part of Hubbard Hamiltonian.
i) Is there a difference in the form of a phase factor?
ii) What happens when tunneling from dot 3 to dot 1 is forbidden ,i.e., t13=0?
