arXiv:0910.1940v1 [cond-mat.mes-hall] 10 Oct 2009
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Suspended graphene films and their Casimir interaction with
I. V. Fialkovsky∗
Instituto de F´ ısica, Universidade de S˜ ao Paulo, S˜ ao Paulo, S.P., Brazil
Department of Theoretical Physics, Saint-Petersburg State University, Russia
We adopt the Dirac model for graphene and calculate the Casimir interaction
energy between a plane suspended graphene sample and a parallel plane ideal
conductor. We employ both the Quantum Field Theory (QFT) approach, and
the Lifshitz formula generalizations. The first approach turns out to be the
leading order in the coupling constant of the second one. The Casimir interac-
tion for this system appears to be rather weak but experimentally measurable.
It exhibits a strong dependence on the mass of the quasi-particles in graphene.
Present article is based on joint works.1,2
Keywords: Casimir energy, graphene, QFT, Lifshitz formula
Graphene is a (quasi) two dimensional hexagonal lattice of carbon atoms.
It belongs to the most interesting materials in solid state physics now due
to its exceptional properties and importance for nano technology.3,4Here
we consider the Casimir interaction between suspended graphene plane and
parallel ideal conductor. This setup was considered in5–7using a hydrody-
namical model for the electrons in graphene following.8,9Later it became
clear that this model does not describe the electronic properties specific to
this novel material.
Here we use a realistic and well-tested model where the quasi-particles in
graphene are considered to be two-component Dirac fermions. This model
incorporates the most essential and well-established properties of the their
dynamics: the symmetries of the hexagonal lattice, the linearity of the spec-
trum, a very small mass gap (if any), and a characteristic propagation ve-
locity which is 1/300 of the speed of light.3,10By construction, this model
∗The author gladly acknowledge the financial support of FAPESP, as well as of grants
RNP 2.1.1/1575 and RFBR 07–01–00692.
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should work below the energy scale of about 1eV , but even above this limit
the optical properties of graphene are reproduced with a high precision.15
The action of the model, therefore, is given by
where ˜ γlare just rescaled 2×2 gamma matrices, ˜ γ0≡ γ0, ˜ γ1,2≡ vFγ1,2,
The value of the mass gap parameter m and mechanisms of its generation
are under discussion.11–14The upper limit on m is about 0.1eV at most.
The propagation of photons in the ambient 3 + 1 dimensional space is
described by the Maxwell action
In the following we shall suppose that the graphene sample occupies the
plane x3= a > 0, and the conductor corresponds to x3= 0.
SD=d3x¯ψ(˜ γl(i∂l− eAl) − m)ψ,l = 0,1,2(1)
0= −(γi)2= 1. In our units, ? = c = 1, and Fermi velocity vF≃ (300)−1.
d4xFµνFµν, µ,ν = 0,1,2,3.(2)
2. QFT approach
In the framework of QFT one evaluates the effective action in a theory
described by the classical action SD+SM. Then the Casimir energy density
per unit area of the surfaces at the leading order in the fine structure
constant α is given by
where T is time interval, and S is the area of the surface. The solid line
denotes the fermion propagator in 2+1 dimensions (i.e., inside the graphene
sample), and the wavy line is the photon propagator in the ambient 3 + 1
dimensional space subject to the perfect conductor boundary conditions at
x3= 0: A0|x3=0= A1|x3=0= A2|x3=0= ∂3A3|x3=0= 0.
The fermion loop in 2 + 1 dimensions has already been calculated in a
number of papers.11,13,14It gives the quadratic order in A of the effective
action for electromagnetic field
Seff(A) = A
gjl−˜ pj˜ pl
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is the polarization tensor in the lowest, one loop, order. Here ηm
diag(1,vF,vF), ˜ p denotes the rescaled momenta ˜ pj = ηk
tion Φ(p) is model dependent, and for graphene it reads Φ(p)
2?2m˜ p − (˜ p2+ 4m2)arctanh(˜ p/2m)?/˜ p. We assume here that all possible
parity-odd parts are canceled in Π. Possible effects invoked by their presence
are considered in.1
To calculate the diagram (3) we only need to couple the kernel (5)
to the photon propagator subject to conducting boundary conditions. In
Fourier representation and for the Euclidean 3-momenta, i.e., after the Wick
rotation p → pE= (ip0,p1,p2), the a-dependent part of the energy reads
jpk, The func-
?+ ˜ p2
where we expanded Πj
j(pE) explicitly with help of (5), and p?≡ |pE|.
