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Modelling Electron States in Silicon-Based Quantum Computers
Dr. Brendon Lovett
University of St. Andrews
In experimental research towards quantum computation, planar silicon devices are made comprised of a metal
gate structure over an insulating oxide layer above a silicon substrate. Voltages are applied to the gates to control
the potential in the silicon and create quantum dots to conﬁne electrons, the spin states of which encode quantum
information to form qubits.
Working with Dr. Brendon Lovett’s Theory of Quantum Nanomaterials group, this project involved theoretical work
complementing these experimental developments: a simulation was produced to ﬂexibly model such devices. Using a
self-consistent method based on a Schr¨odinger-Poisson solver, the wavefunctions of two electrons in quantum dots are
solved for and the interaction size between them determined. Results are compared to experimental measurements and
used to gauge viability of exploiting interactions between electrons in speciﬁc systems for use in quantum computation.
The project objective was to computationally model elec-
tronic states in silicon, allowing calculation of physical
interaction parameters useful in the implementation of
silicon-based quantum computation. Rrequiring micro-
scopic system analysis, the electron wavefunctions had to
be calculated before determination of coupling parameters.
Initially, it was thought computational modelling would
be best achieved using the simulation package NEMO3D.
It was also considered that the exchange coupling would be
the parameter focussed on as manipulation of silicon-based
quantum data is generally theorised to be performed using
this interaction between qubits.
Before the start of the project, a crude Python program
which solved for the potential from a speciﬁed gate struc-
ture then solved for an electronic wavefunction in that re-
gion was shared by collaborators of the research group.
Rather than NEMO3D, this formed the basis of the simu-
lation as it provided greater ﬂexibility in program design.
It also allowed full understanding of program operation,
as opposed to more ‘black box’ packages. While increas-
ing initial diﬃculty of writing the program, this choice also
aided learning and allowed more low-level control.
This code inspired the ﬁnal simulation, with sections
being used and rewritten to provide its starting point.
Quantum mechanical calculations were performed using
the Python package Kwant.
While the project concerns theoretical work, the results
of the simulation can be compared to experimental mea-
surements and used to help design systems for particu-
lar applications. To learn more about the implementation
of quantum computation, the project involved a visit to
University College London to meet Prof. John Morton’s
Quantum Spin Dynamics group. This honed simulation
development to be compatible with current devices under
researched and allowed a feel for which system parameters
should be focussed on to generate a realistic model. The
visit also conﬁrmed the theoretical parameters the experi-
mentalists need access to.
Maturing the Simulation
Following this successful start, the simulation was built up
to the point where the wavefunctions of two interacting
electrons in a silicon substrate could be determined in two
The desired gate structure is deﬁned by creating a GDSII
which the simulation reads. All further settings, such
as solving region boundaries, are set in a conﬁguration ﬁle.
The voltages applied to gates are set, and the potential
due to this gate structure is solved for in a parallel two-
dimensional electron gas at a set distance below the gate
structure, this being the plane in which the rest of the sim-
ulation works. This initial potential landscape is eﬃciently
found using an analytical solution of Poisson’s equation.
A square tight-binding lattice is set up in a region where
an electron would experience conﬁnement, generally a po-
tential well artiﬁcially introduced by the gate structure po-
tentials to form a quantum dot (see ﬁgure 1.). This allows
the Hamiltonian to be found, and thus Schr¨odinger’s Equa-
tion can be solved in this region to ﬁnd the wavefunction
of the ﬁrst electron.
This process is repeated in a separate region in the po-
tential landscape to yield the wavefunction of the second
electron, with the diﬀerence that this solution includes the
Coulomb repulsion exerted on this electron from the ﬁrst.
The simulation then returns to the ﬁrst electron and
solves for its wavefunction under the electrostatic interac-
tion of the second, then repeats this for the second electron
with the new electrostatic interaction of the ﬁrst electron.
This process forms the second step in an iterative calcula-
tion, which continues to cycle until the eigenvalues of the
systems converge to a set accuracy. The results of this
calculation are the self-consistent wavefunctions of the two
electrons (see ﬁgure 2).
