Giant Magnetodrag in Graphene at Charge Neutrality

更新时间:2023-06-28 12:38:34 阅读: 评论:0

Giant Magnetodrag in Graphene at Charge Neutrality律师函英文
M.Titov,1R.V.Gorbachev,2,3B.N.Narozhny,4T.Tudorovskiy,1M.Schu¨tt,5P.M.Ostrovsky,6,5,7I.V.Gornyi,5,8
A.D.Mirlin,5,4,9M.I.Katsnelson,1K.S.Novolov,2A.K.Geim,2,3and L.A.Ponomarenko2
1Radboud University Nijmegen,Institute for Molecules and Materials,NL-6525AJ Nijmegen,Netherlands
2School of Physics and Astronomy,University of Manchester,Manchester M139PL,United Kingdom 3Centre for Mesoscience and Nanotechnology,University of Manchester,Manchester M139PL,United Kingdom 4Institut fu¨r Theorie der Kondensierten Materie and DFG Center for Functional Nanostructures,
Karlsruhe Institute of Technology,76128Karlsruhe,Germany
5Institut fu¨r Nanotechnologie,Karlsruhe Institute of Technology,76021Karlsruhe,Germany
6Max-Planck-Institut fu¨r Festko¨rperforschung,Heinbergstras1,70569,Stuttgart,Germany
7L.D.Landau Institute for Theoretical Physics RAS,119334Moscow,Russia
8A.F.Ioffe Physico-Technical Institute,194021St.Petersburg,Russia
9Petersburg Nuclear Physics Institute,188350St.Petersburg,Russia
(Received1April2013;published14October2013)
We report experimental data and theoretical analysis of Coulomb drag between two cloly positioned
graphene monolayers in a weak magneticfield.Clo enough to the neutrality point,the coexistence of
electrons and holes in each layer leads to a dramatic increa of the drag resistivity.Away from charge
neutrality,we obrve nonzero Hall drag.The obrved phenomena are explained by decoupling of
electric and quasiparticle currents which are orthogonal at charge neutrality.The sign of magnetodrag
depends on the energy relaxation rate and geometry of the sample.
DOI:10.1103/PhysRevLett.111.166601PACS numbers:72.80.Vp,73.22.Pr,73.50.Àh
Recent measurements[1]of frictional drag in graphene-bad double-layer devices revealed the unexpected phe-nomenon of giant magnetodrag at charge neutrality. Applying external magneticfields as weak as0.1–0.3T results in a reversal of the sign and a dramatic enhancement of the amplitude of the drag resistance.If the device is doped away from charge neutrality,the impact of such a weakfield on the drag resistance is very modest.The obrved effect weakens at low temperatures,hinting at the classical origin of the phenomenon.
In this Letter,we report experimental data on longitudi-nal and Hall drag resistivity in isolated graphene layers parated by a1nm thick boron-nitride(hBN)spacer.The obrved effects are explained in terms of coexisting elec-tron and hole liquids in each layer[2,3].This theory is bad on the hydrodynamic description of transport in graphene derived in Refs.[4–6]using the quantum kinetic equation(QKE)framework[7,8].It provides a simplified description of magnetodrag while capturing the esntially classical physics of the phenomenon[9].The effect can be traced back to the fact that the Lorentz force in the electron and hole bands has the opposite sign,which is also the reason for the anomalously large Nernst effect[10,11]and vanishing Hall effect at charge neutrality.
