Levchuk, Elena

Makarenko, Leonid

BSU Digital Library

Proceedings of the 2nd International Conference on Modern Applications of Nanotechnology 
Minsk, Belarus, 6-8 May 2015 
 
P113-1 
 
Modeling of Magnetic Field-Assisted Electron Manipulation 
in Nanogate-Donor System 
Elena Levchuk, Leonid Makarenko   
Department of Applied Mathematics and Computer Science, Belarusian State University, 
Independence Ave. 4, 220030, Minsk, Belarus 
liauchuk@bsu.by 
 
ABSTRACT: Numerical simulations of shallow donor states in external electric and 
magnetic fields near semiconductor surface are carried out. It is considered that the electric 
field is applied by a disc-shaped metallic gate and magnetic field is uniform and 
perpendicular to semiconductor interface. The stationary Schrödinger equation is solved 
using finite element method. The dependences of main characteristics of electron shuttling 
from the donor to the gate on the magnetic field are obtained for different gate diameters 
and donor positions. 
 
Keywords: energy level; wave function; modeling; donor; magnetic field 
 
1 INTRODUCTION 
Single atoms of dopant impurities in 
semiconductors under control of electric and 
magnetic fields are among the most promising 
physical systems for implementing a quantum 
computer (Kane, 1998). This problem has 
stimulated much theoretical studies and 
numerical simulations on the quantum states of 
donors localized near a semiconductor surface in 
a magnetic field (Li et al., 2013, Bruno-Alfonso 
et al., 2010, Calderon et al., 2007). In some 
applications, such as a Si-based quantum 
computer, electronic states are to be controled 
by small gates with nanometer sizes. However, 
in previous works the system has been 
considered without external electric field  (Li et 
al., 2013, Bruno-Alfonso et al., 2010) or under 
uniform electric field (Calderon et al., 2007). So, 
in this work, we assume a disc-shaped gate as a 
source of the external electric filed, and study 
characteristics of electron shuttling from the 
donor to the gate in the presence of uniform 
magnetic field.  
 
2 FORMULATION OF THE PROBLEM 
We consider a donor, positioned at the 
distance z0 from semiconductor surface where 
the disc-shaped gate with diameter d is situated. 
The gate is separated from the semiconductor by 
an infinitely thin and infinitely high dielectric 
barrier. A uniform magnetic field B is applied 
perpendicular to the semiconductor surface. 
Then, the stationary Schrödinger equation for 
the electron wave function Ψ and energy E in 
cylindrical coordinates is given by 
Hˆ E        (1) 
with the Hamiltonian operator 
2 2
ˆ ˆ ˆ ˆ ,
4
D GH T i V V
 



    

 (2) 
2 2
2 2 2
1 1ˆ ,T
z

    
    
    
    
 (3) 
where * 2 2( ) / Ba   with /B eB    the 
magnetic length and a
*
 effective Bohr radius,  
ˆ
DV  is the potential electron energy in the field of 
the donor centre, 0
ˆ ( , )GV f z  describes the 
Proceedings of the 2nd International Conference on Modern Applications of Nanotechnology 
Minsk, Belarus, 6-8 May 2015 
 
 
 
P113-2 
potential landscape due to the gate, 0  is the 
gate potential. Two kinds of boundary 
conditions for the field applied by the gate are 
used. Firstly, we consider that the gate is 
surrounded by a dielectric material (BC-A). 
Secondly, we assume that the gate is surrounded 
by a metal, at which the potential equals zero 
(BC-B). 
Energy in equation (1) is expressed in the 
units of effective Rydberg (Ry
*
). Magnetic field 
µ is expressed in * 20 /( )a e   . 
 
3 NUMERICAL RESULTS 
The problem for the equation (1) has been 
solved using finite element method (FEM). 
Firstly, we consider the effect of the 
magnetic field on the structure of electron 
energy spectrum of near-gate states, i.e. when 
the donor is infinitely distant from the gate. The 
system in this case can be considered as a 
quantum dot induced by external electric field. 
Energies of ground and several lowest excited 
states are presented in Figure 1 (numerical data 
for states localized at near-donor region are 
taken from Makado and McGill, (1986)). 
0,0 0,5 1,0 1,5 2,0
-1,5
-1,0
-0,5
0,0
0,5
1,0
1,5
2s(0)
2p(0) 2(0)
2p(-1)
1s(0)
1(0)
2(-1)
2(+1)
 
 
E
, 
R
y*
, 
0
2p(+1)
 
Figure 1: Energies of near-donor (dashed lines) 
and near-gate states (solid lines) as a function of 
magnetic field. The gate is surrounded with 
dielectric (BC-A) (d = 7a
*, Φ0 = 3.9Ry
*
/e)  
 
