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Compressibility of graphene

D. S. L. Abergel, E. H. Hwang, and S. Das SarmaCondensed Matter Theory Center, Department of Physics,

University of Maryland, College Park, MD 20742

We develop a theory for the compressibility and quantum capacitance of disordered monolayerand bilayer graphene including the full hyperbolic band structure and band gap in the latter case.We include the effects of disorder in our theory, which are of particular importance at the carrierdensities near the Dirac point. We account for this disorder statistically using two different averagingprocedures: first via averaging over the density of carriers directly, and then via averaging in thedensity of states to produce an effective density of carriers. We also compare the results of thesetwo models with experimental data, and to do this we introduce a model for inter-layer screeningwhich predicts the size of the band gap between the low-energy conduction and valence bands forarbitary gate potentials applied to both layers of bilayer graphene. We find that both models fordisorder give qualitatively correct results for gapless systems, but when there is a band gap at chargeneutrality, the density of states averaging is incorrect and disagrees with the experimental data.

I. INTRODUCTION

Recently, graphene has become a highly-studied elec-tronic system with many potential uses in electronic de-vices, optics, and sensing applications.1 While significanttheoretical effort has been concentrated on the transportproperties2 (such as the minimal conductivity, localiza-tion effects, and signatures of relativistic behavior of thecharge carriers), bulk thermodynamic quantities such asthe compressibility of the electron liquid in monolayerand bilayer graphene has also received substantial at-tention. The compressibility is an important quantitybecause it is possible to extract information about theelectron-electron interactions and correlations directlyfrom these measurements, and to gain information aboutfundamental physical quantities such as the pair corre-lation function. The quantum capacitance3,4 is also animportant quantity in the context of device design andfundamental understanding of the graphene system. Infact, the compressibility is often inferred from the mea-surement of quantum capacitance.

Experiments5 have shown that the contribution tothe compressibility of monolayer graphene from electron-electron interactions is small (amounting to a 10 − 15%renormalization of the effective Fermi velocity). This wasexplained within the Hartree–Fock approximation.6 Ca-pacitance measurements of bilayer graphene have alsobeen carried out recently,7,8 and they purport to showdata which indicates that the quadratic approximationof the band structure of this material does not producethe correct predictions for the quantum capacitance ofthe graphene sheet. Other measurements which claimto access the quantum capacitance directly have shownsimilar behavior.9

Theoretical work has been published which consid-ers the compressibility of monolayer6 and bilayer10

graphene at the Hartree-Fock level and in the RPAapproxmation11. They find that interactions are moreimportant in the bilayer system (due to the finite densityof states at charge-neutrality), but that in RPA the con-

tributions from exchange and correlation almost canceleach other out leaving a small positive compressibility atall densities. A consensus has developed that many-bodycorrections to the compressibility in graphene are at mostof the order of 10% at reasonable carrier densities.12

In this article we discuss theoretically the compress-ibility of monolayer and bilayer graphene, and specif-ically employ the four-band model of bilayer graphene(which generates the non-quadratic band structure). Wewill highlight the similarities and differences between thismodel and the linear and quadratic cases, and take intoaccount the possiblility of a gap at the charge-neutralitypoint. We find that the non-quadratic band structure sig-nificantly affects the theoretical predictions for the com-pressibility and brings them into qualitative and semi-quantitative agreement with experimental findings. Wealso include disorder in the form of electron–hole pud-dles created by charged impurities in the vicinity of thegraphene. We discuss two different procedures for includ-ing this effect statistically and determine the validity ofeach model by comparing with experimental data. Thefirst assumes that the disorder creates a spatial variationin the density of carriers which can be modelled by av-eraging a physical quantity such as compressiblity over arange of densities. The second model assumes that dis-order affects the local density of states which can be av-eraged to give an effective bulk density of carriers whichcan then be used to compute the physical quantities ofinterest. We find that for gapless graphenes, inclusion ofdisorder by either method predicts qualitatively correctresults in agreement with experiments, but when a gap isopened, the stage at which the averaging is done makesa critical difference.

