A calculation for polar Kerr effect in high temperature cuprate superconductors
Girish Sharma, Sumanta Tewari, Pallab Goswami, V. M. Yakovenko, Sudip Chakravarty
AA calculation for polar Kerr effect in high temperature cuprate superconductors
Girish Sharma , Sumanta Tewari , Pallab Goswami , V. M. Yakovenko , , and Sudip Chakravarty Department of Physics and Astronomy, Clemson University, Clemson, SC 29634 Department of Physics, CMTC, University of Maryland, College Park, Maryland 20742 Department of Physics, Joint Quantum Institute, University of Maryland, College Park, Maryland 20742 Department of Physics and Astronomy, University of California Los Angeles, Los Angeles, California 90095
A mechanism is proposed for the tantalizing evidence of polar Kerr effect in a class of high temperaturesuperconductors–the signs of the Kerr angle from two opposite faces of the same sample are identical andmagnetic field training is non-existent. The mechanism does not break global time reversal symmetry, as in anantiferromagnet, and results in zero Faraday effect. It is best understood in a phenomenological model of bilayercuprates, such as
YBa Cu O δ , in which intra-bilayer tunneling nucleates a chiral d -density wave such thatthe individual layers have opposite chirality. Although specific to the chiral d -density wave, the mechanismmay be more general to any quasi-two-dimensional orbital antiferromagnet in which time reversal symmetry isbroken in each plane, but not when averaged macroscopically. I. INTRODUCTION.
The origin and nature of the pseudogap phase in the high- T c cuprate superconductors still remains an unresolved prob-lem . The pseudogap phase, which occurs in the under-doped regime of hole doping, and at temperature range T ∗ >T > T c , displays many interesting properties including vari-ous charge, spin, electron nematic, or current ordered statescompeting with superconductivity . Recently a nonzeropolar Kerr effect (PKE) has been observed in the pseudogapphase in a number of recent experiments , but with un-usual characteristics. The effect measures the angle of rotationof linearly polarized light reflected from a medium at normalincidence and typically signals time reversal symmetry (TRS)breaking in the reflecting medium . In a ferromagnetic ma-terial, the signs of the polar Kerr angle from two opposite sur-faces of the same sample are expected to be different. Thisis because the net magnetic moment points in the same direc-tion throughout the sample and hence if it points away fromthe sample on the top surface, it points into the sample on thebottom surface, see Fig. 1. Moreover, It should be possible tochoose (or ‘train’) the direction of the net magnetic moment,and, in turn, the sign of the polar Kerr angle, by cooling thesample in the presence of a magnetic field.In contrast, in high- T c superconductors it has been observedthat the signs of PKE from the two opposite surfaces of thesame sample are identical, and, moreover, the signal cannot betrained by magnetic field. To account for these puzzling ex-perimental observations, time-reversal invariant models withgyrotropic order were employed recently . However, theconcept of gyrotropic order as an explanation for non-zeroPKE in the cuprates were subsequently retracted becauseit does not satisfy Onsager’s reciprocity principle in normalreflection that forbids a non-zero PKE in the absence of TRSbreaking .Here we show that the observations in high- T c can be un-derstood in the framework of a chiral d -density wave state in the presence of interlayer tunneling, which is invariant un-der TRS in the bulk, but can still have a non zero PKE be-cause it is a property of the light reflected from the top surfacewhich breaks TRS locally. Therefore PKE in this mechanism FIG. 1.
Left panel:
Ferromagnetic (FM) ordering where magneticmoments point in the same direction at each plane, thus resulting inthe opposite signs of the Kerr angle from the top and bottom surfaces.
