# Theory and Procedure

In the following analysis we consider a quantum dot based on GaAs/Ga1−xAlxAs heterostructure, but assume that the electron effective mass remains constant across the heterogeneous boundary. This is a reasonable approximation for a potential step of size 100 meV, as it amounts to only a 15% difference in the effective mass for the Ga1−xAlxAs outer region in comparison to that of the inner GaAs region [98]. As the the electron densities for the lower energy states are negligible in the outer region, this difference does not contribute much to our problem. Further, we assume a much stronger confinement in the axial direction of the dot relative to that in its radial direction, thus justifying our two dimen- sional model for the analysis [106]. Under these assumptions, the Hamiltonian of a two electron cylindrical quantum dot system in a constant magnetic field B can be written as

(4.1) H = X

i=1,2

1 2me

~

pi+q ~Ai2

+V (ri) + gµB

¯

h S~i·B~

+ q2

κ|~r1 −r~2|

where A~i = B2 (0, ri,0) and

V (ri) = 0 ; ri ≤r0

V0 ; ri > r0 .

Here~ri and ~pi are the conjugate position and momentum of the ith electron, me is the electron effective mass, q is the electron charge, g is the electron g-factor, µB is the Bohr magneton, S~i is the spin angular momentum of the ith electron, and κ= 4π, in SI units. Here, the value of quantum dot permittivity,, is given an average value of 13.10 throughout the material. In the operator form Eq.(4.1) becomes

(4.2) Hˆ =

2

X

i=1

− ¯h2 2me

2

∂ri2 + 1 ri

∂ri + 1 ri2

2

∂φ2i

+ ωC 2

zi +1

2meωC 2

2

ri2 +V (ri) + ¯hωL

2 σˆz(i)

+ q2

κ|r~1−r~2| where ˆLzi = −i¯h∂φ

i, the z-component of the angular momentum operator of the ith electron, ωC = mqB

e, the cyclotron frequency, and ωL = mqB

0, the Larmor frequency.

By adding and subtracting a harmonic oscillator potential 12meω02ri2 for each electron, we can re-write the above Hamiltonian as

Hˆ = ˆH0(1,2) + ˆH0(1,2) (4.3) The definitions of ˆH0(1,2) and ˆH0(1,2) are as given below.

0(1,2) = ˆHho(1) + ˆHho(2) (4.4) where

(4.5) Hˆho(i) = − ¯h2

2me2

∂r2i + 1 ri

∂ri + 1 ri2

2

∂φ2i

C 2

zi+ 1

2meω2ri2 with ω =

q

ω02+ ω2C2

. Similarly,

0(1,2) = ˆW(1,2) + ˆZ(1,2) + ˆC(1,2) (4.6)

where

Wˆ (1,2) =

2

X

i=1

V (ri)−1

2meω02r2i

, the residue potential term

Zˆ(1,2) = hω¯ L

2

2

X

i=1

ˆ

σ(i)z , the Zeeman term and Cˆ(1,2) = q2

κ|~r1−~r2|, the Coulomb term.

The eigenvalues and eigenfunctions of ˆHhoare well known [26] and are as given below.

ho ϕn,m(r, φ) = En,m ϕn,m(r, φ) (4.7) wheren = 0,1,2. . .and m= 0,±1,±2. . .are the radial and azimuthal quantum numbers respectively, and

En,m = ¯hω(2n+|m|+1) + ¯hωCm

2 (4.8)

ϕn,m(r, φ) = 1 a1+|m|B |m|!

r(|m|+n) ! πn! exp

−r2

2a2B +imφ

r|m|1F1[−n,|m|+1, r2 a2B]

(4.9) where aB =q

¯ h

meω is the effective Bohr radius of the quantum dot.

