Thermodynamics of electron-hole liquids in graphene
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Thermodynamics of electron-hole liquids in graphene
L.A. Falkovsky L.D. Landau Institute for Theoretical Physics RAS, 119334 MoscowL.F. Verechagin Institute of the High Pressure Physics RAS, 142190 Troitsk (Dated: November 13, 2018)The impact of renormalization of the electron spectrum on the chemical potential, heat capacity,and oscillating magnetic moment is studied. The cases of low and high temperatures are considered.At low temperatures, doped graphene behaves as the usual Fermi liquids with the power tempera-ture laws for thermodynamic properties. However, at high temperatures and relatively low carrierconcentrations, it exhibits the collective electron-holes features: the chemical potential tends to itsvalue in the undoped case going with the temperature to the charge neutrality point. Simultane-ously, the electron contribution into the heat capacity tends to the constant value, as in the case ofthe Boltzmann statistics.
PACS numbers: 65.80.+n,71.70.Di, 71.18.+y
I. INTRODUCTION
Optic and magneto-optic experiments with graphenelayers have been successfully interpreted so far in ascheme of massless relativistic particles with a conicalenergy spectrum ε s ( p ) = ∓ vp (1)where v is the constant velocity parameter in two bands, s = 1 ,
2, near the K and K’ points in the Brillouin zone.In pure graphene, the chemical potential is situated atthe charge neutrality point ε = 0. However, it can have anonzero value because of doping or under a gate voltage.Thus, the chemical potential is determined by the totalnumber of carriers (difference of electrons in the upperband and holes in the low band) N = 4 S Z | f ( ε − µ ) − f ( ε + µ ) | d p (2 π ¯ h ) , (2)where f ( ε − µ ) is the Fermi function, S is the surfaceof the graphene layer, and the factor 4 takes the valleyand spin degeneracy into account. The integration isperformed over ε >
0, the chemical potential is positivefor electrons and negative for holes. At the fixed N , thiscondition determines the dependence µ ( T ), shown in Fig.1 for a relatively low electron concentration.For the conical spectrum, Eq. (1), the ratio betweenthe kinetic and Coulomb energies has a constant valueindependently of the carrier concentration and the prob-lem of the phase electron-dielectric transition becomesundefined. It was recently discovered in studying ofthe Shubnikov-de Haas oscillations that electron-electroninteractions are very important for low carrier concen-trations, p →
0. While the electron concentration de-creases from 10 to 10 cm − , the velocity parameter v grows by three times from its ordinary value 1.05 × cm/s. The logariphmic renormalization of the velocityfor the linear electron dispersion was found by Abrikosovand Beneslavsky in the three-dimensional case and inRefs. for two-dimensional graphene. Notice, that no C he m i c a l po t en t i a l ( K ) Temperature (K) n =10 cm −2 FIG. 1: Chemical potential versus temperature for the carrierconcentration 10 cm − ; the exact solution to Eq. (2) isshown by the solid line, the asymptotes are for low, Eq. (4),and high, Eq. (6), temperatures (dashed and dashed-dottedlines, correspondingly); the renormalization is not included. phase transition was revealed even at the lowest carrierconcentration. We can conclude that Coulomb interac-tions do not create any gap in the spectrum.The renormalized electron dispersion can be written inthe form ε s ( p ) = ∓ vp [1 + g ln( p /p )] , (3)where g = e / π ¯ hvǫ is the dimensionless electron-electron interaction and ǫ ≃ . p ≃ . × cm − is the cutoff parameter . In Fig. 2, we consider ascreening effect on the chemical potential at the carrierconcentration n = 10 cm − .Equation (3) is written in the linear approximation in g ln( p /p ) <
