# Tuning confined states and valley g-factors by quantum dot design in bilayer
graphene

###### Abstract

Electrostatically confined quantum dots in bilayer graphene have shown
potential as building blocks for quantum technologies. To operate the dots,
e.g., as qubits, a precise understanding and control of the confined states
and their properties is required. Here, we perform large-scale numerical
characterization of confined quantum states in bilayer graphene dots over an
extensive range of gate-tunable parameters such as the dot size, depth, shape,
and the bilayer graphene gap. We establish the dot states’ orbital degeneracy,
wave function distribution, and valley g-factor and provide the parametric
dependencies to achieve different regimes. We find that the dot states are
highly susceptible to gate-dependent confinement and material parameters,
enabling efficient tuning of confined states and valley g-factor modulation by
quantum dot design.

###### keywords:

Bilayer graphene, Quantum dots, g-factor modulation, Valleytronics

Dennis Mayer* Angelika Knothe

D. Mayer, Dr. A. Knothe
Address: Institut für Theoretische Physik, Universität Regensburg
Universitätsstraße 31, D-93053 Regensburg, Germany
Email Address<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
##  1 Introduction

There has been a long-standing interest in graphene-based nanostructures due
to carbon’s low spin-orbit and hyperfine coupling promising long coherence
times [1, 2, 3]. Recently, advanced fabrication and gating techniques have
enabled the experimental realisation of electrostatically confined
nanostructures in bilayer graphene, i.e., quantum channels [4, 5, 6, 7, 8],
dots [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28, 29], Josephson junctions, interferometers, and SQUIDS [30, 31, 32, 33,
34, 35], in an endeavor to design bilayer graphene-based nanostructures for
future quantum technology applications.

In these electrostatically defined structures, one uses a combination of
multiple gates (including a split gate and a crossing finger gate, c.f. Figure
1 a)) to locally modulate the bilayer graphene gap and the charge carrier
density in order to confine individual electrons. In such setups, the bilayer
graphene band structure, as well as the confinement potential, are
simultaneously tuned by the gates. This gate-tunability enables immense
control and the opportunity to manipulate the confined quantum states.
Therefore, a thorough understanding is required of how the confined states
depend on the gate-tunable parameters and their interplay in order to design
tailor-made bilayer graphene quantum nanostructures, e.g., quantum dots to be
operated as qubits.

We present a large-scale space exploration investigating the dependence of
confined bilayer graphene dot states on the gate-tunable dot and material
parameters. Previous studies of bilayer graphene quantum dots have provided
proof-of-principle calculations [36, 37] or descriptions of single points in
parameters space to characterize specific experiments [16, 15, 17, 14]. Here,
in turn, we offer the full parameter dependence, giving tangible indications
of how to design the confinement in order to tailor the dot states and their
properties to achieve different regimes and functionalities. By numerically
investigating bilayer graphene quantum dots over an extensive range of
differently shaped soft confinements, we study:

  * •

The confined states in circular dots, their orbital degeneracy, and the
corresponding wave functions. We find a regime where the orbital ground state
is singly degenerate, and the wave function is localized around the center of
the valley in momentum space, compared to a regime where the ground state
acquires a threefold degeneracy corresponding to three minivalleys around the
valley center. We give the parametric dependencies on the dot size and depth
and the bilayer graphene gap for achieving either of the regimes.

  * •

The confined states’ valley g-factor, which descends from the bilayer graphene
Bloch bands’ Berry curvature and determines the valley splitting of the
confined states in a weak perpendicular magnetic field. We establish the
dependencies of the valley g-factor on the dot parameters in either of the
before-mentioned regimes and thereby demonstrate how to tune the valley
splitting by quantum dot design, e.g., how to maximize or minimize the valley
g-factor.

  * •

How to modify the dot states and their properties via alterations of the dot
shape. We find that elliptical deformations of the confinement potential allow
one to tune further the wave function distribution, the orbital degeneracy,
and the valley g-factor of the confined states.