The classical mechanism behind giant magnetodrag is illustrated in Fig.1.The upper panel shows two infinite graphene layers at charge neutrality.The driving electric current j1in the active layer corresponds to the counter-propagatingflow of electrons and holes with zero total momentum due to exact electron-hole symmetry(hence, in the abnce of additional correlations there is no drag at the Dirac point[5,12–15]).In a weak magneticfield, electrons and holes are deflected by the Lorentz force and drift in the same direction.The resulting quasiparticle flow P1carries nonzero momentum in the
direction FIG.1(color online).Mechanism of magnetodrag at charge
neutrality.Upper panel:In an infinite system quasiparticle currents P i in the two layersflow in the same direction,leading to positive drag D xx¼V2=j1>0.Lower panel:In a thermally isolated sys-tem no net quasiparticleflow is possible(leading to inhomogene-ities in the quasiparticle density);quasiparticle currents in the two layers have opposite directions yielding negative drag.
perpendicular to j1.Momentum transfer by the interlayer Coulomb interaction induces the quasiparticle current P2 in the same direction as P1.Lorentz forces acting on both types of carriers in the passive layer drive the chargeflow in the direction opposite to j1.In an open circuit,this current is compensated by afinite voltage V2,yielding a positive drag resistivity D xx¼V2=j1>0.
This mechanism of magnetodrag is cloly related to the anomalous Nernst effect in single-layer graphene [4,10,11].At charge neutrality,the quasiparticle current is proportional to the heat current.A similar mechanism, where the role of P i is played by spin currents,has been propod in Ref.[16]as a possible explanation for a giant nonlocal magnetoresistance.
The above argument describes the steady state in infinite systems where all physical quantities are homogeneous in real space.This is not the ca in relatively small,meso-scopic samples.Whether a
alternateparticular sample should be considered‘‘small’’or‘‘large’’is determined by compar-ing the sample size to the typical length scale of the leading relaxation process.At high enough temperatures,energy is most efficiently relaxed by electron-phonon scattering, which we describe in this Letter by the length‘ph[17,18].
In afinite system,quasiparticle currents must vanish at the boundaries.For W)‘ph,the quasiparticle current and density is homogeneous in the bulk and the system remains effectively infinite.
For W(‘ph,the currents P i become y dependent.In this ca,energy conrvation dictates P2ðyÞ¼ÀP1ðyÞ. As a result,electric charge in the passive layer tends toflow in the same direction as j1(e Fig.1),yielding negative drag(similarly to Coulomb drag in single-band systems). In order to test the above ideas,we perform new mea-surements of the drag effect in a magneticfield,illustrated in Fig.2.The experiments[19]are carried out on a gra-phene double-layer structure with a1nm hBN spacer and two electrostatic gates.Despite the small thickness of the spacer,the tunneling resistance between the layers >300k gives only a negligible leakage contribution to the drag<0:5 [19,20].The schematics of the experi-ment are shown in the int of Fig.2(e).The same device was ud in Ref.[1]for drag measurements in zero mag-neticfield.
The map for the drag resistivity, D xxðV T;V BÞ,is shown in Fig.2(a)at T¼240K.The main difference compared to the zerofield experiment reported earlier[1]is
large
(c)
(e)(f)
FIG.2(color online).(a)Longitudinal drag resistivity in magneticfield as a function of the top(V T)and bottom(V B)gate voltages. Lines track positions of maxima in single-layer resistivity in top(open symbols)and bottom(solid symbols)layers.(b)Magnetodrag for n1¼n2¼n at T¼160K.Solid symbols reprent the experimental data.The error bars for the data are within the symbol size. The theoretical lines show solutions to Eqs.(6).(c)Magnetodrag for n1¼n2¼n at T¼240K.Solid symbols:experimental data;
lines:theory(6).(d)Map of Hall drag resistivity as a function of V T and V B.The white diagonal area corresponds to vanishing Hall drag for n1¼Àn2.The lines are the same as in(a).(e)Experimental data(blue squares,left axis)and theory(red solid line,right axis) for the Hall drag resistivity for n1¼n2¼n.The theoretical curve is calculated on the basis of the microscopic theory of Ref.[5].Note the sign change at n%Æ2Â1011cmÀ2.Int:schematics of Hall drag measurements in double-layer system.(f)Hall drag resistivity for n1¼n2¼n.The data(blue squares)are identical to tho in(e);the red
solid line reprents solutions to Eq.(6).
