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The thermopower and Nernst Effect in graphene in a magnetic field Joseph G. Checkelsky and N. P. Ong Department of Physics, Princeton University, New Jersey 08544, U.S.A. (Dated: January 12, 2009) Wereportmeasurementsof thethermopower S andNernstsignal Syx ingraphenein amagnetic field H. Both quantities show strong quantum oscillations vs. the gate voltage Vg. Our measure- mentsfor Landau Levelsof index n6=0 arein quantitativeagreement with theedge-current model of Girvin and Jonson (GJ). The inferred off-diagonal thermoelectric conductivity αyx comes close 9 tothequantumofAmpsperKelvin. AttheDiracpoint(n=0),however,thewidthofthepeakin 0 0 αyx isverynarrow. Wediscussfeatures ofthethermoelectric responseat theDiracpointincluding theenhanced Nernst signal. 2 n a In graphene, the linear dispersion of the electronic ingvalueofα isthenauniversalquantum(k e/h)ln2 yx B J states near the chemical potential µ is well described with units of Amperes per Kelvin (k is Boltzmann’s B 2 by the Dirac Hamiltonian. As shown by Novoselov et constant). In turn, δI produces a quantized Hall volt- y 1 al. [1, 2, 3] and by Zhang et al. [4, 5, 6] quantization of age V = (h/e2)δI that drops ||xˆ. Hence, conflating H y the electronic states into Landau Levels leads to the in- these 2 large off-diagonaleffects, the thermopower S be- ] l teger quantum Hall Effect (QHE). Because of the linear comes very large when µ aligns with E (n 6= 0). By l n a dispersion, the energy E of the Landau Level (LL) of contrast, the transverse (Nernst) voltage is small (in the n h index n varies as E = sgn(n) 2e~v2B|n|, where B is absence of disorder). - n F s themagneticinduction,vF thepFermivelocity,etheelec- In graphene, this picture needs revision when µ is e tron charge, and h is Planck’s constant. The quantized at the Dirac point. For the n = 0 LL, the nature m Hall conductivity is given by of the edge currents is the subject of considerable de- . bate [11, 12, 13, 14]. What are the profiles of S and mat σxy = 4he2 (cid:18)n+ 21(cid:19), (1) oSfyx1?4 TWeath2a0veanmdea5s0uKre.dOSuijr raensdulRtsijretvoeaalmthaaxti,mautm9 TH, - the thermoelectric response in graphene already falls in d where the factor 4 reflects the degeneracy g of each LL the quantum regime at 50 K. The inferred off-diagonal n (2 each from spin and valley degrees). o Detailed investigations of the longitudinal resistance currentresponseαxy isaseriesofpeaksclosetothequan- c R and Hall resistance R have been reported by sev- tumvalue(gkBe/h)ln2(independentofn,BandT). We [ xx xy compare our results with the caculations of Girvin and eral groups [1, 2, 3, 4, 5, 6, 7]. By contrast, the ther- Jonson (GJ), and discuss features specific to the n = 0 2 moelectric tensor S is less investigated. S relates the ij ij v observed electric field E to an applied temperature gra- LL at the Dirac point. 6 ↔ Kim and collaborators [8] have pioneered a litho- dient −∇T, viz. E =S ·(−∇T). On the other hand, 6 graphicdesignfor measuringthe thermopowerofcarbon 8 the charge current density J produced by −∇T is ex- ↔ nanotubes. We have adopted their approach with minor 2 pressedbythethermolelectricconductivitytensor α,viz. . ↔ modifications for graphene. Using electron-beam lithog- 2 J= α ·(−∇T). AlthoughJisnotmeasureddirectly,αij raphy, we deposited narrow gold lines which serve as a 1 may be obtained by measurements of both Sij and the micro-heater to produce −∇T and thermometers (Fig 08 resistivity tensor ρij =Rij. (By convention, −Sxx is the 1d). The latter are also used as current leads when Rxx : thermopowerS;werefertoSyx =Ey/|∇T|astheNernst and Rxy are measured. Above 10 K, the thermometers v signal.) can resolve δT ∼ ±1 mK. The typical δT is ∼10 mK Xi A most unusualfeature of the thermoelectric response between the 2 thermometers. However, the small spac- of a QHE system (for n 6= 0) is that, despite the domi- ing