3. Lifshitz formula approach
One can also consider the system as described by effective theory of the
electromagnetic field with the action SM+ Seff subject to the conduct-
ing boundary conditions at x3= 0. Then at the surface of graphene, the
Maxwell equations receive a singular contribution
∂µFµν+ δ(x3− a)ΠνρAρ= 0(7)
following from Seff. Here we set Π3µ= Πµ3= 0. This contribution is
equivalent to imposing the matching conditions
(∂3Aµ)|x3=a+0− (∂3Aµ)|x3=a−0= Πν
assuming that Aµis continuous at x3= a. Now, one can forget the origin
µand quantize (at least formally) the electromagnetic field subject to
the conditions (8) at x3= a and to the conducting conditions at x3= 0.
The original Lifshitz approach17was generalized18,19for the interactions
between two plane parallel interfaces separated by the distance a and pos-
sessing arbitrary reflection coefficients r(1)
electromagnetic modes on each of the surfaces
16π3ln[(1 − e−2p?ar(1)
For graphene with help of matching conditions (8) we can obtain at the
TE,TMof the TE and TM
TE)(1 − e−2p?ar(1)
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while for the perfect conductor one has r(2)
Φ must be rotated to Euclidean momenta as well. We also note that the
perfect conductor case is recovered from (10) in the formal limit Φ → ∞.
One can show by a direct computation that the energy E1, Eq. (6), coin-
cides with the leading α1order in a perturbative expansion of the Lifshitz
formula (9)-(10), so that the two approaches are consistent.
TE= −1, r(2)
TM= 1. It is clear, that
4. Results and discussion
The formulae (6) and (9)-(10) are suitable for the numerical and asymptot-
ical evaluation. The asymptotic expansion for short and long distances are
readily obtained through uniform expansion of the integrand of (6), (9)
2 + v2
Note that the asymptotics at large separations is of the first order in α
while for small separations, it contains all powers of α through h(α,vF),
for the real values of parameters in graphene h(1/137,1/300) ≈ 0.024.
Therefore we see that at large separations Casimir energy does not turn
into the ideal conductor case, while at small separation this case is indeed
recovered. This is counter-intuitive since the main contribution at short
separations shall come from the high momenta for which one would expect
the graphene film to become transparent. We must also stress that this
behavior is drastically different from that in the hydrodynamic model.5-7
For numerical evaluation we normalize the results to the Casimir energy
720 a3for two plane ideal conductors separated by the same distance
a. The results of calculations are depicted at Fig. 1. The scale is defined by
the mass parameter m. For m of the order of next nearest-neighbor hopping
energy t′, i.e., m = 0.1eV ,4ma = 1 corresponds to a = 1.97 micrometer.
Thus, we can see that the magnitude of the considered Casimir inter-
action of graphene with a perfect conductor is rather small. Actual mea-
surement of such weak forces is a challenging, but by no means hopeless,
experimental problem. Strong dependence on the mass parameter m at large
separation is also a characteristic feature of the Casimir force. Getting an
independent measurement of m may be very important for our understand-
ing of the electronic properties of graphene. The mass of quasi-particles in
graphene is, probably, very tiny, which improves the detectability of the
Casimir interaction since the energy increases with decreasing m.
October 13, 2009 0:3 WSPC - Proceedings Trim Size: 9in x 6inIVFGraph Download full-text
line) as functions of ma. Insert shows a zoom of the small-distances region.
The relative Casimir energy densities E1/EC (solid line) and EL/EC (dashed
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