negative drag at the double neutrality point.A dramatic change in drag resistivity with the applied magnetic field is shown in more detail in Figs.2(b)and 2(c)(at 160and 240K,respectively).To ensure the same charge densities n 1and n 2in the top and bottom layers,we sweep both gates simultaneously along the line connecting the bottom left and top right corners of the map.The experiment shows a large negative drag resistivity clo to the double neutrality point,n 1¼n 2¼0,as expected for a small sample (e above);in our device both layers have the width W %2 m and sufficiently resistive contacts.
In addition to the longitudinal drag resistivity we also measure the Hall drag resistivity, D xy ðV T ;V B Þ,shown in Fig.2(d)at T ¼240K as a function of the top and bottom gate voltages.Becau of the low density of states in graphene and the small paration between layers (d %1nm ,interlayer capacitance $2:2 F =cm 2),the relationship between gate voltages and charge densities is rather nontrivial (e the Supplemental Material [19]).To identify signs of charge carriers at each point in Fig.2(d),we also measured resistivity maps for both layers.Since the resistance of graphene is peaked at charge neutrality,tracking the position of the resistivity maximum gives the lines which divide the map into the electron-and hole-doped parts.Such lines are shown in both maps;e Figs.2
(a)and 2(d).The obrved Hall drag resistance is large when one of the layers is clo to the neutrality point and vanishes if two layers have the same charge densities with opposite signs (a white line running from the top left to bottom right corner).捡便宜
We now turn to the theoretical description of the drag effect.Consider first the Drude model for electrons and holes in two layers,
e E i þe ½v ie ÂB  ¼F ie þe v ie =M i ;(1a)Àe E i Àe ½v ih ÂB  ¼F ih þe v ih =M i ;
(1b)
where i ¼1,2,a ¼e ,h ,v ia stand for the mean velocities of electrons and holes in the layer i ,E i and B are electric and magnetic fields,and M i are the carrier mobilities due to impurity scattering.The electric j i and quasiparticle P i currents are related to v ia by [2]
新视野大学英语听说教程3j i ¼e ðn ie v ie Àn ih v ih Þ;P i ¼n ie v ie þn ih v ih ;(2)
with n ie ðh Þ¼R 10d" ð"Þ½e
ð"Ç i Þ=T
þ1 À1standing for the electron and hole densities, ð"Þ¼2j "j = ð@v Þ2being the density of states in graphene at B ¼0,and  i are the chemical potentials measured from the Dirac point.The total charge and quasiparticle densities are defined as n i ¼n ie Àn ih and  i ¼n ie þn ih .
The frictional force acting on each type of carrier can be reprented as
F ia ¼@X
jb
½ ab ij n jb ðv ia Àv jb Þþ~ ab ij n jb ðv ia þv jb Þ ;
(3)where the coefficients ~
appear in monolayer graphene due to the abnce of Galilean invariance.The expression (3)can be obtained by solving the QKE in the hydrodynamic approximation [4–7,21].
For n i ¼0,the first term in Eq.(3)simplifies to
F 1a ¼ÀF 2a ¼F ¼@ ðP 1ÀP 2Þ;
(4)
where  ¼@=ðT P Þ,with  À1
P being the momentum re-laxation rate.The cond term in Eq.(3)renormalizes the mobilities [5,7,8].The Drude model (1)with the force (4)also describes the ca  i )T ,where  ¼@=ð  P Þ.In both limits,the model (1)is equivalent to the hydrody-namic transport equations derived from the QKE [5,21].For strongly doped graphene, i )T ,the quasiparticle current and density are obsolete:P i ¼j i =e and  i ¼n i .Equations (1)are then reduced to the standard Drude model yielding conventional drag  D xx ¼E 2x =j 1x ¼
À@ =e 2
with negligible dependence on the magnetic field [22]and vanishing Hall drag  D xy ¼E 2y =j 1x ¼0.