1. Because the logarithm is assumed to belarge, the condition g ≪ II. TEMPERATURE DEPENDENCE OF THECHEMICAL POTENTIAL
For low ( µ ≫ T ) and high ( µ ≪ T ) temperatures, theanalytical expressions for µ ( T ) can be obtained from Eq.(2) with the renormalization taken into account.For low temperatures, it is convenient to differentiateEq. (2) with respect the temperature, using df ( ε − µ ( T )) dT = (cid:20) ε − µT + dµdT (cid:21) (cid:20) − ∂f ( ε − µ ) ∂ǫ (cid:21) . Here, we have a sharp function of ( ε − µ ). Therefore, inthe integrand, the momentum p = ε [1 − g ln( p v/ε )] /v should be expand near ε = µ in powers of ( ε − µ ), whichgives a factor proportional to T after the integration. Forinstance, we get in the case of electron doping0 = Z ∞−∞ (cid:20) − ∂f ( ε − µ ) ∂ǫ (cid:21) (cid:20) µ dµdT + ( ε − µ ) T (cid:21) × [1 − g ln( p v/µ )] dε , where we do not differentiate the logarithm because ofthe condition g ≪
1. Integrating, one finds dµdT = − π ε F T , (4)where we denote ε F ≡ µ ( T = 0), positive for electronsand negative for holes. Let us notice that this is theknown temperature dependence of the chemical potentialin the degenerate Fermi system at low temperatures. Weemphasize that the Fermi energy ε F is determined indeedby the carrier concentration n = p F π ¯ h = 1 π (cid:16) ε F ¯ hv (cid:17) [1 − g ln( p v/ | ε F | )] , (5)which introduces the renormalization in Eq. (4) bymeans of ε F .For high temperatures, we can expand the integrandin Eq. (2) in µ . Introducing the new variable x = ε/ T ,we get the integral N = 4 | µ | STπ (¯ hv ) Z ∞ − g ln( p v/ T x )cosh x xdx , which gives the chemical potential | µ | = π n (¯ hv ) T [1 + 2 g ln( p v/ T )] . (6)We see the inverse temperature dependence of the chemi-cal potential, as a collective effect in electron-hole liquids.The renormalization term, correcting the temperaturedependence, is presented here explicitly and illustratedin Fig. 2. III. HEAT CAPACITY
Now we consider the electron contribution in the heatcapacity. The energy of carriers E = 4 S Z ∞ ε | f ( ε − µ ) − f ( ε + µ ) | d p (2 π ¯ h ) (7)differs from the carrier concentrations, Eq. (2), only bythe additional factor ε in the integrand. Therefore, wecan follow the same procedure.For low temperatures, T ≪ ε F , the carrier heat capac- C he m i c a l po t en t i a l ( K ) Temperature (K) n =10 cm −2 with el−el interwitout el−el inter FIG. 2: Chemical potential versus temperature for the car-rier concentration 10 cm − ; the renormalization is included(solid line), the chempotential for noninteracting electrons isshown in the dashed lines; the cutoff parameter p = 0 . × cm − , the dielectric constant ǫ = 2 . ity in the case of electron doping writes as C ( e ) S = 2 Sπ (¯ hv ) Z ∞−∞ " µ dµdT + 2 µ ( ε − µ ) T × (cid:20) − ∂f ( ε − µ ) ∂ε (cid:21) [1 − g ln( p v/µ )] dε = 2 Sπ (¯ hv ) (cid:20) µ dµdT + 2 π µT (cid:21) [1 − g ln( p v/µ )] . Using Eq. (4), we have C ( e ) S = 2 πS | ε F | hv ) T [1 − g ln( p v/ | ε F | )]in both cases of the electron or hole doping.For high temperatures, T ≫ µ , one can perform theexpansion of the energy, Eq. (7), in the first order of µE = 4 S | µ | π (¯ hv ) Z ∞ ε (cid:18) − ∂f ( ε ) ∂ε (cid:19) [1 − g ln( p v/ε )] dε = 2 πS | µ | hv ) T [1 − g ln( p v/ T )] . Using Eq. (6), we find E = π N T and C ( e ) S = π N .
