**5** **10** **15** **20** **25** **30** **B****H****Tesla****L**
**40**

**50**
**60**
**70**
**80**
**90**
**100**
**E****H****m eV****L**

**5** **10** **15** **20** **25** **30** **B****H****Tesla****L**

**-25**
**-20**
**-15**
**-10**
**-****5**
**Μ**_{mag}******Μ**_{B}

**T=4K**
**T=0.2K**

Figure 4.2: (a) Two electron energy spectrum including the multiplicity of triplet states due to Zeeman interaction, (b) Magnetization at T = 0.2K (blue) and 4K (red) are shown as a function of magnetic field strength.

while the dot confinement potential limits the extent to which they can be sep-
arated. Due to the inherent anti-symmetry in the orbital wavefunction of the
triplet states, f_{pc}(0) = 0 as two electrons cannot be in the same place in such
states. In contrast, for all singlet states, there exists a non-zero value for f_{pc}(0).

**5** **10** **15** **20** **r****H****nm****L**
**0.002**

**0.004**
**0.006**
**0.008**
**0.010**
**0.012**

**ED****Hnm**^{-2}**L**

**10** **20** **30** **40** **r****HnmL**

**0.0005**
**0.0010**
**0.0015**
**0.0020**

**CF****Hnm**^{-2}**L**

**(0, 0)**
**(0, 0)**

**5** **10** **15** **20** **r****HnmL**

**0.001**
**0.002**
**0.003**
**0.004**
**0.005**

**ED****H****nm**^{-2}**L**

**10** **20** **30** **40** **r****HnmL**

**0.0002**
**0.0004**
**0.0006**
**0.0008**
**0.0010**

**CF****Hnm**^{-2}**L**

**(-1, 1)**
**(-1, 1)**

**5** **10** **15** **20** **r****HnmL**

**0.001**
**0.002**
**0.003**
**0.004**
**0.005**

**ED****Hnm**^{-2}**L**

**10** **20** **30** **40** **r****H****nm****L**

**0.0002**
**0.0004**
**0.0006**
**0.0008**

**CF****Hnm**^{-2}**L**

**(-2, 0)** **(-2, 0)**

Figure 4.3: Left column: Radial electron density (ED),η(r, φ= 0); Right column:

Pair correlation function (CF), fpc(r, φ= 0). The continuous, dashed and dotted curves corresponds to B = 0, 10 and 20 Tesla respectively. The rows 1, 2 and 3 corresponds to quantum numbers (0,0), (−1,1) and (−2,0) respectively. All results are for an r0 = 15nm.

the free energy and determined the dependency of magnetization with respect to the magnetic field strength. Similarly, we have also plotted radial electron density and pair correlation functions for different quantum numbers and different mag- netic field strengths. We could convincingly obtain all the qualitative and even most of the quantitative features of the exact results obtained by F. M. Peeters et al. [106] where they considered around 1000 basis states for each combination of (M, s) quantum numbers. In comparison, we considered less than 50 basis states to obtain the results displayed. This shows that a very few low-energy harmonic oscillator eigenfunctions are able to effectively capture the quantum dot cylindri- cal step-potential. Our method also eliminates the requirement of finite element

analysis for obtaining one electron wavefunctions. However, our results are valid only under the assumption that the effective mass and permittivity of the dot are almost constant throughout the material. This may be taken as a very good approximation as the effective mass and permittivity changes are known to be very small with respect to the composition of the material.

## Two electrons in laterally coupled quantum dots

### 5.1 Introduction

Coupled quantum dots or artificial molecular hydrogen is an important prob- lem that attracted attention of many researchers during the past two decades.

Some have pursued this to characterize its physical properties, whereas oth- ers approached it from the application point of view. Coulomb blockade and conductance properties [110; 111], electronic structure and optical properties [112;113;114], electron-phonon interactions [115], Kondo resonant spectra [116], etc. are examples for some of the physical properties studied. On the other hand, application of these nanostructures include quantum gates [39], resonant tunnel- ing diode [117], preparation of entangled states [118], vertical cavity lasers [119], quantum cellular automata [120] etc. The coupling of quantum dots are done either laterally [121] or vertically [122]. The confinement potential for a coupled quantum dot could be created out of patterned gate electrodes [43], donor impu- rity placement [48] or by pure band-gap engineering as in self-assembled coupled quantum dots [123]. The advantage of a patterned electrode based potential is that we can electrically control the inter-dot distance as well as the number of electrons inside each dot.

The modeling of the confinement potential depends upon the method of con-

finement. For donor impurities, Coulomb potential is an ideal one. For self- assembled quantum dots, step potential may be a good approximation. In the case of self-assembled quantum dots, experimental results indicate a violation of the generalized Kohn theorem suggesting discontinuities in the confinement potential [124; 125]. The disadvantage of fully heterostructured coupled quan- tum dots is that we cannot manipulate their confinement potential for achieving control over inter-dot interaction. But control over the number of electrons in a heterostructred quantum dot, even to the extent of single electron charging was achieved by fabricating them inside a field effect transistor kind of environment [126]. We believe the same could be achieved in the case of coupled self-assembled QDs also. This opens the possibility of using such structures for building quan- tum gates. The advantage of employing such structures over patterned gate is that we can achieve higher number of those basic units in a given area of two- dimensional semiconductor surface. This is due to the fact that the patterned gate based QDs have radial dimensions of the order of 100nm, whereas the self- assembled QDs can be fabricated in less than 10nm radius. The disadvantage of inability to control inter-dot interaction can be overcome by relying fully on an orthogonal external magnetic field as a handle. For simultaneous control of many quantum gate units, we require a nano-scale manipulation of magnetic induction, which is also a part of present day research [127; 128].

In this chapter, we study two electrons in laterally coupled heterostructured quantum dots with cylindrically symmetric step potential as well as coupled donor impurities in the presence of a magnetic field. We investigate the possibility of controlling exchange interaction in these structures by varying the strength of magnetic field. In Section 5.2, we present the theoretical discussion for the analysis of coupled quantum dots using methods from molecular physics. The results thus obtained such as the variation of exchange interaction with respect to the magnetic field and inter-dot separation etc. are presented and discussed in Section 5.3. Finally the chapter is concluded in Section 5.5.