For the dependencies of the dot states’ characteristics on the confinement and
material properties, we perform statistical analysis to reveal the
correlations and provide a physical picture in terms of the underlying
confined quantum states. The insights above will be relevant for efficient
quantum dot design, e.g., for identifying suitable two-level regimes to
operate the dots as qubits. Being able to modulate the g-factor by quantum dot
design is also of interest for multi-dot setups with varying valley splitting
across the different dots, allowing, e.g., for valley valves and selective
state addressing.

Figure 1: a) A combination of gates (including split gates and a finger gate)
electrostatically confines charge carriers in bilayer graphene into a quantum
dot (top). The gate-induced confinement potential, $U(\mathbf{r})$, is smooth
and gate-tunable (center). The dominant source of state splitting in a weak
perpendicular magnetic field is valley splitting characterized by the valley
g-factor, $g_{v}$ (bottom). b) Scatterplots visualizing the statistical
correlations between the lowest-orbital dot state’s $g_{v}$ and the dot
parameters $L,U_{c}$, and the bilayer graphene gap $\Delta_{0}$. We
distinguish the valley center regime (when the lowest energy orbital is non-
degenerate and the dot wavefunction is peaked within the center of the valley
in momentum space, marked by blue dots) and the minivalley regime (lowest
orbital threefold degenerate with wavefunctions in the minivalleys, marked by
red stars). The examples in the bottom panel were computed using the
parameters $L$=20 nm, $\Delta_{0}$=60 meV, $U_{c}$=20 meV (valley center
regime) and $L$=80 nm, $\Delta_{0}$=150 meV, $U_{c}$=20 meV (minivalley
regime), respectively. c) Electronic and topological properties of the bilayer
graphene Bloch bands as a function of the gap. The bands are almost quadratic
around the Brillouin zone corners (valleys at the K-points) for small gaps,
and the Berry-curvature-induced orbital magnetic moment, $M$, exhibits strong
peaks and dips. At large gaps, three minivalleys develop around each K-point,
while the maximum and minimum value of M is suppressed. Plots for the $K^{-}$
valley where the $K^{+}$ valley is rotated by $\pi$ with the opposite sign of
$M$.

##  2 Microscopic modelling of confined dot states

We describe the gate-defined soft confinement by the spatially varying
confinement potential, $U(\mathbf{r})$, sketched in Figure 1 a), and band gap
profile, $\Delta(\mathbf{r})$,

$\displaystyle U(\mathbf{r})=U_{c}/$
$\displaystyle\mathrm{cosh}\left(\frac{\sqrt{(\frac{x}{a})^{2}+(\frac{y}{b})^{2}}}{L}\right),\,\mathrm{and}$
(1) $\displaystyle\Delta(\mathbf{r})=\Delta_{0}-\Delta_{mod}/$
$\displaystyle\mathrm{cosh}\left(\frac{\sqrt{(\frac{x}{a})^{2}+(\frac{y}{b})^{2}}}{L}\right),$
(2)

where the gate- and fabrication-defined parameters such as the dot depth,
$U_{c}$, width $L$, and ellipticity $\frac{a}{b}$, and the pristine bilayer
graphene gap away from the dot can be varied independently. Further,
$\Delta_{mod}=0.3\Delta_{0}$ describes the gate-modulation of the gap towards
the center of the dot. This specific value of the gate-induced gap modulation
has been chosen in accordance with recent experiments, and we show the
stability of our results with respect to slight variations in the appendix.

Spatially varying potentials of the form as in Equation (2) have proven
successful in providing faithful, realistic models of gate-defined quantum
dots in bilayer graphene in comparison with experiments [36, 16, 15, 17]. The
potentials enter into the four-band Hamiltonian of bilayer graphene [38, 39],

