Magnetism and correlations in fractionally filled degenerate shells of graphene quantum dots.

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Magnetism and correlations in fractionally filled degenerate shells of
graphene quantum dots
A. D. Gu¨c¸lu¨,1 P. Potasz,1, 2 O. Voznyy,1 M. Korkusinski,1 and P. Hawrylak1
1Institute for Microstructural Sciences, National Research Council of Canada , Ottawa, Canada
2Institute of Physics, Wroclaw University of Technology, Wroclaw, Poland
(Dated: August 11, 2009)
We investigate interaction effects and magnetization in triangular graphene quantum dots using
a combination of tight-binding, Hartree-Fock and configuration interaction methods. We show that
electronic correlations play a crucial role in determining the nature of the ground state as a function
of filling fraction of the degenerate shell at the Fermi level. A half-filled charge neutral shell leads
to full spin polarization but this magnetic moment can be completely destroyed by adding a single
electron.
Following the progress in the fabrication of graphene[1,
2, 3, 4, 5] based devices, lower dimensional structures
such as graphene ribbons[6, 7, 8, 9], and more recently
graphene quantum dots[10, 11, 12, 13, 14] are attract-
ing increasing attention due to their non-trivial electronic
and magnetic properties. In particular, it was shown that
when an electron is confined to a triangular atomic thick
layer of graphene with zig-zag edges, its energy spectrum
collapses to a shell of degenerate states at the Fermi level
(Dirac point)[15, 16, 17, 18, 19] isolated from remaining
states by a gap. The degeneracy is proportional to the
edge size and can be made macroscopic. This opens up
the possibility to design a strongly correlated electronic
system as a function of fractional filling of the shell, in
analogy to the fractional quantum Hall effect[20], but
without the need for a magnetic field.
In this work, we present new results demonstrating
the important role of electronic correlations, beyond the
Hubbard model[15, 16, 17] and mean-field density func-
tional theory (DFT)[17, 18]. The interactions are treated
by a combination of DFT, tight-binding, Hartree-Fock
and configuration interaction methods (TB-HF-CI). We
show that a half-filled charge neutral shell leads to full
spin polarization of the island but this magnetic mo-
ment is completely destroyed by the addition of a sin-
gle electron, in analogy to the effect of skyrmions on
the quantum Hall ferromagnet[21, 22, 23, 24] and spin
depolarization in electrostatically defined semiconductor
quantum dots[25, 26, 27, 28]. The depolarization of the
ground state is predicted to result in blocking of current
through a graphene quantum dot due to spin blockade
(SB)[26, 28].
In order to describe the interaction effects of electrons
occupying the edge states (see Fig.1), we first solve the
mean-field problem using a combination of tight-binding
approach with a self-consistent Hartree-Fock method for
the system with empty edge states, i.e. with Nref =
Nsite − Nedge electrons, where Nsite is the number of
atoms, and Nedge is the number of edge states. After
extensive algebra, we arrive at the HF Hamiltonian given
by
HMF =
∑
i,l,σ
τilσc†iσclσ +
∑
i,l,σ
∑
j,k,σ′
(ρjk − ρbulkjk )(〈ij|V |kl〉
− 〈ij|V |lk〉δσ,σ′)c†iσclσ +
∑
i,σ
vgii(qind)c
†
iσciσ (1)
where the operator c†iσ creates a pz electron on site “i”
with spin σ. The tight-binding parameters τilσ are fixed
to their bulk values -2.5eV for nearest neighbours, and
-0.1eV for next nearest neighbours. Within the Hartree-
Fock approximation, electrons interact with each other
via the density matrix ρ = ρ↑+ρ↓ from which we subtract
the graphene bulk density matrix ρbulk already present in
the tight-binding term τilσ . In addition to the on-site in-
teraction term, all scattering and exchange terms within
next nearest neighbours, and all direct interaction terms
are included in the two-body Coulomb matrix elements
〈ij|V |kl〉 computed using Slater pz orbitals. In order to
first empty the degenerate shell and later charge it with
successive electrons, we transfer electrons to a metallic
gate. The electrons in the graphene island interact with
the gate via the term vgii given by
vgii(qind) =
Nsite
∑
j=1
−qind/Nsite
κ
√
(xi − xj)2 + (yi − yj)2 + d2gate
(2)
where (xi, yi) are the coordinates of the atoms. This
model assumes that the induced charge qind = −Nedge is
smeared out at positions (xi, yi) at a distance dgate from
the quantum dot.
