# Presentation: 197983

## Electron Correlation in Weakly Confining Spherical Quantum Dot Potentials

**PRESENTED BY **
**     **
**Peter Kimani**
**University of Nevada Reno**
**& IGT Reno**
**                       **

## Introduction

-      **Nanostructures**: Quantum Wells
-                                    Quantum Wires
-                                    Quantum Dots
-      **Applications**: Biological imaging, labeling
-                              Medical imaging, drug therapy
-                              Lasers
-                              Quantum Computing
-      **Motivation**:   Many applications dependent on
-                            electronic structure and properties

## Methods

**     ****Hartree-Fock (HF) Approximation**
-               Finite Oscillator Potential
-               Correlation – beyond HF
**     ****HF V****N-1****  Potential**
-      **Second-order Green’s Function**
-               Dyson Equation
-               Second Order
-      **Tamm-Dancoff Approximation (TDA)**
**              **2ph – TDA,
-               Extended 2ph - TDA

## Hartree-Fock (HF) Approximation 

-     Confining Potential:  Finite Oscillator Potential,
-        A, B chosen such that all dot electrons bound
-     Properties (appeal):
-                Approximates harmonic oscillator well near origin
-                Convenience - analytical calculation of matrix elements

## VN-1  Potential

- Occupied orbitals reproduced
- Virtual (unoccupied) orbitals calculated in field of N-1 electrons
-      without orbital relaxation
-      Reflect excited orbitals features (excited orbitals behavior not attainable with standard HF
- Can naturally extend to VN-2 and on

## Second-order Green’s Function

-          Dyson Equation,
-                   HF energies,
-                   Diagonal matrices:      ,
-                   Self Energy,
-            Second-order             matrix elements
-                                                              +  Exchange terms

## Tamm-Dancoff Approximation (TDA)  

-        Include more interactions
-        2ph – TDA
-               Includes one more interaction -> 3rd Order
-        Extended 2ph – TDA
-               Theory exact to third order of perturbation

## Atomic Test results

-          **Helium Atom ****-  1s****2**** **                                      (a.u.)
**          ****Beryllium Atom ****-  1s****2**** 2s****2**   ** **                        (a.u.)
**                          **

| Orbital | HF 
VN | 2nd Order
  GF VN | HF 
VN-1 | 2nd Order
 HF VN-1 | 2ph-TDA | Extended
2ph-TDA |
| --- | --- | --- | --- | --- | --- | --- |
| 1s | -0.9180 | -0.9041 | -0.9180 | -0.8940 | -0.8626 | -0.9058 |
| 2s | +0.0028 | +0.0028 | -0.1583 | -0.1622 |  |  |

| Orbital | HF 
VN | 2nd Order
  GF VN | HF 
VN-1 | 2nd Order
 HF VN-1 | 2ph-TDA | Extended
2ph-TDA |
| --- | --- | --- | --- | --- | --- | --- |
| 1s | -4.7331 | -4.6107 | -4.7331 | -4.5702 | -4.5444 | -4.5803 |
| 2s | -0.3092 | -0.3276 | -0.3092 | -0.3522 | -0.2968 | -0.3471 |
| 3s | +0.0006 | +0.0006 | -0.0817 | -0.0999 |  |  |

## 2-e and 4-e QDs Results

**         ****I.1   2-Electron Dot  -  1s****2**                        (10-3 a.u.)
-            **I.2   4-Electron Dot  -  1s****2**** 2s****2**                 (10-3 a.u.)

| Orbital | HF 
VN | 2nd Order
  GF VN | HF 
VN-1 | 2nd Order
 HF VN-1 | 2ph-TDA | Extended
2ph-TDA |
| --- | --- | --- | --- | --- | --- | --- |
| 1s | -0.8708 | -0.8833 | -0.8708 | -0.8582 | -0.8500 | -0.8692 |
| 2s | -0.4295 | -0.4318 | -0.4295 | -0.4317 | -0.4213 | -0.4329 |
| 3s | +0.0011 | -0.0033 | -0.1295 | -0.1357 |  |  |
| 4s | +0.0381 | +0.0381 | +0.0285 | +0.0285 |  |  |

| Orbital | HF 
VN | 2nd Order
  GF VN | HF 
VN-1 | 2nd Order
 HF VN-1 | 2ph-TDA | Extended
2ph-TDA |
| --- | --- | --- | --- | --- | --- | --- |
| 1s | -0.5411 | -0.5433 | -0.5411 | -0.5409 | -0.5233 | -0.5419 |
| 2s | +0.0048 | +0.0010 | -0.1888 | -0.1993 |  |  |
| 3s | +0.0193 | +0.0193 | +0.0097 | +0.0097 |  |  |

## 8-e and 10-e QDs Results

-          **I.3    8-Electron Dot  -  1s****2**** 2p****6**                          (10-3 a.u.)
**        ****I.4    10-Electron Dot  -  1s****2**** 2p****6** **2s****2**                 (10-3 a.u.)