$\displaystyle\text{H}_{\xi}\\!=\\!\\!$
$\displaystyle\begin{pmatrix}U-\xi\frac{1}{2}\Delta&\xi v_{3}\pi&0&\xi
v\pi^{\dagger}\\\ \xi v_{3}\pi^{\dagger}&U+\xi\frac{1}{2}\Delta&\xi v\pi&0\\\
0&\xi v\pi^{\dagger}&U+\xi\frac{1}{2}\Delta&\gamma_{1}\\\ \xi
v\pi&0&\gamma_{1}&U-\xi\frac{1}{2}\Delta\end{pmatrix},$ (3)

with momenta $\pi=p_{x}+ip_{y},\,\pi^{\dagger}=p_{x}-ip_{y}$, and parameters
$v=1.02\cdot 10^{6}\text{ m/s}$, $v_{3}\approx 0.12v$, and $\gamma_{1}\approx
0.38\text{ eV}$. Equation (3) is written for the two valleys $K^{\xi}$ labeled
by the valley index $\xi=\pm 1$ and in the Bloch basis
$\psi_{K^{+}}=(\psi_{A},\psi_{B^{\prime}},\psi_{A^{\prime}},\psi_{B})$ in
valley $K^{+}$, and
$\psi_{K^{-}}=(\psi_{B^{\prime}},\psi_{A},\psi_{B},\psi_{A^{\prime}})$ in
valley $K^{-}$ (in terms of the electron’s amplitudes on the bilayer graphene
sublattices $A$ and $B$ in the top, and $A^{\prime}$ and $B^{\prime}$ in the
bottom layer).

In order to compute the confined dot states, we employ an established
numerical technique described in References [40, 36, 5, 6, 16, 41, 17], where
the Hamiltonian in Equation (3) is diagonalized in a suitable basis of
localized states. From this diagonalization, we obtain the quantum dot
spectrum, i.e., the eigenenergies of the $n$-th orbital dot level,
$E_{\xi,n}$, and the corresponding eigenstates, $\Psi_{\xi,n}$. In the absence
of any symmetry breaking, each orbital state is four-fold degenerate in the
spin- and valley degree of freedom.

One key property of these quantized dot states is their coupling to an
external magnetic field. Over the last few years, it has been shown that the
dominant source of state splitting in a perpendicular magnetic field is the
splitting of the two valleys $K^{+}$ and $K^{-}$. The corresponding valley
g-factor, $g_{v}$, quantifying this valley splitting in weak magnetic fields,
can be orders of magnitude larger than the spin g-factor [9, 6, 36, 15, 17].
The origin of this large valley splitting is the topological properties of
bilayer graphene. Its Bloch bands, $|\alpha\rangle$, feature non-trivial Berry
curvature,

$\Omega_{\xi,\alpha}=-2\hbar^{2}\text{Im}\sum_{\alpha\neq\beta}\frac{\langle\alpha|\partial_{p_{x}}H_{\xi}|\beta\rangle\langle\beta|\partial_{p_{y}}H_{\xi}|\alpha\rangle}{(\varepsilon_{\alpha}-\varepsilon_{\beta})^{2}},$
(4)

entailing an orbital magnetic moment, [42, 43, 44, 45, 46],

$M_{\xi,\alpha}=-e\hbar\text{Im}\sum_{\alpha\neq\beta}\frac{\langle\alpha|\partial_{p_{x}}H_{\xi}|\beta\rangle\langle\beta|\partial_{p_{y}}H_{\xi}|\alpha\rangle}{\varepsilon_{\alpha}-\varepsilon_{\beta}},$
(5)

where, for a band $\alpha$, we sum over the other three bands
$\beta\neq\alpha$ in Equation (3) (For the confined states in the following,
we consider states in the first conduction band, dropping the index $\alpha$).
The orbital magnetic moment, $M_{\xi}$, has opposite signs in the two valleys
$K^{+}$ and $K^{-}$ and couples linearly to weak perpendicular magnetic
fields, leading to sizeable magnetic field valley splittings.

Since lowest orbital confined states are zero angular momentum states having
both radial (principal) and azimuthal (angular momentum) quantum numbers close
to zero (akin to the Fock-Darwin levels [47, 48, 36]), for these states the
orbital magnetic moment of Equation (5) is the dominant quantity coupling to a
perpendicular magnetic field. We can hence quantify the valley splitting of
the lowest orbital confined dot states by integrating over the orbital
magnetic moment and the wave function distribution in momentum space, [6, 15,
36, 17]

$-\xi{g}_{v}=\frac{1}{\mu_{B}}\int
d\boldsymbol{k}M_{\xi}(\boldsymbol{k})|\Psi_{\xi,n=0}(\boldsymbol{k})|^{2},$
(6)

determining how much the states in either valley are pushed up and down in a
weak magnetic field. Equation (6) demonstrates the subtle interplay of
confinement and material properties as it depends on both the distribution of
the confined dot wave functions and bilayer graphenes’ orbital magnetic moment
in momentum space.