Now, in a linear variational theory [107], we consider a trial wavefunction expressed as

|ψi=c11i+c22i+· · ·+ciii+· · ·+cddi (4.10) Here d is the dimension of the basis set, coefficients c1, c2, . . . , cd are variational parameters and|χiiare the orthonormalized two-electron states. If s1 and s2 are the spin quantum numbers of individual electrons (each having a value 12), and if s is the total spin quantum number of the two-electron system, then by angular momentum addition rule s can take only two values viz. s = 0 corresponding to

|s1−s2|, ands= 1 corresponding to|s1+s2|. Fors= 0 state, thez-component of the total spin operator, ˆSz, can have only one value for its quantum number viz.

ms= 0 (singlet), and for the s= 1 state, it can take three values viz. ms = 0,±1 (triplet). The orbital part of the singlet state must be symmetric and that of the triplet states must be antisymmetric. These can be easily constructed out of one- electron wavefunctions by taking Slater permanent and determinant respectively.

For example, if|ϕn1,m1iand|ϕn2,m2iare any two distinct one-electron eigen states, then

χ±n1,m1,n2,m2(1,2)

= |ϕn1,m1(1)i |ϕn2,m2(2)i ± |ϕn2,m2(1)i |ϕn1,m1(2)i

√2 (4.11)

is a valid orthonormalized symmetric (antisymmetric) two-electron orbital wave- function constructed out of them, except for the case (n1, m1) = (n2, m2) where, an additional factor of √

2 must be taken care of. This, multiplied by their appro- priate spin-wavefunction counterpart (|si), forms the required basis wavefunctions in Eq. (4.10), given by

ii= χ±i

|s = 0(1)i (4.12)

Since h

H,ˆ Lˆzi

= 0, the total z-component of the angular momentum, M = m1+m2, must be the same for all terms in the trial wavefuntion. Similarly, since hH,ˆ Sˆ2

i

=

hH,ˆ Sˆz

i

= 0, every term in the trial wavefunction must have the same value for the s and ms quantum numbers. Thus, we can do variational analysis for each combinations of M, sand ms values separately. For each case, the linear variational analysis reduces to solving an eigen value problem of the type

[H]{C}=E{C}. (4.13)

Here [H] is a square matrix of size d×d with elements Hij =D χi

Hˆ χjE

, {C}

is a column matrix of size d×1 with unknown coefficients ci of Eq. (4.10) as elements, and E is their corresponding eigen energy. The matrix [H] can be written as a sum of matrices [H0] and [H0] corresponding to the operators ˆH0 and ˆH0 defined via Eq. (4.3). The elements of [H0] for the case of singlet (+)

and triplet (−) states are given by Eq. (4.14). Due to the orthogonality of one- electron wavefunctions|ϕn,mi, it may be noticed that [H0] has non-zero elements only along its diagonal.

Hij0 = D

χ±n1,m1,n2,m2

ho(1) + ˆHho(2)

χ±n3,m3,n4,m4 E

n1n3δm1m3δn2n4δm2m4(En3,m3 +En4,m4)

(4.14)

Similarly, the elements of [H0] are also evaluated. In the case of triplet states, the result after substituting for |χi from Eq. (4.11) becomes

(4.15) Hij0 =

D

ϕn1,m1(1)ϕn2,m2(2)

0(1,2)

ϕn3,m3(1)ϕn4,m4(2) E

−D

ϕn1,m1(1)ϕn2,m2(2)

0(1,2)

ϕn4,m4(1)ϕn3,m3(2)E

A little care must be given while evaluating the elements of [H0] for singlet states.