Therefore, we have dual complementary theoreticalgoals in this work. Our primary goal is to develop anaccurate (and essentially approximation-free) theory forthe graphene compressibility within the non-interactingelectron model which incorporates all aspects of the re-alistic band structure as well as effects of the disorderwithin reasonable and physically motivated approxima-tions. Our secondary goal is to compare our theory

2

closely and transparently with the existing experimentaldata on the compressibility of bilayer graphene so as toobtain some well-informed conclusions about the possi-ble role of many-body exchange-correlation effects, whichhave recently been studied the the literature10–12 and isbeyond the scope of the current work. We believe that anaccurate calculation of graphene compressibility withinthe free electron theory neglecting all interaction effectsis warranted for (at least) three reasons: (i) without anaccurate one-electron theory, quantitative conclusions re-garding many-body effects are not useful; (ii) many-bodytheory is, by definition, approximate and, as such, sub-ject to doubts; (iii) many-body corrections to graphenecompressibility are claimed to be small with considerablecancellation between exchange and correlation effects11.In comparing our theoretical results with the experimen-tal data, we find that the inclusion of disorder effects inthe theory is of qualitative importance, particularly atlow carrier densities near the charge-neutral Dirac point(where the many-body effects are expected to be largest).The inclusion of disorder through two alternative physi-cal approximations and the eventual validation of one ofthe models of disorder by comparing with the existing ex-perimental data is an important accomplishment of ourtheory.

To outline the structure of this article, we begin in theremainder of this section by giving details of the bandstructures of the three types of graphene which we con-sider. In Section II we introduce a model for determiningthe band gap in bilayer graphene with arbitrary gate po-tentials applied to both layers. Then, in Section III wedescribe the calculation of K = dµ

dn in the homogeneous(non-disordered) situation (where µ and n are the chem-ical potential and carrier density respectively) for eachof these three cases and discuss the main features of thisquantity in relation to the quantum capacitance and com-pressibility. In Section IV we introduce the two modelsfor the inclusion of disorder and present the fundamentaltheoretical results for each. Then, in Section V we com-pare the predictions of our theory to experminental datagiven by capacitance measurements of bilayer graphenewhich include the quantum capacitance of the 2D elec-tron gas. Finally, in Section VI we summarize our resultsand give some general discussion of our findings.

In the following, we discuss three different models forthe single-particle dispersion of graphene which we de-note by Ek. We shall refer to these three cases as lin-ear, quadratic, and hyperbolic graphene to distinguishthem. For the monolayer, we have El

k = h̄vFk as is well-known, and we indicate all quantities which follow fromthis dispersion by attaching a superscript ‘l’ as we havehere. For bilayer graphene, there are two commonly-used approximations for the low-energy single-particledispersion. The simplest is the quadratic approximationEq

k = h̄2v2F k2/γ1 = h̄2k2/(2m) where γ1 is the inter-

layer hopping parameter from the tight-binding theoryand m = γ1/(2v

2F ) is the effective mass of the electrons.

This dispersion comes from a low-energy effective theory

EF

k∗kF− kF+

E = u2

E∗

FIG. 1. Sketch of the ‘Mexican Hat’ dispersion in hyperbolicgraphene with u > 0. The depth of the minimum is ex-agerrated for the purposes of illustration. The various wavevectors and energies that are labeled are defined in the text.

of bilayer graphene13 which essentially discards the twosplit bands and confines electrons to those lattice sitesnot involved in the inter-layer coupling. This quadraticbilayer dispersion is assumed to hold for low carrier den-sities, i.e. at low Fermi energy.2 Alternatively, the fulltight-binding theory14 yields the single-particle disper-sion

Ehk =

√

γ21

2+

u2

4+ h̄2v2Fk

2 −√

γ41

4+ h̄2v2F k

2(γ21 + u2)

(1)which is illustrated in Fig. 1. This dispersion allows forthe inclusion of a band gap parameterized by the energyu which corresponds to the potential energy differencebetween the two layers. The ‘Mexican hat’ shape of thedispersion implies that the minimum gap is found at wave

vector k∗ = u2h̄vF

√

2γ21+u2

γ21+u2 and the energy of the mini-

mum is therefore E∗ = γ1u

2√

γ21+u2

. This non-monotonic

dispersion relation leads to some important features inthe Fermi surface at low density. For medium and highdensity regimes (when kF > u

h̄vF), the Fermi wave vector

is given by kF =√πn and the Fermi surface is disc-

shaped. However, if√πn < u

h̄vF(as sketched in Fig. 1)

then the Fermi energy is in the ‘Mexican Hat’ region andthe Fermi surface is ring-shaped15 with kF− < k < kF+

and

kF± =1

2h̄vF

√

π2h̄4v4Fn2 + γ2

1u2

γ21 + u2

+ u2 ± 2πh̄2v2Fn. (2)