As the properties of the dot states depend so intricately on the material
characteristics of bilayer graphene as the host material, we demonstrate the
electronic and topological properties of pristine bilayer graphene in Figure 1
c) as described by Hamiltonian in Equation (3) in the absence of any
confinement potential and gap modulation, i.e., for $U\equiv 0$ and
$\Delta\equiv\Delta_{0}$. Of the four resulting bands, the two low-energy
bands are trigonally warped, forming three minivalleys around each of the
K-points. The depth and the separation of these minivalleys increase with an
increasing gap, such that the low-energy bands range from almost quadratic at
small gaps to strongly modulated at large gaps. The orbital magnetic moment M
inherits the trigonally warped triplet structure, with three peaks around the
minivalleys. Due to the inverted, hole-like part of the band around the
K-point, M exhibits a region of inverted sign around this area of momentum
space. Both the maximum value of both the peaks and the dips in M increase as
the bilayer graphene band gap decreases.

##  3 Tuning dot states and the valley g-factor

###  3.1 States in a circular dot

Figure 2: Evolution of the lowest orbital dot states’ valley g-factor,
$g_{v}$, over an extensive range of system parameters, $U_{c},L,$ and
$\Delta_{0}$. The dashed white lines indicate the transition from the valley-
center regime (regions of low $g_{v}$, blue) to the minivalley regime (regions
of high $g_{v}$, yellow). On the right, we show examples of the wave function
distribution in momentum space characterizing these regimes (from minivalleys
on the top to valley center on the bottom). We observe non-monotonic
dependencies of $g_{v}$ on the gap size, width, and gap, depending on the wave
functions momentum space distribution in the respective regimes. The filled
pink symbols illustrate the positions in the parameter space for the examples
in Figure 3 while the empty symbols relate to the dot states we elliptically
deform in Figure 4—all results for the $K^{-}$ valley.

We analyze the dot’s ground state and their valley g-factor over an extensive
range of gate-tunable parameters $\Delta_{0},U_{c}$, and $L$ in a circular
dot. Figure 1 b) depicts the statistical correlations between $g_{v}$ and the
gap, the dot depth, and the width. These correlations illustrate what
parametric regimes are most suitable when aiming at, e.g., minimizing or
maximizing the value of $g_{v}$. Furthermore, we visualize the correlation
between smaller/larger values of the valley g-factor and distinct momentum
space distribution patterns of the confined dot wave function. The confined
state’s distribution ranges between being peaked around the K-point (the
”valley-center” regime, which we characterize by non-zero weight at $K$ and a
finite separation of the lowest and the first excited single-particle orbital,
$\Delta E$) and sitting mainly in the three minivalleys around the K-point,
with zero weight at the valley center (the ”mini-valley” regime). In the
latter case, the lowest three orbitals corresponding to the three minivalleys
are degenerate.

In Figure 2, we further characterize these regimes by analyzing the wave
function’s evolution and the valley g-factor across characteristic system
parameter ranges. An overarching trend emerges, where the valley g-factor is
consistently maximal within the minivalley regime. However, the dependence of
$g_{v}$ on the system parameters is not always monotonic.

From Figures 1 and 2, we deduce that the most suitable parameters for the
valley center regime are small and deep dots, while large and shallow dots can
promote the minivalley regime for sufficiently large values of the bilayer
graphene gap.

Figure 3: Gap dependence of the orbital magnetic moment and the lowest orbital
confined dot wave functions in the valley center regime ($L$=30nm,
$U_{c}$=20meV) compared to the minivalley regime ($L$=130nm, $U_{c}$=20meV)
exemplified for a small gap (left, $\Delta_{0}$=80meV), and a large gap
(right, $\Delta_{0}$=120meV) in valley $K^{-}$. Dot wave functions confined in
the center of the valley are sensitive to the inverted part of the bilayer
graphene band structure and hence to the region of negative M. Wave functions
residing in the minivalleys pick up close to the maximal possible value of M
near the band edges. The pink symbols relate to the symbols in Figure 2.