After diagonalizing the Hamiltonian,Eq.(1), we obtain
TB+HF quasi-particles denoted by the creation operator
b†p , with eigenvalues ǫp and eigenfunctions |p > shown in
Fig.1. We then start filling the edge states one by one,
and solve the configuration interaction problem for the
Nadd added electrons using
H =
∑
p,σ
ǫpb†pσbpσ +
1
2
∑
p,q,r,s,σσ′
〈pq|V |rs〉b†pσb†qσ′brσ′bsσ
+
∑
p,q,σ
〈p|vg(Nadd)|q〉b†pσbqσ

240 60 80
0
2
4 TB+HF
TB
DFT
E
ne
rg
y
[e
V
]
Eigenstate index
TB+HF DFT
DFTTB+HF
dgate
(a)
(b)
Gate charge
Graphene island
FIG. 1: (Color online) (a) Electronic density in a triangular
graphene island of 97 carbon atoms where 7 electrons were
moved to the metallic gate at a distance of dgate. (b) Sin-
gle particle spectrum of the structure in (a), obtained by
tight-binding (TB, blue lines) and self-consistent Hartree-
Fock (TB+HF, black lines) methods. The 7 edge states near
the Fermi level are compared to DFT results. In Hartree-
Fock and DFT calculations 7 electrons were removed, leaving
the zero-energy states empty. The dielectric constant κ is
set to 6. Inset compares the structure of corner and side
states obtained using Hartree-Fock and DFT calculations. In
DFT calculations, hydrogen atoms were attached to dangling
bonds.
+ 2
Nref/2
∑
p′
〈p′|vg(Nadd)|p′〉 (3)
where the indices (p, q, r, s) run over edge states, while
the index p′ runs over valence states (below the edge
states).
Fig.1a shows the electronic density of a zig-zag edged
triangular island of N = 97 carbon atoms, separated
by a distance dgate from a metallic gate. At zero ap-
plied voltage the island is charge neutral while applied
voltage leads to removal/addition of electrons to the is-
land. The single-particle energy spectrum obtained us-
ing nearest neighbour tight-binding method (TB, blue
lines) and a combination of the next-nearest neighbour
tight-binding and Hartree-Fock methods (TB+HF, black
lines) is shown in Fig.1b. As was previously shown
by Ezawa[16], Fernandez-Rossier and Palacios[17] and
Wang, Meng and Kaxiras[18] using nearest-neighbour TB
method and ab-initio DFT calculations, the linear spec-
trum of Dirac electrons in bulk graphene collapses to a
shell of degenerate levels at the Fermi energy, well sepa-
rated in energy from the valence and conduction bands.
Similar to edge states in graphene ribbons[6, 7, 8, 9],
zigzag edge breaks the symmetry between the two sub-
lattices of the honeycomb lattice, behaving like a defect.
Therefore, electronic states localized on the zigzag edges
appear with energy in the vicinity of the Fermi level. For
a N = 97 atoms island (see Fig.1) there are Nedge = 7
edge states. For the charge neutral system there is one
electron per each edge state. A non-trivial question ad-
dressed here is the specific spin and orbital configura-
tion of the electrons as a function of the size and the
fractional filling of the degenerate shell of edge states.
Due to the strong degeneracy, many-body effects can be
expected to be important as in the fractional quantum
Hall effect. Previous calculations based on the Hub-
bard approximation[15, 16, 17] and local spin density
functional theory[17, 18] showed that the neutral system
(half-filling) has its edge states polarized.
