Quantum oscillations in electron doped high temperature superconductors
aa r X i v : . [ c ond - m a t . s up r- c on ] A ug Quantum oscillations in electron doped high temperature superconductors
Jonghyoun Eun, Xun Jia, and Sudip Chakravarty
Department of Physics and Astronomy, University of California Los Angeles, Los Angeles, California 90095-1547, USA (Dated: June 26, 2018)Quantum oscillations in hole doped high temperature superconductors are difficult to understand within theprevailing views. An emerging idea is that of a putative normal ground state, which appears to be a Fermi liquidwith a reconstructed Fermi surface. The oscillations are due to formation of Landau levels. Recently the sameoscillations were found in the electron doped cuprate, Nd − x Ce x CuO , in the optimal to overdoped regime.Although these electron doped non-stoichiometric materials are naturally more disordered, they strikingly com-plement the hole doped cuprates. Here we provide an explanation of these observations from the perspective ofdensity waves using a powerful transfer matrix method to compute the conductance as a function of the magneticfield. I. INTRODUCTION
Periodically new experiments tend to disturb the statusquo of the prevailing views in the area of high tempera-ture cuprate superconductors. Recent quantum oscillation(QO) experiments fall into this category. The first setof experiments were carried out in underdoped high qualitycrystals of well-ordered
YBa Cu O δ (YBCO), stoichio-metric YBa Cu O (Y124) and the overdoped single layer Tl Ba CuO δ .More recently oscillations are also observed in electrondoped Nd − x Ce x CuO (NCCO). The measurements inNCCO for 15%, 16%, and 17% doping are spectacular. Thesalient features are: (1) The experiments are performed in therange − T , far above the upper critical field, which isabout T or less; (2) the material involves single CuO plane,and therefore complications involving chains, bilayers, Ortho-II potential, etc. are absent; (3) stripes may not be germanein this case. It is true, however, that neither spin densitywave (SDW) nor d -density wave (DDW) are yet directly ob-served in NCCO in the relevant doping range, but QOs seemto require their existence, at least the field induced variety (see,however Ref. 16); (4) these experiments are a tour de force be-cause the sample is non-stoichiometric with naturally greaterintrinsic disorder. The effect is therefore no longer confined toa limited class of high quality single crystals; (5) The authorshave also succeeded in seeing the transition from low to highfrequency oscillations in NCCO as a function of doping.Here we focus on NCCO. We shall see that disorder playsan important role. Without it it is impossible to understandwhy the slow oscillations damp out below T for 15% and16% doping, and below T for 17% doping, even though thefield range is very high. For 17% doping, where a large holepocket is observed corresponding to very fast oscillations (in-consistent with any kind of density wave order), the neces-sity of such high fields can have only one explanation, namelyto achieve a sufficiently large ω c τ , where ω c = eB/m ∗ c , τ is the scattering lifetime of the putative normal phase, m ∗ the effective mass, and B the magnetic field. Qualitatively,the Dingle factor, D , that suppresses quantum oscillations is D = e − pπ/ω c τ where p is the index for the harmonic. As-suming a Fermi velocity, suitably averaged over an orbit to be v F , the mean free path l = v F τ . Thus D can be rewrittenas D = e − pπ ~ ck F /eBl . A crude measure for k F is given by expressing the area of an extremal orbit, A , as A = πk F . Bysetting m ∗ v F = ~ k F the explicit dependence on the parame-ters m ∗ and v F was eliminated. Assuming that the mean freepaths for the hole and the electron pockets are more or lessthe same, not an unreasonable assumption, the larger pockets,with larger k F , will be strongly suppressed for the same valueof the magnetic field because of the exponential sensitivity of D to the pocket size. This argument is consistent with our ex-act transfer matrix calculation using the Landauer formula forthe conductance presented below.Here we show that the oscillation experiments in NCCOreflect a broken translational symmetry that reconstructs theFermi surface in terms of electron and hole pockets. The em-phasis is not the transfer matrix method itself, but its use inexplaining a major experiment in some detail. We study bothSDW and singlet DDW orders with the corresponding meanfield Hamiltonians. A more refined calculation, beyond thescope of the present paper, will be necessary to see the subtledistinction between the two order parameters.In Sec. II we introduce our mean field Hamiltonians and inSec. III we discuss the transfer matrix method for the com-putation of quantum oscillations of the conductance. Sec. IVcontains the results of our numerical computations and Sec. Vour conclusions.