Finally, C ( e ) S = π N (cid:26) T | ε F | , T ≪ | µ |
12 ln 2 , T ≫ | µ | (8)Thus, we see that the renormalization modifies the heatcapacity at low temperatures, i.e., in the degeneratestatistics. At high temperatures, the heat capacity pos-sesses the constant value and does not reveal any renor-malization at least to a first approximation in g ln( p /p ). IV. MAGNETIC SUSCEPTIBILITY
Magnetic susceptibility is determined by the depen-dence of the thermodynamic potential on the magneticfield Ω( B ) = − eBT Sπ ¯ hc X n,s ln (cid:16) e µ − εsnT (cid:17) in terms of the electron dispersion ε sn for two bands s =1 , n = 0 , , .. We neglect thespin splitting of the levels in comparison with the largeLandau splitting in graphene.Oscillations of the magnetic moment in the semi-classical region can be found applying the Poison formulato the thermodynamic potentialΩ( B ) = − eBT Sπ ¯ hc X k =0 Z ∞ n ln (cid:16) e µ − εT (cid:17) + ln (cid:16) e µ + εT (cid:17)o e πikn dn , where the contributions of two bands are written explic-itly. The integraton by parts givesΩ( B ) = − eBT Sπ ¯ hc X k =0 Z ∞ ik [ f ( ε − µ ) − f ( − ε − µ )] e πikn dε . (9)For the semi-classical region, we use the quantization rulein the Bohr–Zommerfeld form2 πn = cA ( ε ) e ¯ hB with the aria enclosed by the electron trajectory for theenergy ε in the momentum space A ( ε ) = π (cid:16) εv (cid:17) (1 − g ln( p v/ε )according to Eq. (3).The main contribution in the integral (9) comes fromthe vicinity of the point ε = ± µ ( T = 0) = ± ε F forthe positive and negative ε F , correspondingly. Expand-ing the exponent in the integrand near that points andintegrating, one findsΩ( B ) = 2 eBT Sπ ¯ hc X k =0 k sin[ kcA ( ε F ) /e ¯ hB ]sinh(2 π kc | m ( ε F ) | T /e ¯ hB ) , where m ( ε ) = π dA ( ε ) dε is the cyclotron mass. In calculat-ing of the magnetic moment we can derivative only therapid factor in the argument of sin with respect B :˜ M ( B ) = 2 πn STB X k =0 cos[ kcA ( ε F ) /e ¯ hB ]sinh[2 π kc | m ( ε F ) | T /e ¯ hB ] , (10)where the carrier concentration n = A ( ε F ) / ( π ¯ h ) . Thisis the standard Lifshiz-Kosevich formula used in Ref. forthe interpretation of experimental data concerning thevelocity renormalization. There are two important fea-tures: first, the aria A ( ε F ) and the effective mass m ( ε F )should be taken at the renormalized Fermi energy cor-responding to the carrier concentration and, second, thefactor in front of the sum differs from the 3d case sincethe integration over p z is absent now.It is interesting to compare the amplitude of oscilla-tions with the monotonic part of the magnetic moment,Ref. , M = − S π (cid:16) evc (cid:17) BT cosh ( µ/ T ) . Thus, we see that the ration of the oscillating and mono-tonic parts of the magnetic moment has the order | ˜ M /M | ∼ πn (cid:18) cTevB (cid:19) cosh ( ε F / T )sinh[2 π c | m ( ε F ) | T /e ¯ hB ] . To observe the oscillations, the argument of sinh has to besmall or at least on the order of unity. Then, the mono-tonic part of the magnetic moment can be observable onlyat relatively high temperatures, | M / ˜ M | ∼ exp( − ε F /T ). V. CONCLUSIONS
It should be emphasize that such a transport prop-erty as the electronic conductivity is not sensitive to theelectron-electron interaction since the conductivity doesnot depend indeed on the velocity parameter v . Therenormalization of the electron spectrum due to Coulombinteractions in graphene is noticeable in thermodynamicproperties especially at low temperatures and for smallcarrier concentrations n < cm − , as can be seenfrom Eqs. (6), (8), and (10). However, the interest-ing temperature dependences µ ∼ T − for the chemicalpotential at high temperatures T ≫ µ , appears indepen-dently of electron-electron interactions. The detection of the renormalization requires the high accuracy in exper-iments because the renormalization can be concealed byincreasing of the velocity parameter v [see Eq. (5) andFig. 2]. Acknowledgments
We gratefully acknowledge Andrey Varlamov for use-ful discussions. This work was supported by the RussianFoundation for Basic Research (grant No. 13-02-00244A)and the SIMTECH Program, New Centure of Supercon-ductivity: Ideas, Materials and Technologies (grant No.246937). A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S.Novoselov, A.K. Geim, Rev. Mod. Phys.
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