Within these two regimes, the valley g-factor shows distinct dependence on the
gap: For fixed $L$ and $U_{c}$, the valley g-factor can decrease with
increasing $\Delta_{0}$ within the valley center regime, while it generally
increases strongly with an increasing gap in the minivalley regime (cf. the
evolution between the filled pink symbols in the top center panel of Figure
2). As we demonstrate in Figure 3, this non-monotonic dependence of $g_{v}$,
in particular on $\Delta_{0}$, stems from the influence of the hole-like part
of the dispersion around the valley center where $M$ becomes negative. For
small gaps, both the maximum peaks of M around the minivalleys and the minimum
dip in the center increase in magnitude (cf. Figure 1 c)), entailing a partial
cancellation when calculating $g_{v}$ for a wave function in the valley center
regime. Conversely, within the minivalley regime, increasing the bilayer
graphene gap promotes the minivalleys and further pushes the wave function
towards the band edges, where M is peaked. Since, in this regime, the wave
function has no weight around the center of the valley, the corresponding
confined states are obnoxious to any negative values of the orbital magnetic
moment.

As a consequence, while the maximal value of the orbital magnetic moment
generally is a decreasing function of the band gap (Figure 1 c)), the
dependence of a dot state’s $g_{v}$ is more subtle and depends on the
distribution of the dot wave function in momentum space. The valley g-factor
of the confined states in the mini-valley regime is closest to the maximal
value achievable from the band structure’s orbital magnetic moment.

The above discussion demonstrates how one may leverage the different dot
parameters to tune the confined dot wave function, and hence the state orbital
multiplicity and their magnetic field coupling via valley g-factor.

###  3.2 States in elliptical dots

One additional way to tune the wave function distribution and its symmetry is
to alter the shape of the confinement potential compared to a circular quantum
dot. Here, we study elliptical deformations since elliptical dot shapes are
ubiquitous in experimental studies and representative of breaking the
rotational symmetry along the two crystallographic directions of bilayer
graphene.

Figure 4 exemplifies how the confined states and their valley g-factors evolve
with the ellipticity of the dot for two characteristic parameter choices for
the valley center regime and the minivalley regime, respectively. Overall, we
observe an increase in the valley g-factor when the dot is deformed into an
ellipse. This increase occurs as the dot wave function is pushed either into
one of the minivalleys when the dot is deformed along the x-axis or into two
minivalleys for deformations along the y-axis. In addition, breaking the
threefold rotational symmetry between the three minivalleys also affects the
orbital degeneracy of the ground state: States peaked in one minivalley are
orbitally non-degenerate, while states peaked in two minivalleys exhibit a
two-fold orbital degeneracy.

We observe that the states in wide dots and for large gaps are most easily
deformed, while deep confinement makes deforming the states more difficult. As
a consequence, the states in the original minivalley regime, Figure 4 b), are
most susceptible to being deformed in momentum space. The states in the
original valley center regime, Figure 4 a), are more stable against
deformations. The relative deformation-induced increase of the valley
g-factor, however, is larger in the valley center regime as compared to the
original minivalley regime since states with little or no support in the
center of the valley do not suffer from cancellations due to the negative
regions of the orbital magnetic moment, $M$, cf. Figure 3.

The analysis above shows that realizing strongly elliptical bilayer graphene
quantum dots represents one possible route to prepare confined states with
large valley splittings at moderate gaps and dot sizes compared to the large
dots and gaps required for the full minivalley regime in circular dots, cf.
Figure 2.

Figure 4: Top: dependence of the lowest orbital confined state’s valley
g-factor, $g_{v}$, on the elliptic deformation of the confinement potential.
Here, the deformation parameters $a^{2}$ and $b^{2}$ quantify the elongation
along the $x$\- and $y$-axis, respectively, as prescribed by Equation (2). The
parameters are chosen such that states of the circular dot are a) in the
valley center regime ($L$=60 nm, $\Delta_{0}$=80 meV, $U_{c}$=5 meV), and b)
in the minivalley regime ($L$=80 nm, $\Delta_{0}$=150 meV, $U_{c}$=20 meV),
respectively. The empty symbols relate to the symbols shown in Figure 2.
Bottom: Examples for the corresponding dot spectra and the lowest orbital
confined state. All results for the $K^{-}$ valley.