Clearly these characteristic features of the full hyper-bolic bilayer band dispersion are lost in the simple linear(“high-energy”) and quadratic (“low-energy”) apprixi-mations often adopted in the literature for simplicity.The parameter u can be treated in two ways. Initially

(and in Section IV), we consider it to be a phenomeno-logical parameter which can be chosen arbitrarily to rep-resent a finite gap at charge-neutrality. However, in or-der to accurately compare our theory with experimentwe must determine a method of relating u to the exter-nal gate potentials. To do this accurately, the screeningof the external field by the charge on the two layers of

3

bilayer graphene must be taken into account, and we de-scribe this process in Section II.

II. INTER-LAYER SCREENING

In order to make an accurate comparison with exper-imental data, we must properly map the gate voltagesapplied in experiment to the parameters n and u whichappear in our theory. We assume the situation sketchedin Fig. 2(a). The density is easily computed via elemen-tary considerations to be n = ǫ0 (Vbǫb/db + Vtǫt/dt) /ewhere V is the voltage applied to a gate, d is the widthof the gate layer, ǫ is the dielectric constant of the gate,and the subscript b and t respectively denote the bottomand top gates. The zeroeth-order approximation for theon-site potential u is to say that the electric field cre-ated by the gates directly gives the electric field E in thegraphene. The potential is then ut,b = ±eEd/2. How-ever, this approximation does not take into account thescreening effect of the charge on the two graphene lay-ers. Previously, various authors have investigated thisproblem and shown that screening reduces the size of thegap14,16,17. We extend this work to the case where thetotal density is finite and gates are present both aboveand below the graphene layer.Since the screening depends only on the electric field

created by the gates and the charge densities on the twolayers, it is independent of the disorder caused by chargedimpurities. Both the screening and the disorder effectsare determined by the net carrier density, and both leavethis quantity unchanged. Therefore, we can use the fol-lowing analysis to set the size of the gap for any given setof external conditions, and use the resulting band struc-ture to calculate the effect of disorder for that situation.This is allowed because neither disorder nor the screeninginduces any extra charge in the graphene. Additionally,we will assume in Section IV that the disorder potentialis the same in both layers meaning that it does not in-duce any inter-layer scattering and therefore leaves thenet carrier density in each layer unchanged.According to Gauss’ law, the electric field directed

away from an infinite-sized, positively charged sheet isE = qn/2ǫ0, where q = −e is the charge of the carrierson the sheet. Hence, we can write the internal field (i.e.the field between the graphene layers) as

E = Eext +e

2ǫ0(nt − nb). (3)

That this internal field is different from the external onecauses the free charge on the two layers to rearrange,which in turn further modifies the internal field. A self-consistent procedure can be constructed to find the finalsolution for the layer charge densities and the internalelectric field. This additional field causes an additionalHartree energy of ut − ub = edE to be added to the on-site potential in the original Hamiltonian. This modified

dt Et = −Vt

dt

V = Vt

db

nb

E

Eb=Vb

db

d

V = Vb

nt

(a)

-80

-40

0

40

80

-100 -80 -60 -40 -20 0 20 40 60 80 100

u (

meV

)

uext

(b)

|n| = 5×1012 cm-2

|n| = 1×1012 cm-2

-20

0

20

-4 -3 -2 -1 0 1 2 3 4

u (

meV

)

Vt (V)

(c)Vb = -50Vb = -10

Vb = +10Vb = +50

FIG. 2. (a) The geometry of the screening problem. Theexternal field Eext = (Eb+Et)/2 is screened by redistributionof the charge density n = nb+nt between the two layers of thegraphene. (b) The screened gap u as a function of the bare gapuext for various densities. (c) The screened gap as a functionof the top gate voltage for various fixed back gate voltagesand the experimental parameters used in the experiments ofYoung et al.7,18 Back gate voltages are quoted in Volts.