Right panel:
Orbital antiferromagnetic (OAF) ordering, where mag-netic moments switch sign on each plane, so the Kerr angle has the same sign from both surfaces. In contrast to FM, magnetic momentsin OAF point out of the sample at the top and the bottom surfaces,provided the sample is cleaved between the bilayers. would be insensitive to the changing skin depth of the incom-ing light at the top surface, while PKE from a bulk order pa-rameter based description would yield a stronger effect for alonger skin depth. The chiral d -density wave state is definedby adding a small d xy component to the dominant id x − y ,i.e. with the combined order parameter d xy + id x − y . Thenet order parameter breaks TRS at each CuO plane and resultsin a non-zero Hall conductivity σ xy . The addition of a possi-ble d xy component could be a result of microscopic electronicinteractions , or a structural transition that breaks the symme-try between the neighboring plaquettes. In the id x − y state,by itself, spontaneous currents alternatingly circulate aroundplaquettes of the two-dimensional square lattice, thus preserv-ing the macroscopic TRS, but not any associated chirality.We establish that, for our present model, the angle of ro-tation due to one layer is cancelled by its neighbor, result-ing in zero Faraday rotation of the polarization plane of thetransmitted light. However, since PKE is primarily a sur-face phenomenon, where the light reflected from the top (orbottom) surface at normal incidence changes its plane of po-larization, there can be a non-zero PKE. Furthermore, sincebilayer cuprates usually cleave through the reservoir layers a r X i v : . [ c ond - m a t . s t r- e l ] M a r FIG. 2. (Color online)
Left:
Band dispersion as a function of k =( k x , π ) for the two states in a bilayer: d + id/d + id (blue) and d + id/d − id (red). We utilized t (cid:48) = 0 . t, W = . t, ∆ =0 . W / , and t ⊥ = 0 . t all energies measured in the unit of t . Right:
The ground state energy versus hole doping (from 0.08 to0.18) indicating that the d + id/d − id state (in red) has lower energyin a bilayer for any given value of hole doping. separating the CuO bilayers, the magnetizations at the topand bottom surfaces should point opposite to each other (seeFig. 1), giving rise to the same sign of the Kerr angle. Fi-nally, since the system as a whole is an OAF, coupling to asmall external magnetic field should be small, resulting, mostlikely, in a small or non-existent magnetic field ‘training’ ef-fect. Importantly, to the best of our knowledge, the scenariopresented here is the only one consistent with all the puz-zling phenomenology seen in the recent PKE experiments incuprates.It has been argued that much of the phenomenology of thecuprates in the underdoped regime can be unified bymaking a single assumption that the ordered id x − y -densitywave (DDW) state is responsible for the pseudogap. More-over, an extensive Hartree-Fock calculation for id x − y statehas recently been carried out . So far, evidence of mag-netism arising from d -density wave in neutron or NMR mea-surements has been controversial. However, the success ofthe present phenomenological model in explaining PKE mustspeak in favor of the suggested order parameter.This paper is divided as follows: in Section II, we introducethe chiral d -density wave state order parameter, and calculatethe anomalous Hall conductivity of a single layer. In Sec-tion III, we discuss the problem of light propagation througha single cuprate layer, and then calculate the Kerr and Fara-day responses through the bilayer system in Section IV. Weconclude in Section V. II. CHIRAL DDW WITH INTERLAYER TUNNELING.
Consider a combination of the density waves, d xy + id x − y , such that the net order parameter is (cid:104) c † k + Q ,α c † k ,β (cid:105) ∝ [ iW k − ∆ k ] δ αβ , (1)where c † k ,σ is the electron creation operator of momentum k and spin σ , and Q = ( π, π ) is the density wave vector. W k and ∆ k correspond to id x − y and d xy respectively, definedas W k = W (cos k x − cos k y ) , ∆ k = ∆ sin k x sin k y . Weconsider a bilayer system where the id x − y component of theorder parameter, i.e., W k may or may not switch sign betweenthe two layers. The four component mean field Hamiltonianfor the system in the basis ψ † k = ( c † k , c † k + Q , c † k , c † k + Q ) takesthe following form H ( k ) = (cid:15) k g k t ⊥ k g ∗ k (cid:15) k + Q t ⊥ k + Q t ⊥ k (cid:15) k g k t ⊥ k + Q g ∗ k (cid:15) k + Q , (2)where (cid:15) k is the energy dispersion for a two-dimensionalsquare lattice. (cid:15) k = − t (cos k x + cos k y ) + 4 t (cid:48) cos k x cos k y , (3)where t and t (cid:48) are the nearest and next-nearest hopping inte-grals in the tight-binding Hamiltonian, g k = iW k − ∆ k and t ⊥ k = t ⊥ (cos k x − cos k y ) / describes the tunneling be-tween the two layers appropriate for tetragonal systems. Thesuperscript (1,2) on the electron operator in ψ † k is the layer in-dex. Note that g k = g k represents a d + id/d + id bilayerconfiguration and g ∗ k = g k is a d + id/d − id configuration.We find that when g ∗ k = g k the system is energetically morefavorable than the case when g k = g k . This is observedby diagonalizing the Hamiltonian and obtaining the groundstate energy for a given doping concentration, as displayed inFig. 