of the voltageleads (∼2 µm) leads to an uncertainty r a nanceoftheoff-diagonal(Hall-like)currentresponse,the of ±10% in estimating δT between them. A slowly os- thermopower displays a large peak at each LL whereas cillating current at frequency ω is applied to the heater the Nernst signal is small. In the geometry treated by and the resulting thermoelectric signals are detected at Girvin and Jonson [9, 10] (Fig. 1a), the 2D sample is of 2ω and −90◦ out of phase. finite width alongxˆ, but is infinite alongyˆ (with the ap- Inourgeometrywith−∇T||xˆ,H||ˆz,thechargecurrent plied magnetic field H||ˆz). As we approach either edge, density J is (summation over repeated indices implied) the LL energy E rises very steeply (bold curve). At n T = 0, edge currents I exist at the intersections (open y J =σ E +α (−∂ T) (i=x,y). (2) circles)ofE withthechemicalpotentialµ. Inagradient i ij j ij j n −∇T, the magnitude of I is larger at the warmer edge y than at the cooler edge because of increased occupation The (2D) conductivity and thermoelectric conductivity of states above µ. The difference |δIy| is a maximum tensors are often written as σij = L1ij1(e2/T) and αij = when µ is aligned with E in the bulk. The correspond- L12/T2, respectively. Setting J = 0, we have for the n ij 2 (a) En -20 -10 0 10 20 30 40 30 f(E) y m Iy f(E) 6 5 T J3, T = 2(0a K) 20 D x - T 4 10 0 2 (b) (c) -10 m S 0 -20 6 9 T (b) 40 ) ) K T(K) 2 e/h 4 20 V / ( G xx 0 S ( 2 (d) -20 0 6 14 T (c) 40 50 K 20 4 20 K FIG. 1: (Color online) (Panel a) The effect of −∇T on the edge currents Iy in a QHE system (n 6= 0). The energy 0 En of a LL (bold curve) increases very steeply at the sam- 2 ple edges, with µ the chemical potential (dashed line). If -20 Hz > 0, Iy is negative (positive) at the left (right) edge, as 0 indicatedbyopencircles. Themagnitude|Iy|islarger atthe -20 -10 0 10 20 30 40 warmer edge. Fermi-Dirac distributions f(E) are sketched VG (V) at the sides. (Panel b) Curves of thermopower S = −Sxx vs. gate voltage Vg in Sample J10 at selected T. The curves areantisymmetricabouttheDiracPointwhichoccursat the FIG. 2: (Color online) Variation of thermopower S vs. Vg offset voltage V0 = 15.5 V. The peak value Sm is nominally (bold curves) and conductance Gxx vs. Vg (thin curves) in linear in T from 25 to 300 K (Panel c). Less complete data Sample J3 at H = 5, 9 and 14 T (Panels a, b and c, respec- from sample K59 are also plotted. A photo of Sample J10 tively). The offset V0 = 12.5 V. All curves were measured at (faint polygon)isshown inPanel(d). Amicro-heateraswell 20K exceptfor thecurveat 50K in Panel(c). Verticallines asthermometers(therm)andsignalleadsarepatternedwith locate themaxima of Gxx. electron-beam lithography. Theblack scale bar is 3 µm. Underfieldreversal(H→-H),Sissymmetricwhereas observed E-fields S is antisymmetric. For each curve taken in field, we yx E =−ρ α (−∂ T)=S (−∂ T), (3) repeatthemeasurementwithHreversed. AllcurvesofS i ik kj j ij j andS reportedherehavebeen(anti)symmetrizedwith yx with ρij = Rij the 2D resistivity tensor. The ther- respect to H. As for charge-inversion symmetry, we ex- mopowerS =−Ex/|∇T|equalsρxxαxx+ρyxαxy (S >0 pectthesignofS tochangewiththeshiftedgatevoltage forholedoping),whiletheNernstsignalisgivenby(with Vg′ ≡Vg−V0 (i.e. between hole and electronfilling), but ρxx =ρyy) the sign of Syx stays unchanged (V0 is the offset voltage). However, we have not imposed charge-inversion S =ρ α −ρ α . (4) yx xx xy yx xx symmetrization constraints on the curves. Apart from The2termstendtocancelmutually,exceptattheDirac field (anti)symmetrization, all the curves are the raw point where ρ vanishes (see below). data. yx Inverting Eq. 3, we may calculate the tensor αij from Figure 1b shows traces of S vs. Vg at selected T. The measured quantities. We have thermopower S changes sign as Vg crosses the charge- neutralpoint(DiracPoint),assumingpositive(negative) αxx = −(σxxEx+σxyEy)/|∇T| values