on-site potential will be smaller than in the unscreenedcase, and therefore the gap will also be smaller.We implement this procedure for a given pair of gate

voltages Vb and Vt as follows. The first step is to computethe unscreened on-site potential in the two layers, whichis given by the expression above with nt = nb so that E =Eext. This gives uext = edEext = ed(Vb/db − Vt/dt)/2.We assume that the potential is symmetrical so thatut = u/2 and ub = −u/2. This determines the single-particle Hamiltonian, from which we can compute thewave functions. The layer density is then found by sum-ming the wave function weight in each layer:

nt =∑

states

(

|φAt|2 + |φBt|2)

where A and B label the two inequivalent lattice sites ineach layer. The sum runs over all valley, spin and wavevector states. A similar expression is given for the lowerlayer by substituting t → b throughout. Computing thesedensities gives all the information required to evaluatethe internal electric field, from which we can find themodified on-site potentials via Eq. (3). These potentialsare then used to calculate the modified layer densities,and this cycle is repeated until convergence is reached.

4

Figure 2(b) shows the screened gap as a function of theexternal gap for various density of carriers. The densityplays a role because the more free carriers there are, themore effectively the external field can be screened and thesmaller the screened gap is. This is shown in the figurebecause the magnitude of the gap for |n| = 5× 1012cm−2

is always smaller than that for |n| = 1×1012cm−2. Figure2(c) shows the screened gap as a function of top gate volt-age assuming that the back gate voltage is fixed. We haveused the parameters from the experiments by Young et

al.7,18. The peak in each trace is caused by the approachof the system to charge neutrality where there are noexcess carriers and hence the external electric field is un-screened. At this point, the screened gap is equal to thebare gap. When there is a finite density of excess car-riers, the external field is screened and the gap reducedaccordingly. As the back gate voltage is changed, theposition of the charge neutrality point shifts and the sizeof the gap at charge neutrality increases. Inclusion ofinter-layer screening turns out to be important for un-derstanding the experimental data.

III. CALCULATION OFdµ

dn

In this section, we describe the calculation of dµdn for

the homogeneous (non-disordered) system. The resultsderived here will be used later to compute the samequantity for a disordered system. The single-particle dis-persion Ek of monolayer and bilayer graphenes are wellknown (as discussed in Section I). From these relations,we can compute the total kinetic energy for excess chargecarriers by summing the energy of carriers in filled states:E =

∑

states Ekf(k) where f(k) is the occupation numberof a state with wave vector k = |k|. Then, the chemicalpotential is defined to be the change in energy with theaddition of a particle, which is expressed as µ = 1

A

dEdn

where n is the carrier density and A is the area of thegraphene flake. From this we can compute K = dµ

dn , andthe quantum capacitance CQ and compressibility κ arethen linked to this quantity via

CQ =ǫ0e2

dµ

dn, κ−1 = n2 dµ

dn. (4)

The linear and quadratic band structures lead straight-forwardly to the following expressions

K l =h̄vF

√π

2√n

, and Kq =h̄2v2Fπ

γ1.

The linear dispersion leads to a square-root divergence atzero density while the quadratic version gives a constantfor all densities. It is possible to obtain an analytical ex-pression for Kh, but it is very lengthy, not particularlyinformative, and not suitable for reproduction in this ar-ticle. Instead, we describe a process for writing down theanswer. Starting from the expression for the total energy,

the sum is evaluated by transforming to an integral overk. The indefinite integral in question is given by

I(k) =2Aπ

∫

kEhk dk

=Aγ3

1

πh̄2v2F

1

24√∆

{√2∆

(

8x− Φ + 1− 4δ2)

Ξ+

+ 12δ2 log[

2∆(

−∆+Φ+√2∆Ξ

)]

}

where x = (h̄vFk/γ1)2, δ = u/(2γ1), Φ =

√1 + 4x∆, and

Ξ =√2x− Φ+ 1 + 2δ. To find the actual total energy,

one must substitute the appropriate limiting values of thewave vector for the density as discussed previously in thecontext of the topology of the Fermi surface. ThereforeE = I(k = k+) − I(k = k−) and the chemical poten-tial µ = dE/dN can be calculated by transforming thederivative into one for the limiting k:

µ =1

AdEdn

=1

Adk+dn

dI(k+)

dk+− 1

Adk−dn

dI(k−)

dk−.

