Realistic confinement potential for a square-patterned two-dimensional semiconductor quantum dot and its approximated circular counterpart

Confinement potential due to a square patterned two-dimensional semiconductor quantum dot

Depending on the placement of the gate potentials in a 2D semiconductor quantum dot, electrons might get confined in a non-circular region with sharp geometric features. In this work, we consider the area depleted of electrons to be a uniformly positively charged square. The confinement potential felt by electrons with charge, − q ₀ ( q ₀ >0) in presence of a uniformly charged finite square (the depleted region) with length, L , and total positive charge, Q , is:

where V ( x , y )is the electrostatic potential created by a the uniformly charged square at the point r→=(x,y) on the 2D plane. To calculate such a potential, we choose the origin of the system of coordinates at the center of the square. This implies that the positive charge is all found in the domain D:−L2≤x′≤+L2;−L2≤y′≤+L2 . The uniform charge density of such distribution is written as ρ ₀ = Q / L ² . The total charge of the domain, Q , and the length, L , can both be tuned experimentally. The calculation of the electrostatic potential, V ( x , y ), is simplified using the transformation:

The remaining steps are more straightforward with the final result for the potential written in integral form as:

where k is Coulomb’s electric constant, X = x / L , Y = y / L , and t = u L are dimensionless variables. The function f ( t , X )(as well as f ( t , Y )) is given by the following expression:

where erf(z)=2/Π∫0zdxe−x2 is the error function[ ²⁹ ]. The above expression is extremely useful since it allows us to calculate the electrostatic potential, thus the resulting confining energy, at any location on the 2D quantum-dot region. In dimensionless units, the confining potential, U ( X , Y )=− q ₀ V ( X , Y ), can be written as:

In Figure ¹ , we plot the confining potential as a function of the dimensionless distance, X2+Y2 , from the center of the square ( X =0, Y =0) for two chosen directions, U ( X , Y =0)and U ( X , Y = X ), along the square. For the same distance from the center of the square, we verified that the strongest confinement is achieved along the Y = X direction. The overall strongest confinement energy felt by electrons was found to be:

and corresponds to electrons placed at the center of the uniformly charged square.

Figure 1

Approximated circular confinement potential

The confinement potential created by a uniformly charged square is complicated given the dependence of the potential on each of the coordinates, x and y , and not simply the distance, X2+Y2 , from the center of the quantum dot. Thus, it looks very appealing to explore possibilities in which the realistic confinement potential for a square-patterned quantum dot is replaced with an approximated circularly symmetric one as done in this work. To this effect, we assess the quality of such an approximation and also discuss instances in which one can reliably use the approximated simplified circular potential instead of the computationally unyielding exact one. To gain back the circular symmetry lost in the square, let us consider a circular domain in the form of a disk having the same area as the original square region:

The idea is to compare the confinement potential created by a uniformly charged disk to the one created by a uniformly charged square under the constrain that the same total charge, Q , is uniformly distributed over two different domains with equal area. In this scenario, one has to calculate the resulting confining potential energy felt by an electron with charge, − q ₀ ( q ₀ >0), in a 2D uniformly charged disk. The calculation of such charged disk potential[ ³⁰ , ³¹ ] is lengthy with the final result:

where R is the radius of the disk, F ( r , R ) is a function that depends only on the ratio r / R (given in integral form), J _n ( x ) are n -th order Bessel functions, r=x2+y2 is the distance of the electron from the center of the disk, V ₀ =(2 k Q )/ R is the electrostatic potential created by the disk at its center ( r =0), and k is Coulomb’s electric constant. The uniformly charged circular-disk region with an area identical to the square is shown in Figure ² . The square domain with area L ² = Π R ² is shown in Figure ³ . Since the disk has the same area as the square, the radius of the disk is fixed to R=L/Π . As before, the most strongly confined electrons are those localized at the center of the disk ( r =0):

where the final value above was obtained after replacing R=L/Π . Compare the above approximated result in Equation ( ¹⁰ ) to the exact value, U ( x =0, y =0)≈−3.52549 k Q q ₀ / L from Equation ( ⁶ ). While such an initial result is encouraging, the overall quality of the approximation employed should be more thoroughly assessed by comparing the values of the exact confining potential, U ( X , Y ), at arbitrary points ( x , y )to the corresponding values obtained from the approximated circular confining potential, UD(r=x2+y2,R=L/Π) , at that location. For the sake of a direct comparison, as before, we measure the confinement energies in units of k Q q ₀ / L . After straightforward transformations, we rewrite:

where X = x / L and Y = y / L are the dimensionless coordinates suitable for the square case. In Figure ⁴ , we plot the circular-disk confining potential, U _D ( X , Y ), as a function of the dimensionless distance, X2+Y2 .

Figure 2

Figure 3

Figure 4

Based on simple arguments, we expect the approximated circular-disk potential to be a very good description of the exact non-circular potential in the central region of the domain and far away from the center of the domain. The most substantial differences are expected to occur at locations that correspond to edges such as sides of the square or corners of the square. We gauged the quality of U _D ( X , Y ) by comparing it to the exact square potentials, U ( X , Y =0)and U ( X , Y = X ), that represent the two boundary directions along the square’s surface. The data clearly show that the agreement between the exact confinement potential and its approximated circular counterpart is very good everywhere with more pronounced variations at the edges of the domain. In Figure ⁵ , we plot the realistic potentials, U ( X , Y =0) and U ( X , Y = X ), together with their approximated circularly symmetric disk confining potential, U _D ( X , Y ).

Figure 5