on the hole (electron) side. The peak value Sm is α = (−σ E +σ E )/|∇T|. (5) nominally T-linear from ∼20 K to 300 K (Fig. 1c). xy xy x xx y 3 -20 -10 0 10 20 30 40 -20 -10 0 10 20 30 40 20 10 (a) 6 5 T 8 (a) K) 6 xy 10 A/ 4 (n 4 xy 0 , xx 2 2 0 -2 J3, T = 20 K xx T = 20 K 0 -10 -4 H = 9 T 6 9 T (b) 30 -6 K) 10 2 e/h) 4 20 V / 8 (b) G (xx 2 010 S (yx (nA/K) 46 xy xy 2 -10 , xx 0 0 6 14 T (c) 40 -2 xx T = 20 K 30 -4 H = 14 T 20 -6 4 -20 -10 0 10 20 30 40 10 Vg (V ) 2 0 -10 0 -20 FIG.4: (Coloronline) Thethermoelectricresponsefunctions -20 -10 0 10 20 30 40 VG (V) α14xxT(f(abi)ntatcuTrv=e)2a0ndKαcxaylcu(blaotledd) vfrso.mVgSaatn9dTSy(xPa(Eneql.a5)).anIdn (a),thewidthofthepeakatn=0isnarrowerthantheothers byafactorof5. Inbothpanels,verticallineslocatethepeaks F(bIoGld. 3c:ur(vCeosl)oarnodnlcionned)uVcatarinacteioGnxoxfvNse.rVngst(tshiginnacluSrvyexs)vsin. VJ3g of Gxx. Thehorizontal dashed line is (4kBe/h)ln2. at the 3 field values H = 5, 9 and 14 T (Panels a, b and c, respectively). Allcurvesweremeasuredat20K.Verticallines locate the maxima of Gxx. The sign of Syx was incorrectly profiles of S. Moreover, S is smaller in magnitude by yx assigned in a previous version of this paper [15]. a factor of 4-5. For the n = 0 LL, however, the profiles change character, with S displaying a large positive yx peak. The reason for the enhancement of S at the yx In sharp contrast to the smooth variation in Fig. 1, Dirac point is discussed below. The positive sign of Syx thecurvesofS vs. Vg showpronouncedoscillationswhen at the n=0 LL implies that the Nernst E-field EN is H is finite, reflecting Landau quantization of the Dirac parallel to H×(−∇T) [15]. states. Figures2a,bandcdisplayS vs. Vg (boldcurves) In the theory of GJ [9], valid for GaAs-based with H fixed at the values 5, 9 and 14 T, respectively. devices, the edge current difference is δI = y For comparison, we have also plotted (as thin curves) (ge/h) dk(∂ǫ/∂k)(∂f/∂T)δT,with f(ǫ) the Fermi- n the corresponding conductance Gxx = σxx. (With the DiracdPistriRbution. Theoff-diagonaltermαxy isgivenby exception of the curve at 50 K in Panel (c), all curves (e/hT) ∞ dǫ(ǫ−µ) −∂f . When µ=E , α at- were measuredat 20 K.) Whereas at large|n|, the peaks tains aPpenakREvnalue, corres(cid:16)pon∂dǫi(cid:17)ngto a quantizend cuxryrent in S are aligned with those in G (vertical lines), at xx ′ per Kelvin, given by (g=1) n =±1, they disagree. In Panel a, the peaks for V <0 g (hole doping) decrease systematically in magnitude as n increases from 1 to 4. k e αmax = B ln2 (∼2.32nA/K). (6) The curves of the Nernst signal S are displayed in xy h yx Fig. 3. For n 6= 0, S displays a dispersive profile yx centered at the vertical lines, in contrast with the peak GJ find that S displays a series of peaks, with the peak 4 value at LL n given by within 30% of the quantized value of Eq. 6 with g = 4 (horizontal dashed line). The largest uncertainty in our k ln2 B measurement is in estimating the gradient between the S (n)= . (7) peak e (n+ 1) voltage leads. For the n = 0 LL, the shortfall may also 2 reflect incipient splitting of the Landau sublevels (com- At low T, S is independent of H and T. pare 9 and 14 T traces). The uniformity of the peaks in In Fig. 2, the peak value of S at the n = −1 LL in- α accountsfortheobservedenhancementoftheNernst xy creases from 25 at 5 T to 41 µV/K at both 9 and 14 peak at the Dirac point. By Eq. 4, S is the difference yx T. Moreover, as T increases from 20 to 50 K (Panel c), of 2 positive terms. For n6=0, the 2 terms are matched, the peak increases only weakly (41 to 48 µV) in sharp and the partial cancellation leads to a dispersive profile. contrast with the T-linear behavior at H=0 (Fig. 1, in- However, for n = 0, the vanishing of ρ as V′ → 0 yx g set). We thus confirm the prediction of GJ that S (at strongly suppresses the second term −ρ α . The re- yx xx the peak) saturates to a value independent of T and H mainingtermρ α