This is because, we can build symmetric orbital states out one-electron eigenstates even when n1 =n2 and m1 =m2, and those states are different from the general symmetric states given by Eq. (4.11). For the general states,

(4.16) Hij0 =D

ϕn1,m1(1)ϕn2,m2(2)

0(1,2)

ϕn3,m3(1)ϕn4,m4(2)E +D

ϕn1,m1(1)ϕn2,m2(2)

0(1,2)

ϕn4,m4(1)ϕn3,m3(2)E

When n1 =n2 and m1 =m2 and n3 =n4 and m3 =m4, the elements of [H0] for singlet states becomes

(4.17) Hij0 =

D

ϕn1,m1(1)ϕn1,m1(2)

0(1,2)

ϕn3,m3(1)ϕn3,m3(2) E

The final case is when quantum numbers are equal only on one side, say n1 =n2 and m1 =m2, the elements of [H0] for singlet states are given by

(4.18) Hij0 =√

2D

ϕn1,m1(1)ϕn1,m1(2)

0(1,2)

ϕn3,m3(1)ϕn4,m4(2)E

All the elements of [H0] were found out through numerical integration in the Mathematica software. To simplify the case, we can split and write [H0] as a sum of three matrices corresponding to the terms defined in Eq. (4.6) as

[H0] = [W] + [Z] + [C] (4.19)

Now, the matrix [Z] is diagonal due to the orthonormality of orbital part of the basis wavefunctions. Furthermore, these diagonal elements are also zero for ms = 0 states

|↑↓i±|↓↑i 2

. But for ms = ±1 states (|↑↑i,|↓↓i), it give rises to a constant value, ±¯hωL for all the diagonal elements. Similarly, many of the elements in [W] are zeros due to the othogonality condition when m1 6= m3 or m2 6= m4. Again, because of the azimuthal and exchange symmetries of the Wˆ (1,2) term, all the non-zero [W] elements require only one–variable numerical integrations. The evaluation of elements in the [C] matrix involves calculation of terms like

C12,34=D

ϕn1,m1(1)ϕn2,m2(2)

Cˆ(1,2)

ϕn3,m3(1)ϕn4,m4(2)E

. (4.20)

If we substitute ϕn,m(r, φ) = ρn,m(r)eimφ

in Eq. (4.20) we get,

(4.21) C12,34= q2

2πκ Z

r1=0

r1dr1ρn1,m1(r1n3,m3(r1) Z

r2=0

r2dr2ρn

2,m2(r2n4,m4(r2)I(r1, r2) where

I(r1, r2) = Z

φ=0

1 Z

φ=0

2 exp[i(−m1+m31+i(−m2+m42]

|~r1−~r2| . Using the multipole expansion of |~r 1

1−~r2| and simplifying further,I(r1, r2) becomes (4.22) I(r1, r2) = 4π2 1

r>

X

l=0

(r<

r>)l(l−m1+m3)!

(l+m1−m3)!(Plm1−m3(0))2

where,r>=r1 andr<=r2whenr1 > r2and vice versa, andPlm is the associated Legendre function. Substituting this expression forI(r1, r2) in Eq. (4.21), we get

C12,34= 2πq2 κ

X

l=|m1−m3|

(l−m1+m3)!

(l+m1 −m3)!(Plm1−m3(0))2 Z

r1=0

r1dr1ρn1,m1(r1n3,m3(r1)

"

Z r1

r2=0

dr2ρn2,m2(r2n4,m4(r2) r2

r1 l+1

+ Z

r2=r1

dr2ρn2,m2(r2n4,m4(r2) r1

r2 l#

(4.23) Thus we have replaced the original four-variable integration given in Eq. (4.20) with an infinite sum of two-variable integrations. But since the result is pretty accurate with only first few (say 30) terms of Eq. (4.23), this new formula helps us to achieve fast computation of the elements in [C].

Once all the elements of [H] are evaluated, we proceed to solve the eigen value problem of Eq. (4.13) which will then give us the energy spectrum and corresponding electron wavefunctions. We have done this by using the subrou- tine Eigensystem available in the mathematica software. This process is done separately for each value of quantum numbers (M, s) and it is then repeated for values of B ranging from 0 to 30 Tesla. The electron density, η(r), and pair cor- relation function, fpc(~r), are then evaluated using these resultant wavefunctions, following the definitions mentioned in reference [106] as given below

η(~r) =

2

X

i=1

hδ(~r−r~i)i fpc(~r) =hδ(~r−r~1+r~2)i

(4.24)

Outline

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