II. MEAN FIELD HAMILTONIAN
We suggest that the experiments in NCCO can be under-stood from a suitable normal state because the applied mag-netic fields between 30-65 T are so far above the upper criticalfield, which is less than 10 T, that vortex physics and the su-perconducting gap are not important. Our assumption is thata broken translational symmetry state with an ordering vector Q = ( π/a, π/a ) ( a being the lattice spacing) can reconstructthe Fermi surface resulting in two hole pockets and one elec-tron pocket within the reduced Brillouin zone, bounded bythe constraints on the wave vectors k x ± k y = ± π/a . Onechallenge here is to understand why the large electron pock-ets corresponding to 15 and 16% doping resulting from theband structure parameters for NCCO defined below are notobserved, but the much smaller hole pockets are. Anotherchallenge is to understand why the large Fermi surface at 17%doping is not observed until the applied field reaches about 60T. The reason we believe is the existence strong cation dis-order in this material. It is therefore essential to incorporatedisorder in our Hamiltonian. For the Hamiltonian itself, weconsider a mean field approach, and for this purpose we con-sider two possible symmetries, one that corresponds to a sin-glet in the spin space (DDW) and one that is a triplet in thespin space (SDW). Note that these are particle-hole conden-sates for which orbital function does not constrain the spinwave function unlike a particle-particle condensate (supercon-ductor) because there are no exchange requirements betweena particle and a hole.We believe that it is reasonable that as long as a system isdeep inside a broken symmetry state, mean field theory andits associated elementary excitations should correctly capturethe physics. The fluctuation effects will be important close toquantum phase transitions. However, there are no indicationsin the present experiments that fluctuations are important. Themicroscopic basis for singlet DDW Hamiltonian is discussedin some detail in Refs. 19,20 and in references therein. So,we do not see any particular need to duplicate this discussionhere. The mean field Hamiltonian for the singlet DDW inreal space, in terms of the site-based fermion annihilation andcreation operators of spin σ , c i ,σ and c † i ,σ , is H DDW = X i ,σ ǫ i c † i ,σ c i ,σ + X i , j ,σ t i , j e ia i , j c † i ,σ c j ,σ + h.c., (1)where the nearest neighbor hopping matrix elements are t i , i +ˆ x = − t + iW − ( i x + i y ) , (2) t i , i +ˆ y = − t − iW − ( i x + i y ) , (3)Here W is the DDW gap. We also include the next nearestneighbor hopping t ′ , whereas the third neighbor hopping t ′′ isignored to simplify computational complexity without losingthe essential aspects of the problem. The parameters t and t ′ are chosen (see Table I) to closely approximate the more con-ventional band structure, as shown in Fig. 1. We have checkedthat the choice t ′′ = 0 provides reasonably consistent resultsfor the frequencies in the absence of disorder. For example,for DDW, and doping, the hole pocket frequency is 185T, and the corresponding electron pocket frequency is 2394 T.Similarly, the SDW mean field Hamiltonian is H SDW = X i ,σ (cid:2) ǫ i + σV S ( − i x + i y (cid:3) c † i ,σ c i ,σ + X i , j ,σ t i , j e ia i , j c † i ,σ c j ,σ + h.c. (4)and the spin σ = ± , while the magnitude of the SDW am-plitude is V S . In both cases a constant perpendicular mag-netic field B is included via the Peierls phase factor a i , j = πeh R ij A · d l , where A = (0 , − Bx, is the vector potentialin the Landau gauge. We note that usually a perpendicularmagnetic field, even as large as T , has little effect on the TABLE I: The band parameters, the chemical potential, and the meanfield parameters for DDW and SDW used in our calculation. F inTesla corresponds to the calculated oscillation frequencies of the holepocket, the so-called slow frequencies. The measured F for 15%doping is ± T and for 16% doping is ± T. The calcu-lated magnitude of F does depend on the neglected t ′′ .Order t (eV) t ′ W V S µ V F (T)DDW 15% 0.3 . t . t * − . t . t . t . t * − . t . t . t * . t − . t . t . t * . t − . t . t (0,0) (0,0)( ) FIG. 1: (Color online) The solid curve represents the t − t ′ − t ′′ band structure ( t = 0 . , t ′ = 0 . t, t ′′ = 0 . t ′ ), and thedashed curve corresponds to t − t ′ band structure, (see Table I). Thequasiparticle energy is plotted in the Brillouin zone along the triangle (0 , → ( π, → ( π, π ) → (0 , . In the inset the chemicalpotential, µ , was adjusted to obtain approximately 15% doping. DDW gap, except close to the doping at which it collapses,where field induced order may be important.We have seen previously that the effect of long-rangedcorrelated disorder is qualitatively similar to white noise inso-far as the QOs are concerned. The effect of the nature of disor-der on the spectral function of angle resolved photoemissionspectroscopy (ARPES) was found to be far more important.The reason is that the coherence factors of the ARPES spec-tral function are sensitive to the nature of the disorder becausethey play a role similar to Wannier functions. In contrast, theQOs are damped by the Dingle factor, which is parametrizedby a single lifetime and disorder enters in an averaged sense.Thus, it is sufficient to consider on-site disorder. The on-site energy is δ -correlated white noise defined by the disorderaverage ǫ i = 0 and ǫ i ǫ j = V δ i , j . For an explicit calculationwe need to choose the band structure parameters, W , V S , andthe disorder magnitude V . When considering the magnitudeof disorder one should keep in mind that the full band widthis t . The magnetic field ranges roughly between T and T , representative of the experiments in NCCO. The mag-netic length is l B = p ~ /eB , which for B = 30 T is approxi-mately a , where the lattice constant a is equal to . ˚A.The effect of potential scattering that modulates charge den-sity is indirect on two-fold commensurate SDW or DDW or-der parameter, mainly because SDW is modulation of spinand DDW that of charge current. Thus, the robustness of theseorder parameters with respect to disorder protects the corre-sponding quasiparticle excitations insofar as quantum oscil-lations are concerned, as seen below in our exact numericalcalculations. Thus we did not find it important to study thisproblem self consistently. III. TRANSFER MATRIX METHOD
The transfer matrix method and the calculation of the Lya-punov sketched elsewhere is fully described here for thecase of singlet DDW; for SDW the generalization is straight-forward, where the diagonal term must be modified becauseof V S , and the term W will be absent. Consider a quasi-1Dsystem, L ≫ M , with a periodic boundary condition alongy-direction. Let Ψ n = ( ψ n, , ψ n, , . . . , ψ n,M ) T be the ampli-tudes on the slice n for an eigenstate with a given energy, thenthe amplitudes on three successive slices satisfy the relation (cid:20) Ψ n +1 Ψ n (cid:21) = (cid:20) T − n A n − T − n B n (cid:21) (cid:20) Ψ n Ψ n − (cid:21) = T n (cid:20) Ψ n Ψ n − (cid:21) (5)where T n , A n , B n are M × M matrices. The non-zero matrixelements of the matrix A n are ( A n ) m,m = ǫ n,m − µ, ( A n ) m,m +1 = (cid:20) − t + i W − m + n (cid:21) e − i nφ , ( A n ) m,m − = (cid:20) − t + i W − m + n (cid:21) e i nφ . (6)where φ = 2 πBa e/h is a constant. For the matrix B n : ( B n ) m,m = − (cid:20) − t − i W − m + n (cid:21) , ( B n ) m,m +1 = − t ′ e i( − n + ) φ , ( B n ) m,m − = − t ′ e i( n − ) φ , (7)For the matrix T n , we note that T n = B † n +1 .The M Lyapunov exponents, γ i , of lim N →∞ ( T N T † N ) / N , where T N = Q j = Nj =1 T j , are de-fined by the corresponding eigenvalues λ i = e γ i . AllLyapunov exponents γ > γ > . . . > γ M , are computed bya procedure given in Ref. 23. The modification here is thatthis matrix is not symplectic. Therefore all M eigenvalueshave to be computed. The remarkable fact, however, is thatexcept for a small fraction, consisting of larger eigenvalues,the rest do come in pairs ( λ, /λ ) , as for the symplecticcase, within numerical accuracy. We have no analytical proof of this curious fact. Clearly, larger eigenvalues con-tribute insignificantly to the more general formula for theconductance: σ ( B ) = e h Tr M X j =1 T N T † N ) + ( T N T † N ) − + 2 . (8)When the eigenvalues do come in pairs, the conductance for-mula simplifies to the more common Landauer formula: σ xx ( B ) = e h M X i =1 ( M γ i ) . (9)The transfer matrix method is a very powerful method andthe results obtained are rigorous compared to ad hoc broaden-ing of the Landau levels, which also require more adjustableparameters to explain the experiments. Once the distributionof disorder is specified there are no further approximations.We note that the values of M were chosen to be much largerthan our previous work, at least 128 (that is a in phys-ical units) and sometimes as large as 512. The length of thestrip L is varied between and . This easily led to anaccuracy better than 5% for the smallest Lyapunov exponent, γ i , in all cases.We have calculated the ab -plane conductance, but the mea-sured c -axis resistance, R c , is precisely related to it, at leastas far as the oscillatory part is concerned. This can be seenfrom the arguments in Ref. 26. Although the details can beimproved, the crux of the argument is that the planar densityof states enters R c : the quasiparticle scatters many times inthe plane while performing cyclotron motion before hoppingfrom plane to plane (measured ab -plane resistivity is of the or-der µ Ω -cm as compared Ω -cm for the c -axis resistivity evenat optimum doping ). It is worth noting that oscillations of R c also precisely follows the oscillations of the magnetizationin overdoped Tl Ba CuO δ . IV. RESULTS
There are clues in the experiments that disorder is veryimportant. For 15 and 16% doping the slow oscillations inexperiments, of frequency − T , are not observed un-til the field reaches above T , which is much greater than H c < T . For 17% doping the onset of fast oscillationsat a frequency of , T are strikingly not observable untilthe field reaches T . The estimated scattering time from theDingle factor at even optimal doping and at K is quite short.For 17% doping corresponding to µ = − . t and theband structure given in Table I, a slight change in disorderfrom V = 0 . t to V = 0 . t makes the difference betweena clear observation of a peak to simply noise within the fieldsweep between − T , as shown in Fig. 2 and Fig. 3. Sincein this case W = V S = 0 , there is little else to blame forthe disappearance of the oscillations for fields roughly below T . The results are essentially identical for small values of W , such as . t . F ( T ) I n t en s i t y ( a r b i t r a r y un i t s ) -- FIG. 2: (Color online)The main plot shows the Fourier transform ofthe field sweep shown in the inset. The peak is at , T . Theinset is a smooth background subtracted Shubnikov-de Haas oscil-lations, as calculated from the Landauer formula for 17% doping asa function of /B . The disorder parameter is V = 0 . t .The bandstructure parameters are given in Table I. ----- FIG. 3: (Color online)The same parameters as in Fig. 2 but V =0 . t . The background subtracted conductance is simply noise to anexcellent approximation. For 15% and 16% dopings we chose V to simulate thefact that oscillations seem to disappear below T . The fieldsweep was between − T . The results for DDW orderare shown in Fig. 4 and Fig. 5. The most remarkable featureof these figures is that disorder has completely wiped out thelarge electron pocket leaving the small hole pocket visible. Toemphasize this point we also plot the results for 15% dopingbut with much smaller disorder V = 0 . t ; see Fig. 6. Now wecan see the fragmented remnants of the electron pocket. Withfurther lowering of disorder, the full electron pocket becomesvisible. It is clear that disorder has a significantly stronger ef-fect on the electron pockets than on the hole pockets. This,as we noted earlier, is largely due to higher density of statesaround the antinodal points, which significantly accentuatesthe effect of disorder. We have done parallel calculationswith SDW order as well. The results are essentially identi-
T ) I n t en s i t y ( a r b i t r a r y un i t s )
80 0.6
F ( 1/B ( T ) -1 --- FIG. 4: (Color online)The same plot as in Fig. 2, except for 15%doping and DDW order. The parameters are given in Table I. I n t en s i t y ( a r b i t r a r y un i t s ) F ( T ) -- FIG. 5: (Color online)The same plot as in Fig. 2, except for 16%doping and DDW order. The parameters are given in Table I. I n t en s i t y ( a r b i t r a r y un i t s ) F (T)1/B (T ) -1 --- FIG. 6: (Color online) The same plot as in Fig 4, except that V = 0 . t instead of . t . There is now a fragmented electron pocketcentered around T and the main peak is at T . The rest ofthe parameters are given in Table I. I n t en s i t y ( a r b i t r a r y un i t s ) F ( T ) -- FIG. 7: (Color online) The same plot as in Fig. 4 for 15% dopingbut using SDW order. The main peak is at T . The rest of theparameters are given in Table I. I n t en s i t y ( a r b i t r a r y un i t s ) F ( T ) ---