2. The d + id state spontaneously breaks time-reversalsymmetry ( T ) as well as the in-plane reflection symmetryabout the principal axes and exhibits anomalous Hall effectwith a non-zero value of the Hall conductivity σ xy . However,the value of σ xy reverses sign for the d − id state. Thus theground state of the bilayer breaks TRS in each plane, but sincethe inversion symmetry ( P ) about the mid point between theplanes is also broken, the product P T is conserved, allowingthe system to have a nonzero polar Kerr effect despite conserv-ing the global TRS and being an OAF . The magnetoelec-tric effect and PKE in another antiferromagnet Cr O werepredicted theoretically in , and subsequently observed inexperiments .The Hall conductance of a single layer described by a d + id mean field Hamiltonian can be calculated using theformalism of linear response theory and Kubo formula . Thetwo-component mean field Hamiltonian describing a d + id density-wave state in the ψ † k = ( c † k , c † k + Q ) basis is given by: H s ( k ) = (cid:18) (cid:15) k g k g ∗ k (cid:15) k + Q (cid:19) . (4)At a finite frequency ω and in the limit q → , the anomalousHall conductivity at any finite temperature is given by : σ xy ( ω ) = 2 e (cid:126) (cid:90) dk (2 π ) B ( k ) f ( E + ( k )) − f ( E − ( k )) w ( k )[ z − w ( k )][ z + 2 w ( k )] , (5)where B ( k ) = 4 t ∆ W (sin k y + cos k y sin k x ) is theBerry curvature, w ( k ) is the modulus of a three compo-nent vector w ( k ) = [ − ∆ k , − W k , ( (cid:15) k − (cid:15) k + Q ) / , f is theFermi distribution function, µ is the chemical potential, and z = ω + iδ , with δ a positive infinitesimal. E ± ( k ) =( (cid:15) k + (cid:15) k + Q ) / ± w ( k ) − µ describe the two energy bandsobtained by diagonalizing the Hamiltonian in Eq. (4). Thesign of σ xy is determined by the sign of the product ∆ W ,so the d ± id states have opposite signs of σ xy . III. TRANSMISSION AND REFLECTION OF LIGHTFROM A SINGLE LAYER.
We now study propagation of an electromagnetic wavethrough a layered system with chiral DDW using standardelectrodynamics formalism . First we consider anelectromagnetic wave incident normally on a single two-dimensional layer of a material in the xy plane. The electricfield components of the wave in a medium are given by ¯ E = e ikz (cid:20) E t + E t − (cid:21) + e − ikz (cid:20) E r + E r − (cid:21) (6) E t + and E t − are the transmitted components of right and leftcircularly polarized (CP) light respectively and similarly E r + and E r − are the reflected components, and k is the wavevec-tor. The corresponding magnetic field components can befound using Maxwell’s equation, ¯ k × ¯ E = ω ¯ H . The com-ponents of the electromagnetic field satisfy standard electro-dynamic boundary conditions at the material layer, which weassume is located at z = h : ¯ E >h = ¯ E
T R (cid:48) (cid:21) = R ++ R + − T (cid:48) ++ T (cid:48) + − R − + R −− T (cid:48)− + T (cid:48)−− T ++ T + − R (cid:48) ++ R (cid:48) + − T − + T −− R (cid:48)− + R (cid:48)−− (8)This scattering matrix S describes reflection and transmissionof electric field components from the top surface of the slab.We have also defined in Eq. (8) two-component matrices R , T (cid:48) , T and R (cid:48) , whose components are given by the corre-sponding block entries. Matching the boundary conditions at FIG. 3. (Color online) Schematic diagram showing multiple reflec-tions and transmissions through the top and bottom layers, which weuse to calculate the Kerr and Faraday angles. Note that we have as-sumed normal incidence for the incoming light in our calculations. z = h , we find that R ++ = e ik > h (cid:0) − n − (4 πσ xy ) + i πσ xy (cid:1) (1 + n ) + (4 πσ xy ) R −− = e ik > h (cid:0) − n − (4 πσ xy ) − i πσ xy (cid:1) (1 + n ) + (4 πσ xy ) T ++ = e i ( k > − k < ) h (cid:0) (cid:0) n (cid:1) + i πσ xy (cid:1) (1 + n ) + (4 πσ xy ) T −− = e i ( k > − k < ) h (cid:0) (cid:0) n (cid:1) − i πσ xy (cid:1) (1 + n ) + (4 πσ xy ) , where n is the refractive index of the medium. R (cid:48) ++ , R (cid:48)−− , T (cid:48) ++ ,and T (cid:48)−− can be obtained in a similar fashion. The othercomponents of S which couple right and left CP componentsi.e. R + − , T (cid:48)− + and so on, all vanish. We note that when σ xy (cid:54) =0 , R ++ (cid:54) = R −− and T ++ (cid:54) = T −− which is a signature ofbroken time-reversal symmetry. IV. POLAR KERR AND FARADAY EFFECTS IN BILAYERCHIRAL DDW.