thendictatesthesizeandprofileof xx xy at sufficiently low T. This saturation contrasts with the the Nernst peak. T-linear behavior of S in H=0 (Fig. 1c). We note that the sign of α is a direct consequence In graphene, however, Berry phase effects lead to a xy the 1-integer shift in Eq. 1 [4]. In evaluating σ ∼ of the edge-currents [9]. When Hz > 0, Iy <0 on the 2 xy warmeredgeandI <0onthecooleredge,asdepictedin 1 y f(E ), the -integer shift implies that S (n) de- n n 2 peak Fig. 1a. As a result, αxy >0 for both hole-andelectron- Pcreases as kBln2/(en), instead of Eq. 7. The measured doping. values S = 41 µV/K at n = -1 at 9 T already ex- peak Our measurements are consistent with the GJ theory ceeds slightly the predicted value 39.7 µV/K in Eq. 7. for n 6=0. For the n=0 LL, however, there is consider- Future experiments on cleaner samples may yield values able uncertainty about the nature of the edge states in closer to the predicted value 59.6 µV/K for Dirac sys- graphene (or whether they exist in large H). The sim- tems. Regardless, for LL with n 6= 0, our observations ple edge-current picture for understanding the peaks in aregenerallyconsistentwiththeGJtheory. Inprinciple, α may need significant revision for n = 0, despite the xy Eq. 7 provides a way to measure δT on micron-scales similarity of the peak magnitude. We also note that the with a resolution approaching voltage measurements. peak at n =0 is much narrower (by a factor of 5) than Themostinterestingquestionisthe thermoelectricre- the other peaks. This is also not understood. By going sponse of the n=0 LL. This is easier to analyze using to cleaner samples and higher fields, we hope to exploit the pure thermoelectric currents α and α (obtained xx xy this narrow width to resolve splitting of the 4 sublevels using Eqs. 5). In Fig. 4a, α and (α ) is plotted as xy xx at n = 0. bold (thin) curves for H=9 T and T = 20 K. Panel (b) WearegratefultoP.A.Leeformanydiscussions. The shows the curves at 14 T. Compared with S vs. V , the g research is supported by NSF through a MRSEC grant peaksinα aremuchnarrowerandclearlyseparatedby xy (DMR-0819860). intervals in which α is nominally zero. Likewise, the xy purely dispersive profile of α is also more apparent. Note added After we completed these experiments, we xx Consistent with the GJ theory, the overall magnitude of learned of 2 thermopower and Nernst experiments on α (for n6=0) is larger than that of α . graphenepostedrecently (YuriM. Zuevet al., cond-mat xy xx A striking feature of α is that its peaks are inde- arXiv: 0812.1393 and Peng Wei et al., cond-mat arXiv: xy pendent of n. Their average value ∼75 nA/K reaches to 0812.1411). [1] K.S. Novoselov et al. Science 306, 666-669 (2004). (1984). [2] K. S. Novoselov et al. Proc. Natl. Acad. Sci. USA 102, [11] H. A. Fertig and L. Brey, Phys. Rev. Lett. 97, 116805 10451-10453 (2005). (2006). [3] K.S. Novoselov et al. Nature 438, 197-200 (2005). [12] D. A. Abanin, P. A. Lee and L. S. Levitov, Phys. Rev. [4] Y. Zhang, J. Tan, H. L. Stormer and P. Kim, Nature Lett. 96, 176803 (2006). 438, 201-204 (2005). [13] K. Nomura and A. H. MacDonald, Phys. Rev. Lett. 96, [5] Y.Zhang et al. Phys.Rev.Lett. 96, 136806 (2006). 256602 (2006). [6] Z. Jiang, Y. Zhang, H. L. Stormer and P. Kim, Phys. [14] D.A.Abaninet al. Phys.Rev.Lett.98,196806 (2007). Rev.Lett. 99, 106802 (2007). [15] Inapreviouspostedversionofthismanuscript(cond-mat [7] Joseph G.Checkelsky,LuLi,and N.P.Ong,Phys.Rev. arXiv:0812.2866.v1), the sign of Syx was incorrectly as- Lett.100, 206801 (2008). signed.Aftercarefulcheckingandfurthermeasurements, [8] J.P.Small,K.M.Perez,andP.Kim,Phys.Rev.Lett.91, we have determined that Syx >0 and αxy >0 at n=0 256801 (2003). as reported here. The sign of Ey is the same as that for [9] S.M.GirvinandM.Jonson,J.Phys.C:SolidStatePhys. the vortex-Nernsteffect in superconductors. 15, L1147 (1982). [10] M. Jonson and S. M. Girvin, Phys. Rev. B 29, 1939