During the course of the project, ongoing research specu-
lated that a more robust method of qubits interaction to
exploit than exchange interaction was the quadrupole in-
teraction. A post-doc in the group, Giuseppe Pica, was in-
volved in this research. With his collaboration, the project
focus shifted from calculating the exchange coupling to in-
vestigating the quadrupole interaction, EQ, between the
two electrons, given by:
where α and β deﬁne system axes, V is an externally ap-
plied electric ﬁeld gradient and Q is the quadrupole mo-
ment of the system.
1A database ﬁle format used for data exchange of integrated circuit
layouts in industry.
Figure 1: Potential landscape at a depth of 50nm from
two square gates of side length 2nm set 100nm apart with
applied voltages of -1V and -2V respectively. This produces
two conﬁnement regions, on which square solving regions
have been centred, with the tight-binding lattices shown.
(a) Region 0 (b) Region 1
Figure 2: Self-consistent probability densities of the two
electrons. The simulation ﬁrst solves for the wavefunction
of one electron in isolation. Then it solves for the sec-
ond including the Coulomb interaction from the ﬁrst, and
proceeds to alternate this step between the electrons until
convergence in the eigenenergies is reached.
Exchange energy is a quantum mechanical eﬀect arising
from the requirement of a two electron wavefunction to be
antisymmetric under particle exchange (because electrons
are fermions). As this exchange symmetry concerns the
product of the spatial and spin parts of the wavefunction,
diﬀerent spatial conﬁgurations - with diﬀerent Coulomb
interactions - and spin states aﬀect it. The exchange cou-
pling therefore depends on the overlap of the wavefunctions
of the two electrons, which are exponentially decaying in
the overlap region and thus the parameter is highly sensi-
tive to changes in potential. The quadrupole interaction,
however, does not depend on wavefunction overlap. Early
research indicates that it should therefore be more robust
as a method of manipulating quantum information through
the interaction of physical qubits. 
Finally, the wavefunction results are used to calculate the
quadrupole moment of the system under application of an
electric ﬁeld gradient, allowing calculation of quadrupole
interaction energy between the electrons.
The results of simulations run at the end of the project us-
ing realistic system parameters correlate with current re-
search into the quadrupole interaction in quantum com-
putation, suggesting as a preliminary outcome that the
quadrupole interaction is viable for use in manipulation
of quantum information. This presents a positive outlook
towards its future use in quantum computation.
With the project concluding on this promising result, the
group and collaborators will continue investigation into im-
plementation of silicon-based quantum computers and the
use of quadrupole interaction in quantum computation.
Moreover, the completed simulation will see more use in
this research as a valuable tool for the involved scientiﬁc
There is also potential for others to continue this project
with extensions such as expanding the simulation to func-
tion in three dimensions.
My personal experience of the project has been hugely re-
warding and motivating. My initial intent to follow my
undergraduate Master’s degree in Theoretical Physics with
a PhD has blossomed from an interested but uninformed
aim to a passionate and excited purpose.
Having the opportunity to immerse myself in the re-
search of quantum computation, and quantum mechanics
and solid state physics in general, was completely captivat-
ing. Already topics of great interest to me, my knowledge,
understanding and enthusiasm in them has explosively in-
creased, which will be extremely useful for future study as
these will remain core topics. Perhaps most thrillingly, I
worked on and contributed to cutting-edge research in the
forefront of these ﬁelds, producing academically useful and
I gained ﬁrst-hand research experience in both theo-
retical and experimental environments in diﬀerent lead-
ing UK universities and participated in various meetings
and talks. Furthermore, I worked with some not only bril-
liant, but also exceedingly encouraging and friendly people,
not to mention the outstanding mentorship of Dr. Brendon
Lovett. I have learnt and honed valuable skills while forg-
ing links I can return to, opening up exciting future op-
portunities. The project has given me a fantastic insight
into postgraduate academic life and professional research,
a career path I am eager to embark on.
 C. P. Slichter, “Electrical Quadrupole Eﬀects” in Prin-
ciples of Magnetic Resonance, vol. 1, Springer Se-
ries in Solid-State Sciences, 3rd Ed. Berlin, Germany:
Springer, 1989, ch. 10, sec. 2, pp. 486-489.
 P. A. Mortemousque et al., Quadrupole Shift of
Nuclear Magnetic Resonance of Donors in Sili-
con at Low Magnetic Field, arXiv:1506.04028v1
[cond-mat.mes-hall], Jul 2015.