To discuss scattering from the bilayer, we consider two suchinterfaces at z = + h and z = − h as depicted in Fig. 3. Sinceeven in the presence of the interlayer coupling t ⊥ ( k ) the sys-tem breaks P and T while P T is conserved, allowing a non-zero PKE , in the following we will ignore t ⊥ ( k ) for sim-plicity, expecting it to modify our results only quantitatively.The Hall conductivity σ xy reverses it sign at the bottom layerat z = − h . One can then appropriately define the scatter-ing matrix elements for the bottom layer taking into accountthe opposite sign of the Hall conductivity and the position ofthe bottom plane to be − h instead of h . We denote the two-component matrices defined in Eq. (8) for the top layer by thesubscript T and by subscript B for the bottom layer. Thusreflection and transmission through the bilayer as whole aredescribed by tensors R and T given by, R = R T + T (cid:48) T R B ( − R (cid:48) T R B ) − T T T = T B ( − R (cid:48) T R B ) − T T (9)We now switch basis from CP light to linearly polarized (LP)light for convenience of the following discussion. Denotingthe electric field of the light incident on the sample by ¯ E I , ¯ E R = R ¯ E i and ¯ E T = T ¯ E i give the reflected and the trans-mitted electric fields. When linearly polarized light is incidenton the sample, the Kerr and Faraday angles are determined bythe difference between right and left CP light: θ F = 12 ( arg [ E + T ] − arg [ E − T ]) θ K = 12 ( arg [ E + R ] − arg [ E − R ]) , (10)where E ± R,T = E xR,T ± iE yR,T , for ¯ E R,T = [ E xR,T , E yR,T ] . Forthe bilayer system discussed above, the R has non-zero offdiagonal elements (in LP basis) and T is diagonal, which is aclear signature of a non-zero Kerr response and the absence ofthe Faraday effect. (We do not state the analytic expressionsfor these matrices here, as they are too cumbersome.)We now make a rough estimate for the polar Kerr anglefor a bilayer system using Eqs. (9) and (10). Measuring allthe energies in units of t we use t (cid:48) = 0 . , µ = − . , n ≈ . , the interlayer distance h = 3 . ˚A, the strength of the id x − y component of the order parameter W ( p ) = 0 . − p/p c ) , where p is the hole doping concentration and p c =0 . , ∆ = 0 . W / and the frequency of measurement ω = 1500 nm. In Fig. 4, we have plotted the polar Kerr angle θ K as a function of hole doping and we obtain a non-zero Kerrangle of the order of 100 nrad. The estimated Faraday anglefrom our formalism turns out to be zero, again from Eqs. (9)and (10). Since the chiral DDW with interlayer tunneling is anOAF, the angle of rotation of the plane of polarization of lightdue to one layer is cancelled by its neighbor, resulting in zeroFaraday rotation of the transmitted light. However, since PKEis primarily a surface phenomenon, where the light reflectedfrom the top surface changes its plane of polarization, thereis a non-zero PKE. Further, since the magnetizations at thetop and bottom surfaces should point opposite to each other(see Fig. 1), the two surfaces give rise to the same sign ofthe Kerr angle. Finally, since the system as a whole is anOAF, coupling to a small external magnetic field should besmall, leading to small or non-existent ‘training’ effect. It isimportant to note that all of these conclusions are consistentwith the phenomenology of the recent PKE measurements inthe cuprates. V. CONCLUSIONS.
To conclude we considered the chiral DDW state in a bi-layer where the sign of the id x − y component of the orderparameter changes between the layers which is an energeti- cally more favorable configuration. This also leads to the re-versal of sign of σ xy in the bottom layer, thus breaking inver-sion symmetry. The calculations presented here are consis-tent with the unusual PKE observed in high- T c materials. Ourcalculations, although applied here specifically to the chiralDDW state, are more generally valid for any OAF with TRSbroken at each plane. In Ref. similar ideas were applied toa tilted loop current model. In addition, the ideas presentedhere also apply to the bi-axial density wave recently seen in FIG. 4. Estimated Kerr angle in nrad as function of hole doping p . The strength of the id x − y is assumed to vary with doping as W ( p ) = 0 . − p/p c ) eV, where p c is chosen to be 0.17. The am-plitude ∆ of the d xy component is assumed to be of W . Whilea non-zero PKE is a robust consequence of our model the precisevalues of W , ∆ , θ K in this figure are for illustrative purposes only. the pseudogap phase if they are accompanied by sponta-neous currents . However, the theories are currentlyformulated for a single layer, and it remains to be seen whetherthey can be generalized to a multilayer model with alternatingsign of σ xy similar to the present work.Another important class of high- T c materials is single layercompounds, such as Bi-2201 ( Bi Sr − x CuO δ ) and Hg-1201 ( HgBa CuO δ ). Although the detailed results are notyet published, it is known that such materials also show simi-lar PKE, as discussed here . At the level of order parametersymmetry, there is no difference, in the sense that one can eas-ily envision CuO-layers alternating between d + id and d − id .In addition, recent X-ray measurements indicate that the unitcell in the c -direction is doubled, bringing it closer to the bi-layer problem. Until PKE measurements in single layer ma-terials are published in detail, it is probably prudent to refrainfrom further speculations. Acknowledgement.
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