However, it is clear from the fundamental theorem ofcalculus and the definition of I that dI/dk = 2AkEk/π.Hence, the inverse density of states is now easily writtenas

Kh =2

π

{

d2k+dn2

k+Ek++

(

dk+dn

)2 (

Ek++ k+

dEk+

dk+

)

− d2k−dn2

k−Ek−−(

dk−dn

)2 (

Ek−+ k−

dEk−

dk−

)}

.

The remaining derivatives are straightforward to com-pute from Eqs. (1) and (2). As described previously,when EF < u/2 the values of k± are given by Eq. (2).When EF > u/2 then k− = 0 and k+ =

√πn and only

the first two terms contribute to Kh.The three versions of K are plotted in Figure 3(a)

where the density is assumed to be a tunable external pa-rameter and the gap is fixed. In the limit of small n, thegapless hyperbolic band structure is roughly quadraticand so Kh is very close to Kq. However, the linearityof the hyperbolic dispersion quickly asserts itself to re-duce the size of Kh until, at high density, it approachesthe value of linear graphene. Thus Kh in the ungappedregime interpolates between these two limiting cases asone expects. We also show Kh for the gapped case (redline) where we take u = 50meV. When the density is highand the Fermi energy is well above the gap region, Kh

tends to the same functional form as for the ungappedscenario. However, when the density is low, Kh goes tozero for any finite u. The large reduction in Kh by thegap will be an important feature for the comparison withexperiment. In Figure 3(b) we show the Kh at low car-rier density for several different values of the gap. Oneimmediately notices a discontinuity when n = u/(h̄vF )which is caused by the change in topology of the Fermisurface from a ring to a disc. When a gap is present,

5

0

2

4

6

8

2 4 6 8 10

dµ/d

n (

eV n

m2 )

n (1012 cm-2)

(a)LinearQuadraticHyper., u = 0Hyper., u = 50 meV

0

1

2

3

0 0.2 0.4 0.6

dµ/d

n (

eV n

m2 )

n (1012 cm-2)

(b)

u = 10meVu = 30meVu = 50meVu = 70meV

FIG. 3. (a) The inverse density of states, K = dµ

dnfor

linear, quadratic, gapless hyperbolic and gapped hyperbolic(u = 50meV) dispersions. (b) K for different gap sizes at lowcarrier density.

there is also a δ-function divergence at n = 0 (not shownin the plots) caused by the step in the chemical potentialas the Fermi energy goes from the valence band to theconduction band. The prefactor of this divergence is thesize of the gap so that it is not present when u = 0.

IV. MODELS FOR DISORDER

We now discuss the effects of disorder caused byelectron-hole puddles which are formed by charged im-purities in the environment.5,19 We will describe two in-equivalent methods of averaging over disorder and high-light the differences between them. In what follows, wemust be careful to distinguish global (averaged) quanti-ties from their local counterparts. Therefore, we use anoverbar to denote a quantity which has been averagedover disorder and which is therefore global. We start byassuming that charged impurities (which may be locatedin the substrate or near the graphene) create a local elec-trostatic potential which fluctuates randomly across thesurface of the graphene sheet. In the case of the bilayer,we assume that this potential is felt equally by both lay-ers. We then make the transition from the spatial varia-tion of the potential to an average by assuming that thevalue of this potential at any given point can be describedby a statistical distribution.20 This distribution can thenbe used to average some pertinent quantity to obtain thebulk result for the disordered system. But it is not cleara priori what the optimum method of performing thisaverage is. In order to discuss this question, we use twodifferent methods and (in the next section) compare theresults of each to experimental data.The first model assumes that the disorder potential

directly affects the density of carriers in the graphene sothat the density is distributed according to a Gaussiandistribution P parameterized by width ν

P (n) =1

ν√2π

exp

(

− n2

2ν2

)

.

0

0.5

1

1.5

2

-50 0 50

Den

sity

(10

12 c

m-2

)

EF (meV)

Linear

-50 0 50EF (meV)

Quadratic

-50 0 50EF (meV)

Hyperbolic

ElectronHoleTotalTotalu = 50meV

FIG. 4. Electron, hole and total carrier densities for linear,quadratic and hyperbolic graphene for s = 20meV.