In order to study many-body effects within the charged
shell of the edge states via the configuration interaction
method, we first perform a Hartree-Fock calculation for
the charged system of N − Nedge electrons, with empty
edge states and Nedge electrons transferred to the gate,
shown in Fig.1a. The spectrum of HF quasi-particles
is shown with black lines in Fig.1b. Due to the mean-
field interaction with the valence electrons and charged
gate, the degeneracy of the edge states is lifted, and the
electron-hole symmetry is broken. The main effect is
that a group of three states is now separated from the
rest by a gap of ∼ 0.2 eV. The three states correspond
to HF quasiparticles localised in the three corners of the
triangle. The same physics occurs in density functional
calculation within local density approximation (LDA),
shown with red lines in Fig.1b. Hence we see that the
shell of edge states with a well defined gap separating
them from the valence and conduction bands exists in
the three approaches.
The wave functions corresponding to the band of
nearly-degenerate edge states obtained from TB-HF cal-
culations are used as a basis set in our configuration inter-
action calculations where we add Nadd electrons from the
gate to the shell of degenerate states. In Fig.2, total spin
S of the ground state as a function of the filling number
of the edge states is shown for different sizes of quantum
dots. Two aspects of these results are particularly inter-
esting. For the charge neutral case (Nadd−Nedge = 0),for
all the island sizes studied (Nedge = 3−7), the half-filled
shell is maximally spin polarized as indicated by red ar-

3FIG. 2: (Color online) Spin phase diagram from TB-HF-CI
method as a function of the size of the triangular dot charac-
terized by the number of edge states Nedge, and the filling of
the zero-energy states Nadd-Nedge. Charge neutral case cor-
responds to Nadd −Nedge = 0, for which the total spin of the
zero-energy electrons are always maximized (S = Nedge/2,
indicated by red arrows). On the other hand, if the quantum
dot is charged by 1 electron (Nadd −Nedge = 1) then the total
spin has minimum value, i.e. S = 0 if Nadd is even, S = 1/2
if Nadd is odd (indicated by blue arrows).
rows. This is in agreement with our DFT calculations
and reproduces previous work[17, 18]. However, the spin
polarisation is extremely fragile. If we add one extra
electron (Nadd − Nedge = 1), magnetisation of the island
collapses to the minimum possible value, as indicated by
blue arrows in Fig.2. We note that full or partial depolar-
ization occurs for other filling numbers but we focus here
on the spin depolarisation at half-filling. The spin de-
polarisation was found to be insensitive to the screening
of electron-electron interactions, as values of dielectric
screening constant κ between 2 and 8 led to the same be-
haviour of the charge neutral and single electron charged
cases (not shown).
In order to illuminate the depolarization process as an
electron is added to the charge neutral maximally spin
polarized system, in Fig.3a,b we show the orbital occu-
pancy of up-spin edge states at Nadd − Nedge = 1, for
the fully polarized state S = 6/2 (upper panel) and for
the ground state, S = 0, (lower panel) for the Nsite = 97
atoms quantum dot with 7 edge states shown in Fig.1.
For the large spin S = 6/2 case, the added spin up elec-
tron simply occupies the orbital 1 and its spin is oppo-
site to the spins of other 7 electrons. However, the true
ground state has S = 0, with spin occupancy shown in
the lower panel. The added electron causes electrons al-
ready present to partially flip their spin, with spin up
0 1 2 3 4 5 6 7 8
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
O
cc
up
an
cy
O
cc
up
an
cy
Edge state index
(b)
(a)
(d)
(c)
FIG. 3: (Color online) Left panel: Orbital occupancy of the
7 edge states by spin up electrons, for the charged (Nadd −
Nedge = 1) system, for (a) S = 6/2 and (b) S = 0 total
spin states. The ground state is S = 0 (see Fig.2). Right
panel: Corresponding spin up-up pair-correlation functions
〈ρ↑(r0)ρ↓(r)〉. The fixed spin-up electron is represented by a
red arrow, and its position r0 by a red circle.