Two main characteristics of electron 
shuttling from the donor to the gate have been 
studied: critical gate potential at which the 
shuttling takes place and minimum energy gap 
(gmin) between the ground and first excited σ-
states, which can be used to estimate the 
tunnelling probability between these two states. 
As one can see from Figure 1, energies of 
all σ-states become higher for larger magnetic 
fields if d > a
*
. Energy level for 2π(–1) 
asymptotically approaches 1σ(0)-level for 
magnetic fields µ > 1, which does not happen for 
near-donor states 1s(0) and 2p(–1). The effect of 
the magnetic field on ground state energy is 
stronger for near-gate 1σ(0)-state than for near-
donor 1s(0)-state. This means that magnetic field 
increases the critical potential and can be used to 
tune this characteristic. 
Figure 2 demonstrates difference of the 
effect of the magnetic field for different gate 
diameters and boundary conditions. The effect 
of the magnetic field is stronger for the larger 
gate diameters and for the BC-A boundary 
conditions as in these cases wave function is less 
localized in ρ direction. Using variational 
method, it has been shown that the presence of 
the magnetic field can be described by 
multiplying trial wave function of near-gate state 
ground by the factor 2exp( ) , where   is 
variational parameter. 
0,0 0,5 1,0 1,5 2,0 2,5 3,0
-1,5
-1,0
-0,5
0,0
0,5
 
 
E
0
, 
R
y*
, 
0
1 - d = 9a
*
, 
0
 = 4.85, BC-A
2 - d = 20a
*
, 
0
 = 2.33, BC-A
1' - d = 6a
*
, 
0
 = 5.58, BC-B
2' - d = 9a
*
, 
0
 = 4.07, BC-B
3 - Isolated Donor
1'
2'
1
2
3
 
Figure 2: Ground state energy as a function of 
magnetic field for the electron in the field of 
isolated donor and in the field of the gate for 
different gate diameters and boundary conditions 
 
We also consider a uniform external 
electric field, which is a limiting case of 
Proceedings of the 2nd International Conference on Modern Applications of Nanotechnology 
Minsk, Belarus, 6-8 May 2015 
 
 
 
P113-3 
infinitely large gate diameter. The dependence 
of the critical potential on μ for both finite and 
infinite d can be approximated by a linear 
function: a + bμ. It has been found that the value 
of b remains almost the same for different d and 
z0. 
The dependence of gmin in uniform external 
field on μ has been found to reveal 
nonmonotonic behaviour: the value of gmin 
increases on ~20% for μ < 0.2 and decreases for 
larger magnetic fields (Figure 3).  
0,0 0,5 1,0 1,5 2,0
0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
 
 
g
m
in
 /
 g
m
in
(0
)
, 
0
 z
0
 = 10a
*
, Uniform
 z
0
 = 8a
*
, Uniform
 z
0
 = 8a
*
, d = 6a
*
 z
0
 = 8a
*
, d = 12a
*
 
Figure 3: The dependence of the minimum gap 
on magnetic field for uniform external field and 
finite gate diameter. The value of the gap is 
normalized on the value of the gap on µ = 0. 
Calculations are carried out for BC-A boundary 
conditions 
At the same time, the minimum gap 
decreases for all values of the magnetic field 
when the gate diameter is finite. This fact 
can be explained as the result of less wave 
function overlapping for higher magnetic 
fields. 
 
4 CONCLUSION 
We have studied the effect of the magnetic 
field on donor electron energy spectrum in 
external electric field using the results of finite 
element calculations. It has been found that the 
effect of the magnetic field on the gate-induced 
electronic states is different as compared to 
electronic states hydrogenic donor. 
It has been determined that the use of an 
external magnetic field leads to increasing of the 
critical potential. The effect of the magnetic 
field on the minimum gap between ground and 
first excited states depends on the gate size. If 
the gate diameter is finite, the minimum gap 
gradually gets smaller for larger magnetic fields. 
If the gate diameter is large enough (i.e. the 
electric field can be considered as uniform), the 
dependence of the minimum gap on magnetic 
field is nonmonotonic. 
The results of the calculations can be used 
for the optimization of nanodevices that are 
based on the manipulation of shallow donor 
states by external fields. 
 
 
 
REFERENCES 
1. Bruno-Alfonso A., Candido L., Hai G.-Q. 
(2010). Two-dimensional electron states 
bound to an off-plane donor in a 
magnetic field. J. Phys.: Condens. 
Matter, 22, 125801. 
2. Calderon M.J., Koiller B., Das Sarma S. 
(2007). External field control of donor 
electron at the Si/SiO2 interface. Phys. 
Rev. B, 75,125311. 
3. Kane B.E. (1998). A silicon-based nuclear 
spin quantum computer. Nature, 393, 
133–137. 
4. Li B. et al. (2013). Effect of a 
perpendicular magnetic field on the 
shallow donor states near a 
semiconductor-metal interface. Phys. 
Rev. B, 87, 075313. 
5. Makado P.C., McGill N.C. (1986). 
Energy levels of a neutral hydrogen-like 
system in a constant magnetic field of 
arbitrary strength. J. Phys. C.: Solid 
State Phys., 19, 873–885. 
 
 


Modeling of Magnetic Field-Assisted Electron Manipulation in Nanogate-Donor System

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Modeling of Magnetic Field-Assisted Electron Manipulation in Nanogate-Donor System

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