Then, the overall value of K = dµdn can be found by

evaluating the convolution of the homogeneous (non-disordered) K with this distribution:

dµ

dn=

∫ ∞

−∞

dµ(n− n′)

d(n− n′)P (n′)dn′. (5)

This has the effect of broadening any sharp features inthe homogenous K, such as the δ-function at n = 0 ina gapped system or the step associated with the changein topology of the Fermi surface in the hyperbolic band.Specifically, evaluating this convolution gives the follow-ing contribution to K for the hyperbolic band from theδ-function:

uγ1√

u2 + γ21

exp(

−n2/2(δn)2)

δn√2π

so that this spike is broadened into a peak where theheight is determined by the size of the effective gap 2E∗.The second model assumes that the disorder potential

introduces variation in the density of states (DoS). Thepotential itself is assumed to follow a Gaussian distribu-tion P (V ) parameterized by width s, and therefore thedisordered DoS is given by20

D(E) =

∫ ∞

−∞

D0(E − V )P (V )dV. (6)

This modified DoS then gives the zero-temperature elec-tron density by integrating over energy so that

n(EF ) =

∫ ∞

−∞

D(E)Θ(EF − E)dE (7)

where EF is the global Fermi energy and Θ(x) is theHeaviside step function. A similar expression exists forthe density of holes. This density is assumed to holdthroughout the graphene so that the averaged density can

now be used to evaluateK = dµ(n)dn . Since we assume that

the disorder does not induce any additional charge in the2D system, the net carrier density ne − nh is conserved.Therefore, to compute the effective Fermi energy EF inthe presence of disorder we solve the equation ne(EF )−

6

nh(EF ) = ne0(EF )− nh

0 (EF ) for EF . However, the totalcarrier density ne+nh is not conserved, and the primaryeffect of disorder in this model is to introduce a finitetotal density for any finite amount of charged impurities.

The integrals in Eqs. (6) and (7) can be computed an-alytically for the linear and quadratic bands if we assumethat P (V ) is a Gaussian distribution. They are

nl =1

πh̄2v2F

[

E2F + s2

2erfc(∓y)± EF s√

2πe−y2

]

and

nq =γ1

2πh̄2v2F

[

s√

2π e

−y2 ± EF erfc(∓y)

]

.

where the upper sign corresponds to the electron densityand the lower sign to the hole density, erfc is the com-plementary error function, and y = EF /(

√2s) with s the

standard deviation of the distribution of the disorder po-tential. We see that the effect of disorder is to introducea tail in the electron density which extends into the con-duction band, the size of which increases with increasingdisorder strength. In the hyperbolic case, the integralsin Eqs. (6) and (7) cannot be evaluated analytically, so anumerical comparison must be done with the other twocases.

The expressions which must be computed for the hyperbolic band are

nhe =1

s√2π

1

πh̄2v2F

∫ EF

−∞

dE

[

∫ E−E∗

−∞

dV (E − V ) [2 + F(E − V )] e−V 2/(2s2)

−∫ E−E∗

E−u

2

dV (E − V ) [2−F(E − V )] e−V 2/(2s2)

]

and

nhh =1

s√2π

1

πh̄2v2F

∫ ∞

EF

dE

[

−∫ ∞

E+E∗

dV (E − V ) [2 + F(E − V )] e−V 2/(2s2)

+

∫ E+u

2

E+E∗

(E − V ) [2−F(E − V )] e−V 2/(2s2)

]

where ∆ = 1 + u2/γ21 , F(x) = ∆/

√

∆x2/γ21 − δ2 and in

both cases the second term comes from the inner Fermisurface for E∗ < |E| < u/2.Figure 4 shows the density of carriers in these three

cases for a modest value of disorder (s = 20meV) andalso for u = 50meV in hyperbolic graphene. The smalldensity of states in linear graphene gives a small carrierdensity at low Fermi energy. Quadratic and hyperbolicgraphene have finite density of states at charge-neutralityso the density is higher in this case. The density in hyper-bolic graphene increases slightly quicker than quadraticgraphene because of the linear increase in the densityof states compared to the constant density of states inquadratic graphene. Also, the finite density of states atthe charge-neutrality point in quadratic and hyperbolicgraphene means that the effect of disorder is significantlystronger in these systems than in linear graphene. In fact,substituting EF = 0 into the total density gives the de-pendence of the density at the charge-neutrality point asa function of the disorder strength s as

nl(EF = 0) =s2

πh̄2v2F, nq(EF = 0) =

√

2

π

γ1s

πh̄2v2F

so that the linear increase in n(EF = 0) is faster thanthe quadratic one for small s. The density at charge-neutrality in hyperbolic graphene is identical to that inquadratic graphene because the density of states is thesame in both cases. We emphasize that both models ofdisorder considered here are physically motivated and a

priori there does not appear to be any practical reasonto choose one over the other. In the next section, wecompare theory with recent graphene compressibility ex-perimental data to establish the comparative validity ofour disorder models.