density being delocalized over all the 7 orbitals in anal-
ogy to skyrmion-like excitations in quantum dots and
quantum Hall ferromagnets[21, 22, 23, 24]. The corre-
lated nature of the S = 0 spin depolarised ground state
is illustrated by the up-up spin pair correlation functions
given by 〈ρ↑(r0)ρ↓(r)〉, shown in Fig.3c,d. The location
of the fixed up-spin electron at site r0 is schematically
shown with an up arrow. The up-up spin correlation
function for S = 6/2 spin polarized system is strictly
zero as there are no other spin up electrons. The spin
correlation function for the spin depolarized ground state
with S = 0 shows exchange hole at r0 , which extends to
the nearest neighbours, and, more interestingly, for larger
|r0 − r| , spin pair correlation function reveal a spin tex-
ture: Beyond the exchange hole there is the formation
of an electronic cloud with positive magnetization which
decreases and changes sign at even larger distance, again
consistent with the skyrmion picture[21, 22].
Experimentally, spin properties of quantum dots
can be probed using Coulomb and spin blockade
spectroscopy[28]. By connecting graphene quantum dot
to leads and measuring the conductance as a function
of gate voltage, one obtains a series of Coulomb block-
ade peaks. The relative position of these peaks and their
height reveal information about the electronic properties
of the system as the number of electrons is increased.
The amplitude of the Coulomb blockade peak is given by
the conductivity Gi of the graphene quantum dot con-
nected to leads via atom “i”[29] as shown schematically
in Fig.4a. Spin and correlation effects are reflected in
the weight of Coulomb blockade peak proportional to the
matrix element |〈N + 1, J ′, S′|c†iσ|N, J, S〉|2 which gives
the transition probability from state (N, J, S) to state
(N + 1, J ′, S′) when additional electron is added to the
site “i” of the graphene quantum dot from the lead. The
ground state configuration (N, J, S) is controlled by the

4lead 1 lead 2
0.00
0.05
0.10
0.15
0.20
0.25
G
i[a
rb
.u
ni
ts
] N=1
N=3
N=4
N=5,6
N=7
N=8,9,10 N=11
N=14
-11.5 -11.0 -10.5 -10.0 -9.5
0.00
0.05
0.10
0.15
0.20
0.25
G
i [
ar
b.
un
its
]
V
g
[eV]
(a)
(b)
(c)
FIG. 4: (Color online) (a) Schematic representation of the
graphene island connected to the leads through a side site.
(b) Conductivity as a function of applied gate voltage Vg due
to an electron entering the dot through the side atom. (c)
Same as (b) but without the site dependence of the incoming
electron. The oscillations of the spectral weight in (c) are
purely to due correlation effects and spin blockade.
gate voltage. For our model graphene quantum dot with
N = 7 degenerate zero energy states, we can add a total
of 14 electrons. Hence, one expects to obtain 14 peaks.
In Fig.4b some of the peaks have zero height due to spin
blockade phenomenon. For instance, the transitions from
(N = 7, S = 7/2) states to (N = 8, S = 0) states are
spin blocked since it is not possible to change the spin of
the system by ∆S = −7/2 by adding one electron with
S = 1/2. Similarly, transitions from (N = 9, S = 1/2)
states to (N = 10, S = 4/2) states are spin blocked.
Besides the spin blockade, one sees a strongly oscillating
feature of the spectral function heights. This is due to (a)
strongly correlated nature of the states |N,S〉 , and (b)
specific choice of the site “i”, where the lead is attached
to. Here, we chose a site close to the middle of one of the
sides of the triangle. The overlap of the site wavefunction
is strongly dependent on the nature of the edge-states. In
particular, the existence of corner states, as discussed in
Fig.1, strongly affects the transition probabilities. To
isolate the effect of correlation to the lead’s position, in
Fig.4c we plot the conductivity assuming that the weight
of the site “i” is a constant independent of the edge-state.
As a result, the weights of spectral peaks are different,
except for N = 8, 9, 10 where the spin blockade occurs.
These results show how to detect the spin depolarization
in transport experiments. Ultimately, we show here that
one can design a strongly correlated electron system in
carbon based material whose magnetic properties can be
controlled by applied gate voltage.
Acknowledgement. The authors thank NRC-CNRS
CRP, Canadian Institute for Advanced Research, Insti-
tute for Microstructural Sciences, and QuantumWorks
for support.
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