##  4 Conclusion

Overall, we have presented the confined bilayer graphene dot states, their
spectra, and valley g-factors over an extensive range of gate-tunable
parameters, such as the confinement, width, gap, shape, and the bilayer
graphene gap. We find that the ground state can range from being non-
degenerate (and peaked in the center of the valley in momentum space) to
threefold degenerate (and sitting in the minivalleys) in circular dots and
peaked in one or two minivalleys (with corresponding single or two-fold
degeneracy) in elliptically deformed dots. Alongside the distribution of the
confined dot wave functions, the corresponding topological valley g-factor
changes in the different regimes as a function of the dot parameters and
shape. The largest valley g-factors are observed when the wave function is
peaked in the minivalleys, both in circular and elliptical dots. We note that
we expect many of our observations, as long as they are based on wave function
distributions, to generalize to higher dot levels with more complex wave
function shapes. However, for higher confined dot states with non-zero angular
momentum quantum numbers, the intrinsic angular magnetic moment needs to be
considered in conjunction with the topological orbital magnetic moment when
studying the coupling of the states to a magnetic field and the corresponding
valley g-factors [36].

Figure 5: a) Double-dot setup of two electrostatically confined quantum dots
defined by two split gates and two finger gates where the two dots have very
different size and geometry. b) The different size and shape of the left and
right quantum dots imply different valley g-factors and, hence, different
valley splittings in a perpendicular magnetic field. Whether the g-factor
decreases (for small gaps) or increases (for significant gaps) with dot size
(cf. Figure 2) may be detected in finite bias spectroscopy from the relative
sign and ratio of the finger gate voltages on the left and right dot,
$V_{FGL}$ and $V_{FGR}$, needed to re-establish tunnel transport when a weak
magnetic field is applied.

Our results demonstrate how the dot states and their properties can be
manipulated by virtue of gate-tunable parameters. Such an understanding is
required for tailoring specific, confined states, e.g., with the desired
degeneracy, by quantum dot design.

For example, our findings of valley g-factor manipulation by quantum dot size
may be exploited in multi-dot systems as the one sketched in Figure 5 a),
where different dot sizes and shapes imply different valley g-factors for the
states in the left and the right dot. Hence, valley g-factor modulation by
quantum dot design allows for controllably induced dot-dependent state
splittings in a constant perpendicular magnetic field, Figure 5 b). Similar to
setups of semiconductor quantum dots with slanted magnetic fields [49, 50,
51], device designs like the one in Figure 5 may enable, e.g., selective
blocking and transport of specific states and single-site valley control
without the need of an inhomogeneous B-field. We hence demonstrate how the
simultaneous tuning of material properties and confined quantum states offers
immense control over bilayer graphene quantum dots and holds great potential,
e.g., for future valley-logic and valleytronics applications.

Acknowledgements

We thank Samuel Möller, Max Ruckriegel, Eike Icking, Katrin Hecker, Christoph
Stampfer, Vladimir Fal’ko, and Christoph Schulz for fruitful discussions. We
acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - SFB 1277 (Project No. 314695032).

## Appendix: Data for extended parameters of the gap variation

In Figure 6, we demonstrate the stability of our results with respect to
variations of the parameter $\Delta_{mod}$ in the gap profile in Equation (2).
This parameter describes the modulation of the bilayer graphene gap underneath
the gates and can differ slightly between experiments, depending on the exact
device architecture and screening. While varying $\Delta_{mod}$ leads to
shifts of the exact quantitative values of the boundary between the minivalley
and valley center regime, see Figure 6, there are no qualitative changes. We
expect our results and conclusions to hold as long as the potential is smooth
and of the same symmetry.

Figure 6: Lowest orbital dot states’ valley g-factor over a range of the
system parameters $L$ and $\Delta_{0}$ for different values of the gap profile
parameter $\Delta_{mod}$. Here, $U_{c}=5$ meV for all three cases.

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