V. EXPERIMENTAL COMPARISON: THE

EFFECT OF DISORDER

We now discuss our results in the context of recentlypublished experiments. In particular, two measurementshave been reported where the capacitance of bilayergraphene has been measured and the compressibility ex-tracted from those measurements.7,8 The quantum ca-pacitance is linked to dµ

dn since CQ = Ae2 dndµ where A is

7

(a)

CSCQ

Ct

Young, et al.

(b)

CSCbCQ

Ct

Henriksen, et al.

FIG. 5. Classical circuit diagrams for the experiments by (a)Henriksen et al.8, and (b) Young et al.7

the surface area of the sample. The two experiments ac-tually measure slightly different capactitances which arerepresented by the two circuit diagrams in Fig. 5 andcorrespond to the following expressions for the measuredcapacitances CH and CY :

CY =Aǫ0ǫt

ǫtdg + dt+ CS Young Ref. 7

CH =ǫ0ǫbǫtdgA

dg(ǫbdt + ǫtdb) + dbdt+ CS Henriksen Ref. 8

The notation follows that defined in Fig. 2(a), exceptthat dg is an effective length associated with the quantumcapacitance as

CQ =Aǫ0dg

⇒ dg =ǫ0e2

dµ

dn.

The quantity CS is the background (or stray) capaci-tance associated with, for example, the substrate andexperimental equipment. For relevant experimental pa-rameters, dg ≪ db, dt so that CH ∝ dg ∝ dµ

dn . We find

the relevant experimental parameters from the papers18

and use them to produce the plots shown in Figs. 6 and7.Figure 6 shows the comparison of data from both dis-

order models to the experiments7 of Young et al. In thiscase, the back gate is kept at a fixed potential while thetop gate is varied. This has the effect of simultaneouslychanging the size of the gap and the carrier density so inthis case we treat these two quantities seperately and usethe full theory of the inter-layer screening described inSection II to compute the size of the effective gap. Whenthe gap is small at low density (i.e. for Vb = 10V, plot(a)) the both theories of disorder predict qualitativelycorrect answers. The difference between the experimen-tal and theoretical curves can be reduced by changingthe value of the stray capacitance as a fitting parameter.For very low values of disorder, a small spike is presentat zero density in the density averaged data. This is dueto the very small band gap that is present and the asso-ciated δ-function in dµ

dn . Larger values of disorder smearthis divergence out into a dip, the shape of which matchesthe experimental data well. The DoS averaged plots donot show this dip because carrier density is always fi-nite in this model, and the effect of increasing disorder is

110

115

120

-4 -2 0 2 4

CY

(fF

)

Vt (V)

(b)Vb = +70 V

Exp. data

den. av., ν = 0.1×1011

den. av., ν = 6×1011

DoS av., s = 40meV

110

112

114

116

118

CY

(fF

)

(a)

Vb = +10 V

-4 -2 0 2 4Vt (V)

(c)Vb = -80 V

FIG. 6. Comparison of theoretical results including disorderfor the measured capacitance CY with experimental data byYoung et al.7

to increase the minimum value of the carrier density thusdecreasing the depth of the dip. When the gap is large atzero density (i.e. for large back gate voltage, Vb = 70V,plot (b) and Vb = −80V, plot(c)) then the DoS aver-aging procedure predicts qualitatively wrong results forlow values of disorder. There are two main features inthe low density part of these plots. First, for finite gap,Fig. 3 shows that dµ

dn tends to zero as density decreases.This gives a peak in the measured capacitance (because

CY ∝ 1/(dµdn + dt

ǫt)) which is blurred by finite disorder.

