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Suspended graphene films and their Casimir interaction with

ideal conductor

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

1. Introduction

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,

γ2

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

SM= −1

4

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

E1= −1

TS

, (3)

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

A =1

2

?

d3p

(2π)3Aj(p)Πjl(p)Al(p), (4)

where

Πmn=αΦ(˜ p)

v2

F

ηm

j

?

gjl−˜ pj˜ pl

˜ p2

?

ηn

l, (5)

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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

Πj

j(pE)

p?

j

=

jpk, The func-

=

E1= −1

4

?

d3pE

(2π)3

e−2ap?= −

?

d3pE

(2π)3

α(p2

?+ ˜ p2

4p?˜ p2

?)Φ(pE)

?

e−2ap?.

(6)

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

of Πν

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

d3pE

16π3ln[(1 − e−2p?ar(1)

For graphene with help of matching conditions (8) we can obtain at the

Euclidean momenta

−αΦ

2p?+ αΦ,

µAν|x3=a.(8)

TE,TM, r(2)

TE,TMof the TE and TM

EL=

?

TEr(2)

TE)(1 − e−2p?ar(1)

TMr(2)

TM)]. (9)

r(1)

TE=

r(1)

TM=

αp?Φ

?+ αp?Φ,

2˜ p2

(10)

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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)

EL

∼

a→∞

−

α

24π2

2 + v2

ma4,

f

EL

∼

a→0

1

16πa3h(α,vF) (11)

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

EC= −

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.

π2

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Fig. 1.

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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