FIG. 8: (Color online) The same plot as in Fig. 7, except for 16%doping and using SDW order. The main peak is at T . The rest ofthe parameters are given in Table I. cal. They are shown again for 15 and 16% doping in Fig. 7and Fig. 8. We have kept all parameters fixed, while adjust-ing the the SDW gap to achieve as best an approximation toexperiments as possible.It is important to summarize our results in the context ofexperimental observations. First, we were able to show thatthe electron pocket frequencies are strikingly absent becauseof disorder and the slow frequencies corresponding to the holepocket for 15% and 16% doping damp out below about 30 T,even though H c is less than 10 T. Similarly, that the high fre-quency oscillations at 17% doping do not arise until about 60T has a natural explanation in terms of disorder, although inthis case some magnetic breakdown effect, which was not ex-plored, can be expected. This requires both further experimen-tal and theoretical investigations. The calculated frequency ofthe high frequency oscillations, , T is remarkably closeto experimental value of , ± T. As to the magni-tude of the slow oscillations, the calculated values are givenin Table I, which are reasonable in both magnitude and trendwhen compared to experiments. The small discrepancies in the magnitude of F are due to our neglect of t ′′ in the bandstructure. This can be, and was, checked by checking the purecase, that is, without disorder. V. CONCLUSIONS
In the absence of disorder or thermal broadening, the oscil-lation waveforms are never sinusoidal in two dimensions andcontain many Fourier harmonics. At zero temperature moder-ate disorder converts the oscillations to sinusoidal waveformwith rapidly decreasing amplitudes of the harmonics. Furtherincrease of disorder ultimately destroys the amplitudes alto-gether. Many experiments exhibit roughly sinusoidal wave-form at even ultra low temperatures, implying that disorderis important. The remarkably small electronic dispersion inthe direction perpendicular to the CuO-planes cannot aloneaccount for the waveform.For NCCO it is no longer a mystery as to why the fre-quency corresponding to the larger electron pocket is not ob-served. As we have shown, disorder is the culprit. Neither isthe comparison with ARPES controversial, as in the case ofYBCO, since there is good evidence of Fermi surface cross-ing in the direction ( π, → ( π, π ) , which is a signature ofthe electron pocket. The crossing along ( π, π ) → (0 , canbe easily construed as an evidence of a small hole pocket forwhich half of it is made invisible both from the coherencefactors and disorder effects. For electron doped materials,such as NCCO and PCCO, it is known that the Hall co-efficient changes sign around 17% doping and therefore thepicture of reconnection of the Fermi pockets is entirely plau-sible, with some likely magnetic breakdown effects. The realquestion is what is the evidence of SDW or DDW in the rele-vant doping range between 15% and 17%. From neutron mea-surements we know that there is no long range SDW orderfor doping above 13.4%. We cannot rule out field inducedSDW at about T . For DDW, there are no correspondingneutron measurements to observe its existence. Given thatDDW is considerably more hidden from common exper-iments, it is more challenging to establish it directly. NMRexperiments in high fields for suitable nuclei can shed lighton this question. The unavoidable logical conclusion from theQO measurements is that a density wave that breaks transla-tional symmetry must be present. We suggest that motivatedfuture experiments will be necessary to reach a definitive con-clusion. Finally, at the level of mean field theory we havebeen unable to decide between SDW and singlet DDW. At themoment the best recourse is to experimentally look for spinzeros in the amplitude of quantum oscillations in a tilted mag-netic field. A theoretical discussion of this phenomenon thatcan potentially shed light between a triplet order parameter(SDW) and a singlet order parameter, the singlet DDW dis-cussed here, was provided recently. So far experiments arein conflict with each other in YBCO: one group suggests atriplet order parameter and the other a singlet order pa-rameter. It is unquestionable that the QO experiments are likely tochange the widespread views in the field of high tempera-ture superconductivity. Although the measurements in YBCOare not fully explained, the measurements in NCCO appear tohave a clear and simple explanation, as shown here. However,given the similarity of the phenomenon in both hole and elec-tron doped cuprates, it is likely that the quantum oscillationshave the same origin.
Acknowledgments
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