In contrast,at charge neutrality the quasiparticle and charge degrees of freedom are inequivalent.The quasipar-ticle density for n i ¼0is determined by the temperature, i ¼ 0¼ T 2=3ð@v Þ2,while the currents j 1and P i become orthogonal;e Fig.1.
Rewriting Eqs.(1)and (4)in terms of currents,we obtain the resistivity tensor.For n i ¼0,the longitudin
chatal drag resistivity is given by
D xx ðn i
¼0Þ¼@ e 2
B 2M 1M 2
1þ@  0ðM 1þM 2Þ=e
情人节快乐用英语怎么说;(5)
which describes positive drag in an infinite system in agreement with the qualitative picture;e Fig.1,upper panel.In the limit of weak interaction, MT 2(@ev 2,the result (5)can be obtained from the standard perturbative approach [13]modified for graphene in the classical mag-netic field.
The large negative peak in  D xx at the double neutrality point [Fig.2(b)]suggests that the sample width W %2 m is relatively small as compared to ‘ph (Fig.1,lower panel).To account for the finite sample width,we rewrite the equations (1)in terms of the currents j i and P i and allow for the spatially varying quasiparticle density, i ðy Þ.The resulting model for the first layer reads
ÀK 1r  1þen 1E 1þ½j 1ÂB  ¼ 1F 1þe P 1=M 1;
(6a)e 1E 1þe ½P 1ÂB  ¼n 1F 1þj 1=M 1;
(6b)r ÁP 1¼Àð 1À 0Þ= ph Àð 1À 2Þ=ð2 Q Þ:
(6c)
Here K i ¼ð @2v 2=2Þð@n i =@ i Þ¼2T ln ð2cosh  i =2T Þis the mean quasiparticle kinetic energy.In the cond layer the force F enters with the opposite sign.The continuity equation for the quasiparticle current [Eq.(6c )]includes
relaxation by the electron-hole recombination [2],with  À1
ph describing the energy loss from the system,which is domi-nated by phonon scattering (e the Supplemental Material
[19]); À1Q characterizes quasiparticle imbalance relaxation
due to interlayer Coulomb interaction.For  À1ph ¼0,the
hard-wall boundary conditions in y directions require P 1þP 2¼0.Near the Dirac point,energy and momentum relaxation rates coincide ( Q $ P ).In doped graphene,recombination rates are exponentially suppresd [19].The continuity equation for the electric current simply reads r Áj i ¼0;hence,j i ¼ðj i ðy Þ;0Þ.Within linear respon,the equilibrium density  0has to be substituted into products  i F and  i E .This way we obtain the linear system of differential equations for P iy ðy Þ,j 1x ðy Þ,and  i ðy Þ.Since the charge current acquires the depen-dence y coordinate,we define  D xx ¼E 2x =h j 1x i ,where h j 1x i ¼W À1R W 0j 1x dy .
The model (6)with the frictional force (4)admits a full analytic solution (e the Supplemental Material [19])in
terms of  À1Q , À1ph ,and  À1
P .The resulting behavior crucially depends on the rates:in particular,in the abnce of phonons ( ph !,in a thermally isolated system)drag at the Dirac point is always negative;e Fig.1.For vanishing sample width (W !0),we find  D xx %ÀB 2W 2=ð24 0K Q Þ.In general,the rates depend on n i and have to be determined by the microscopic theory [5].Relegating further mathematical details to the Supplemental Material [19],we prent the results of our calculations in Fig.2alongside experimental data.The drag resistivity  D xx is plotted in Figs.2(b)and 2(c)for T ¼160K and T ¼240K ,respectively.The expo-nential collap of theoretical curves at high carrier density is an artifact of our phenomenological model [17,19].At higher temperature [Fig.2(c)],the drag resistivity exhibits qualitatively new features near charge neutrality which can be physically attributed to higher efficiency of relaxation process.The sign of  D xx at the Dirac point is then determined by the relation between the typical relaxation
length ‘ph ¼2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K ph M=e q and the sample width.This is illustrated in Fig.3,where we plot  D xx as a function of magnetic field for different values of W choosing realistic values for T ¼240K ,M ¼4m 2=Vs ,and ‘ph ¼1:2 m .Bad on the above results,we predict that in wider samples giant magnetodrag at the Dirac point should become positive.We also speculate that magnetodrag at charge neutrality may become
positive in stronger fields due to the magnetic-field dependence of the scattering times  Q , P ,and  ph .
