Effective mass theory of a two-dimensional quantum dot in the

presence of magnetic field

 

HIMANSHU  ASNANI1, RAGHU MAHAJAN2, PRAVEEN

PATHAK3 and VIJAY  A  SINGH3,*

1Electrical   Engineering Department, Indian Institute of Technology  Bombay,

Mumbai 400 076, India

2Department of Physics,     Massachusetts Institute of

Technology, 77 Massachusetts Avenue Cambridge, MA 02139-4307, USA

3Homi  Bhabha Centre  for  Science Education  (TIFR),  V.N. Purav  Marg,

Mankhurd,  Mumbai 400 088, India 

*Corresponding author

E-mail: ksingh@mailhost.tifr.res.in; praveen@hbcse.tifr.res.in

 

Abstract. The effective  mass of electrons in low-dimensional

semiconductors is position-dependent.   The standard kinetic energy

operator of quantum mechanics for this position-dependent mass is

non-Hermitian and needs to be modified. This is  achieved by

imposing the BenDaniel--Duke (BDD) boundary condition.   We have

investigated the role of  this boundary condition for semiconductor

quantum dots (QDs) in one,  two and three dimensions. In these

systems the effective mass $m_{\rm i}$  inside the dot of size $R$

is different from the  mass $m_{\rm o}$ outside.   Hence a crucial

factor in determining   the   electronic  spectrum   is  the   mass

discontinuity  factor  $\beta=m_{\rm i}/m_{\rm o}$.~We  have

proposed a novel quantum   scale,  $\sigma$,   which  is   a

dimensionless parameter proportional  to $\beta^2R^2V_0$, where

$V_0$ represents the barrier height.   We  show  both  by  numerical

calculations and  asymptotic analysis that the ground state  energy

and the surface charge density, ($\rho(R)$), can be large and

dependent on $\sigma$.  We also show that the dependence  of the

ground state energy  on the size of  the dot is infraquadratic.  We

also study the  system in the presence of magnetic field $B$.   The

BDD condition introduces a  magnetic length-dependent term

($\sqrt{\hbar/eB}$) into  $\sigma$  and hence  the ground  state

energy.   We demonstrate  that the  significance of  BDD  condition

is pronounced at large $R$ and large magnetic fields. In many cases

the results using the BDD condition is significantly different from

the non-Hermitian treatment of the problem.

 

Keywords. Effective mass theory; BenDaniel--Duke; quantum dot;

electron; magnetic field.

 

PACS Nos 73.21.La;  73.21.-b;  85.75.-d