The second feature is the δ-function at the band edge(or, equivalently, at zero density) which is broadened byfinite disorder in the density averaging model, but whichplays no role in the DoS averaging model because thismodel always gives rise to a finite density of carriers.This means that the DoS average cannot predict correctresults in the low-density, finite gap regime. However,the density averaging model does include the effects ofthe δ-function spike by sampling many values of density,and crucially including n = 0. This spike in dµ

dn mani-fests as the dip in the measured capacitance and param-eters can be chosen so that the size and shape of the dipmatch the experimental data quite closely. We thereforebelieve the density-averaging procedure of Eq. (5) to bethe preferable model for including disorder over the theDoS-averaging model.

This general picture is also seen in the comparison ofthe two disorder theories with data by Henriksen et al.

as shown in Fig. 7. Panels (c) and (d) show the compari-son of the experimental data to the DoS averaged model.In this experiment, the measured capacitance is propor-tional to dµ

dn , so in the gapped case in panel (d), the step

8

55

56

57

58C

H (

fF)

(a)

Density average

Exp. data

ν = 0.1×1011

ν = 3×1011

u = 25meV

(b)

Densityaverage

55

56

57

58

-4 -2 0 2 4

CH

(fF

)

Density (1012 cm-2)

(c)

u = 0

DoS average

s = 0s = 30 meV

-4 -2 0 2 4Density (1012 cm-2)

(d)

s = 30meVDoS average

(b) and (d)u = 40 meVu = 140 meV

FIG. 7. Comparison of theoretical results including disorderwith experimental data by Henriksen et al.8 (a) uext = 0compared with the density averaged model for zero gap anda small gap; and u = 25meV for ν = 3× 1011cm−2. (b) uext

finite, compared with the density averaged theory for severalvalues of the gap. We have taken ν = 3× 1011cm−2 and thelegend is as shown in (d). (c) DoS averaged theory comparedwith uext = 0 expermimental data for s = 0 and s = 30meV.(d) uext finite compared with DoS averaged theory for twovalues of the gap. s = 30meV in each case.

in dµdn due to the change in topology of the Fermi surface

is clearly seen for large gap. The zero gap data in panel(c) shows qualitatively the correct features, although thestray capacitance must be altered to get a better fit andthe peak is taller in the experimental data than in the the-ory. But when the gap is opened as shown in part (d), theDoS averaged theory predicts qualitatively wrong resultsbecause it does not account for the broadened δ-functionassociated with the band edges. The experimental dataalso shows a distinct asymmetry between the electronand hole density regimes. This is not explained in ourmodel, but can be added by allowing for an asymmetry inthe underlying band structure, for example by includingspecific next-nearest neighbor hops in the tight binding

model. On the other hand, the density-averaging theory(shown in panels (a) and (b)) do predict the correct be-havior. For the zero-gap experiment, the variation of thepredicted CH is too small, and better agreement can befound by allowing for a small gap to open as shown bythe dashed line. In panel (b), it is clear that a gap size of≈ 40meV gives a semi-quantitative fit to the experiment.

VI. CONCLUSION AND DISCUSSION

In conclusion, we have presented a full calculation of dµdn

(and hence the quantum capacitance and compressibil-ity) for graphene and have discussed two different meth-ods for including disorder effects. These models requireaveraging the effect of the disorder potential in eitherthe density of states or directly as a variation in the localelectron density. We find that when there is no gap inthe system, these two models predict qualitatively similarresults. But when there is a gap present, the DoS averag-ing method fails to account for the step in the chemicalpotential due to the band edges and therefore predictsqualitatively incorrect results. So, we present two conclu-sions: First, that the single-particle theory for the com-pressibility of graphenes appears to allow qualitativelygood predictions for thermodynamic quantities such asthe compressibility. Second, the inclusion of disorder viastatistical averaging must be done carefully because thechoice of where the averaging is done critically affects theoutcome. We find that the averaging should be carriedout at the level of the experimental quantity rather thanthe density of states.

We believe that the effects left out of the theory, par-ticularly the many-body exchange-correlation correctionsare likely to be qualitatively unimportant for under-standing the experimental data for currently availablegraphene samples which are dominated by disorder. Inparticular, disorder effects are strongest at the lowest car-rier densities where exchange-correlation effects becomemore important quantitatively. This makes an observa-tion of many-body effects through compressibility mea-surements quite challenging.

We thank J. P. Eisenstein, E. A. Henriksen, P. Kim,and A. Young for discussions of their experimental re-sults. This work was supported by US-ONR and NRI-SWAN.

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