The model (6)allows us to calculate the Hall drag resistivity  D xy .The result is shown in Fig.2(f).The theory
also predicts vanishing Hall drag for the ca of oppositely doped layers,n 1¼Àn 2.Interestingly enough,the data
show a sign change of  D xy at n %Æ2Â10
11
cm À2.At that point the effect is rather weak and requires a more accurate consideration.Using the microscopic theory of Ref.[5],we have evaluated the Hall drag resistivity for an infinite sample with an energy-independent impurity scat-tering time  .The value of  was determined from the measured single-layer resistivity and we have ud the most plausible estimate for the effective electron-electron interaction parameter in graphene on hBN, %0:2.The result is shown in Fig.2(e)along with the corresponding experimental data without any fitting.
In conclusion,we have measured the longitudinal and Hall drag resistivity in double-layer graphene and pro-vided a theoretical description of the obrved effects.Giant magnetodrag at the neutrality point appears due to the prence of two types of carriers (electrons and holes),which in weak magnetic fields experience a uni-directional drift orthogonal to the driving current.This effect is specific to the neutrality point,where nonzero drag appears despite the exact electron-hole symmetry.Our theory does not rely on the Dirac spectrum in gra-phene,but is equivalent to the microscopic theory [5,9]at and far away from charge neutrality,capturing the esntial physics of magnetodrag.For a more accurate description of the effect at intermediate densities,the microscopic theory should be formulated on the basis of the QKE [21].
We are grateful to the Royal Society,the Ko
¨rber Foundation,the U.S.Office of Naval Rearch,the U.S.Air Force Office of Scientific Rearch,the Engineering and Physical Sciences Rearch Council (UK),the EU IRSES network InterNoM,Stichting voor
Fundamenteel
FIG.3(color online).The magnetic field dependence of the
longitudinal drag resistivity at the neutrality point.The positive sign of the magnetodrag in weak fields corresponds to the limit W )‘ph ,where ‘ph %1:2 m for the parameters of the plot.The magnetic-field dependence of scattering rates is disregarded in the plot.
Onderzoek der Materie(FOM,Netherlands),DFG SPP 1459,and BMBF for support.
Note added.—Recently,we became aware of a related work by Song and Levitov[23].
[1]R.V.Gorbachev,  A.K.Geim,M.I.Katsnelson,K.S.
Novolov,T.Tudorovskiy,I.V.Grigorieva,  A.H.
MacDonald,K.Watanabe,T.Taniguchi,and L.A.
Ponomarenko,Nat.Phys.8,896(2012).
[2]M.S.Foster and I.L.Aleiner,Phys.Rev.B79,085415
(2009).
[3]  D.Svintsov,V.Vyurkov,S.Yurchenko,T.Otsuji,and V.
Ryzhii,J.Appl.Phys.111,083715(2012).
[4]M.Mu¨ller and S.Sachdev,Phys.Rev.B78,115419
(2008);M.Mu¨ller,L.Fritz,and S.Sachdev,ibid.78, 115406(2008).
[5]M.Schu¨tt,P.M.Ostrovsky,M.Titov,I.V.Gornyi,B.N.
Narozhny,and A.D.Mirlin,Phys.Rev.Lett.110,026601 (2013);M.Schu¨tt,Ph.D.thesis,Karlsruhe Institute of Technology(KIT),2013,digbib.ubka.uni-karlsruhe .de/volltexte/1000036515.