| Orbital | HF 
VN | 2nd Order
  GF VN | HF 
VN-1 | 2nd Order
 HF VN-1 | 2ph-TDA | Extended
2ph-TDA |
| --- | --- | --- | --- | --- | --- | --- |
| 1s | -0.5552 | -0.5538 |  |  |  |  |
| 2s | -0.0111 | -0.0043 |  |  |  |  |
| 2p | -0.3179 | -0.3109 |  |  |  |  |
| 3p | +0.0614 | +0.0614 |  |  |  |  |

| Orbital | HF 
VN | 2nd Order  GF VN | HF 
VN-1 | 2nd Order HF VN-1 | 2ph-TDA | Extended
2ph-TDA |
| --- | --- | --- | --- | --- | --- | --- |
| 1s | -0.6678 | -0.6752 | -0.6678 | -0.6784 |  |  |
| 2s | -0.2604 | -0.2728 | -0.2604 | -0.2810 |  |  |
| 3s | 1.3054 | 1.2844 | 0.9886 | 0.9474 |  |  |
| 2p | -0.4487 | -0.4488 | -0.4487 | -0.4473 |  |  |
| 3p | 0.0017 | -0.0008 | -0.1750 | -0.1760 |  |  |

## 18-e and 20-e QDs Results

-           **I.5    18-Electron Dot  -  1s****2**** 2p****6** **3d****10****    **                 (10-3 a.u.)
-           **I.6    20-Electron Dot  -  1s****2**** 2p****6** **2d****10**** 2s****2**** **              (10-3 a.u.)

| Orbital | HF 
VN | 2nd Order
  GF VN | HF 
VN-1 | 2nd Order
 HF VN-1 | 2ph-TDA | Extended
2ph-TDA |
| --- | --- | --- | --- | --- | --- | --- |
| 1s | -0.5536 | -0.5509 |  |  |  |  |
| 2s | +0.0061 | -0.0029 |  |  |  |  |
| 2p | -0.3657 | -0.3587 |  |  |  |  |
| 3p | +0.1604 | +0.1595 |  |  |  |  |
| 3d | -0.1733 | -0.1719 |  |  |  |  |
| 4d | +0.1631 | +0.1630 |  |  |  |  |

| Orbital | HF 
VN | 2nd Order
  GF VN | HF 
VN-1 | 2nd Order
 HF VN-1 | 2ph-TDA | Extended
2ph-TDA |
| --- | --- | --- | --- | --- | --- | --- |
| 1s | -0.6423 | -0.6399 |  |  |  |  |
| 2s | -0.0773 | -0.866 |  |  |  |  |
| 3s | +0.2087 | +0.2021 |  |  |  |  |
| 2p | -0.4534 | -0.4462 |  |  |  |  |
| 3p | +0.0950 | +0.9418 |  |  |  |  |
| 3d | -0.2594 | -0.2579 |  |  |  |  |
| 4d | +0.1575 | +0.1572 |  |  |  |  |

## Conclusion

- Not much experimental data for comparison
- “Ballpark” comparison – level of agreement not so good, but encouraging
-     Few Electron Quantum Dots Ground state Energies (10-3 a.u.)
-     Approach  - merit, applicable to quantum dots

| Number of Electrons | Finite Oscillator Potential  | Muller & Koonin
 (1996  -  Parabolic) | Kim & Zubarev (2001  -  Parabolic) |
| --- | --- | --- | --- |
| 2 | -1.449 | -1.473 | -1.532 |
| 4 | -4.170 | -3.545 | -3.756 |
| 8 | -9.255 | -9.100 |  |
| 10 | -12.460 | -12.492 |  |
| 18 | -31.953 | -29.442 |  |
| 20 | -33.685 | -34.470 |  |

## Sample References

- [1]     B. Szafran, J. Adamowski, and S. Bednarek, “Electron-electron correlation in quantum dots,”
-          *Physica E*: **5**, 185-195 (2000).
- [2]      B. Szafran, J. Adamowski, and S. Bednarek, “Ground and excited states of a few-electron
-           systems in spherical quantum dots,”  *Physica E*: **4**, 1-10 (1999).
- [3]     Peter Winkler, *, *“Electron Interaction in Weakly Confining Quantum Dot Potentials,”* *
*         **Int. J. Quant. Chem. : ***100**, 1122-1130 (2004).
- [4]      Peter Winkler, “Use of VN-1 Potentials in Green’s Function Calculations,” *Int. J. Quant, Chem.*:* *
*          ***XXIII**, 1571-1577 (1983).
- [5]      O. Walter and J. Schirmer, “The two-particle-hole Tamm-Dancoff approximation (2ph-TDA)
-           for atoms,” *J. Phys. B*: *At. Mol. Phys. ***8**, 3805-3826 (1981).
- [6]      Yeong E. Kim  and Alexander L. Zubarev,  “Interacting electrons in quantum dots,”
-           *Phys. Lett. A: ***289**, 155-159 (2001).
- [7]    H. M. Mϋller and S. E. Koonin, “Phase transitions in quantum dots,” *Phys. Rev. B*: **56**,
-           14532-14539 (1996).
- [8]    Jimmie D Doll and William P. Reinhardt, “Many-Body Green’s Functions for Finite, Nonuniform
-           Systems: Applications to Closed Shell Atoms,” *J. Chem. Phys.*:*  ***57**, 1169-1184 (1972).