[6]J.Lux and L.Fritz,Phys.Rev.B86,165446(2012).
[7]L.Fritz,J.Schmalian,M.Mu¨ller,and S.Sachdev,Phys.
Rev.B78,085416(2008);M.Mu¨ller,J.Schmalian,and L.
Fritz,Phys.Rev.Lett.103,025301(2009).
[8]  A.B.Kashuba,Phys.Rev.B78,085415(2008).
[9]The hydrodynamic description of drag in graphene derived
in Ref.[5]was justified by the singular behavior of the collision integral due to kinematics of Dirac fermions.This singularity leads to the fast unidirectional thermalization and allows one to lect the relevant eigenmodes of the collision integral[4,7].Projecting the collision integral onto the modes,one arrives at the effective model,which is equivalent to Eq.(1)with the generalized force(4).[10]Y.M.Zuev,W.Chang,and P.Kim,Phys.Rev.Lett.102,
096807(2009).
[11]P.Wei,W.Z.Bao,Y.Pu,C.N.Lau,and J.Shi,Phys.Rev.
Lett.102,166808(2009).
[12]W.K.T,Ben Yu-Kuang Hu,and S.Das Sarma,Phys.
Rev.B76,081401(2007).
[13]  B.N.Narozhny,M.Titov,I.V.Gornyi,and P.M.
Ostrovsky,Phys.Rev.B85,195421(2012).
[14]M.Carrega,T.Tudorovskiy,A.Principi,M.I.Katsnelson,
and M.Polini,New J.Phys.14,063033(2012).
[15]  B.Amorim and N.M.R.Peres,J.Phys.Condens.Matter
24,335602(2012).
[16]  D.A.Abanin,S.V.Morozov,L.A.Ponomarenko,
R.V.Gorbachev,  A.S.Mayorov,M.I.Katsnelson,K.
Watanabe,T.Taniguchi,K.S.Novolov,L.S.Levitov,
and A.K.Geim,Science332,328(2011).
汽车装潢培训[17]The microscopic theory[5,21]includes thermoelectric
effects formulated in terms of energy currents.The
corresponding hydrodynamic description yields only theielts
power-law decay of the magnetodrag at i)T,in contrast to the exponential collap shown in Figs.2(b)and2(c).
datat
At the Dirac point the energy current is equivalent to the
quasiparticle current P.
[18]J.C.W.Song,M.Y.Reizer,and L.S.Levitov,Phys.Rev.
Lett.109,106602(2012).
[19]See Supplemental Material at link.aps/
supplemental/10.1103/PhysRevLett.111.166601for details.
[20]Y.Oreg and    B.I.Halperin,Phys.Rev.B60,5679
(1999).
[21]M.Schu¨tt et al.(to be published).
[22]  A.-P.Jauho and H.Smith,Phys.Rev.B47,4420(1993);
K.Flensberg,Ben Yu-Kuang Hu,A.-P.Jauho,and J.M.
2259
Kinaret,Phys.Rev.B52,14761(1995);A.Kamenev and
Y.Oreg,Phys.Rev.B52,7516(1995).
[23]J.C.W.Song and L.S.Levitov,Phys.Rev.Lett.111,
126601(2013).

本文发布于:2023-06-28 12:38:34,感谢您对本站的认可!

本文链接:https://www.wtabcd.cn/fanwen/fan/78/1058900.html

版权声明:本站内容均来自互联网,仅供演示用,请勿用于商业和其他非法用途。如果侵犯了您的权益请与我们联系,我们将在24小时内删除。

标签:新视野   装潢   教程   汽车   大学   培训   听说
相关文章
留言与评论(共有 0 条评论)
   
验证码:
推荐文章
排行榜
Copyright ©2019-2022 Comsenz Inc.Powered by © 专利检索| 网站地图