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Quantum magnetic oscillations and angle-resolved photoemission from impurity bands in cuprate superconductors

Present-day angle-resolved photoemission spectroscopy (ARPES) has offered a tremendous advance in the understanding of electron energy spectra in cuprate superconductors and some related compounds. However, in high magnetic field, magnetic quantum oscillations at low temperatures indicate the existence of small electron (hole) Fermi pockets seemingly missing in ARPES of hole (electron) doped cuprates. Here ARPES and quantum oscillations are reconciled in the framework of an impurity band in the charge-transfer Mott-Hubbard insulator.

Present-day angle-resolved photoemission spectroscopy (ARPES) has offered a tremendous advance in the understanding of electron energy spectra in cuprate superconductors and some related compounds. However, in high magnetic field, magnetic quantum oscillations at low temperatures indicate the existence of small electron (hole) Fermi pockets seemingly missing in ARPES of hole (electron) doped cuprates. Here ARPES and quantum oscillations are reconciled in the framework of an impurity band in the charge-transfer Mott-Hubbard insulator. ARPES of cuprate superconductors [2] proved to be particularly instrumental in modeling the electron energy spectrum of these charge transfer Mott-Hubbard insulators showing that below optimal doping the singleparticle Fermi surface is reduced to four disconnected peculiar-shaped spots. While on the overdoped side of the phase diagram a large Fermi surface is expected, a few holes doped into the insulator would naturally give rise to four hole nodal Fermi pockets with a small area proportional to the doping on the underdoped side of the phase diagram. Another possibility is a truncation of the large hole Fermi surface giving rise to four nodal Fermi arcs due to a highly anisotropic quasiparticle life-time and/or a d-wave-like pseudogap.
Remarkably, magnetic quantum oscillations (MQO) in kinetic and magnetic response functions of oxygenordered ortho-II YBa 2 Cu 3 O 6.5 (YBCO6.5) and of some other cuprates [3] revealed small electron Fermi pockets, rather than hole pockets or arcs seemingly in disagreement with ARPES results [2].
A number of Fermi surface reconstructions [4] and non-Fermi-liquid models, including our modulated vortex lattice scenario [5], have tempted to account for the nature of these unusually slow MQO. Further careful experiments have found the Zeeman splitting in MQO, which separates spin-up and spin-down contributions, indicating that electrons in cuprates behave as nearly free spins, which rules out most of the reconstruction and non-Fermi liquid scenarios [6]. This observation as well as excellent fits of MQO data with field-independent oscillation frequencies support the view [3] that MQO are a signature of the true zero-field normal state. This view is also supported by the observation of similar slow oscillations in electron-doped cuprates Nd 2−x Ce x CuO 4 [7], where due to their lower critical fields, the normal state is reliably accessed for any doping level. In contrast with the electron pockets in hole doped cuprates, MQO reveal small hole pockets around certain doping level x ≈ 0.165 in electron doped cuprates [8].
Until recently these two different measurement techniques, ARPES and MQO, were carried out on different materials. Aiming to resolve the outlined puzzle Hossain et al. [9] succeeded to control the surface doping and follow the evolution of ARPES from the overdoped to the underdoped regime through an in situ deposition of potassium atoms on cleaved YBCO. Ref. [9] did not find any sign of the electron pockets in the ARPES data from underdoped YBCO, nor any sign of extra zone folding due to the kind of density wave instabilities that might give rise to such a Fermi surface reconstruction. Hossain et al. argued that if any pocket had to be postulated on the basis of their ARPES data, the most obvious possibility would be that the Fermi arcs are in fact nodal Fermi pockets. However, these are hole, not electron pockets.
Apparently without a detailed microscopic theory both ARPES and MQO data remain a mystery. Here I reconcile these two rather precise techniques by introducing an impurity band in the doped charged transfer Mott-Hubbard insulator. Small electron (hole) pockets in the impurity band account for MQO observed in hole (electron) doped cuprates [3,7], and for the half-moon spots in ARPES [9].
Cuprate superconductors are strongly correlated electron systems where the ab-initio local-densityapproximation (LDA) fails especially in the undoped regime. Fortunately, adding the on-site Coulomb repulsion U to the LDA analysis within the LDA+U algorithm [10] or using LDA plus the tight-binding (cluster) approximation (LDA+GTB algorithm [11]), one can reproduce the correct magnetic ground state and the charge-transfer gap in parent insulators such as La 2 CuO 4 (LCO). The latter and some other schemes [12] for the electronic structure found the charge-transfer gap also in the paramagnetic state of cuprates, pointing to a persistent Mott physics also at finite doping [13].
Different from the reconstruction models, proposed so far, I suggest that YBCO6.5 is the Mott-Hubbard insulator where the Fermi level is pinned within the chargetransfer gap as in its parent insulator YBCO6.0 [4]. This assumption is supported by experiments with ultrathin insulating La 2 CuO 4 and doped superconducting La 2−x Sr x CuO 4 (LSCO) layers [14] and by earlier optical spectroscopies of YBCO [15,16] indicating that doped electronic states appear within the charge-transfer gap. These states, localized in band-tails by disorder, readily account for sharp quasiparticle peaks and a rapid loss of their intensities in some directions of the Brillouin zone observed in ARPES of lightly doped LSCO [17]. The same band-tail model explains two energy scales, their temperature and doping dependence, the asymmetry and inhomogeneity of tunneling spectra of cuprate superconductors [18]. Pinning the Fermi level within the impurity band-tail is also compatible with the insulating-like low-temperature normal state resistivity and many other kinetic properties of lightly doped cuprates [17,19,20].
In the LDA picture the extra oxygen in the full chains of YBa 2 Cu 3 O 6.5 , Fig.(1) splits the planar CuO 2 metallic band in two with a gap opening at the new Brillouin zone boundary k x = π/2a, estimated from local-density approximation calculations between 120 and 160 meV [21]. Quite a different band structure emerges in the chargetransfer insulator. Our key point is that, at variance with other doping levels, 0.5 (per unit cell) extra oxygen creates a coherent band within the charge-transfer gap rather than the localized band-tail, since YBCO6.5 is perfectly ordered, Fig.(1). As shown below, this coherent band accounts for the observed quantum oscillations, and its spectral function combined with the matrix ele- ments also explains ARPES.
To get an insight into the coherent "impurity" band dispersion of YBCO6.5 we employ the LDA+U band structure of the parent insulator YBCO6.0 [4]. In such a framework, the CuO 2 in-plane states are found about 0.5 eV below the Fermi level and the first electron-removal (i.e valence) state has a BaO − Cu chain character (see Fig.5 in Ref. [4]). This valence band has its maximum at Γ point of the Brillouin zone. Spin fluctuations and significant electron-phonon interactions [19] could push the in-plane states closer to the Fermi energy, thus back into the first ionization energy range, due to polaronic level shifts missing in the LDA+U band structures. The edge of the upper in-plane band is also found at Γ point in YBCO6 [4], therefore which particular orbitals form the valence band is not an issue in our further analysis. In any case we are dealing with quasi-two dimensional carriers.
A single extra oxygen ion in the chain creates the Coulomb attractive potential for a hole, about −2e 2 /ǫρ, at a large enough distance ρ from the ion. Solving the 2D Coulomb problem in the effective mass (m) approximation yields an estimate of the localized state ionization energy, E im = 8me 4 / 2 ǫ 2 . E im is about 87 meV with m = 2m e measured in MQO experiments [3] and with ǫ = √ ǫ ab ǫ c , where ǫ ab = 500 [22] and ǫ c = 10 [23] are the in-plane and c-axis dielectric permittivities, respectively, measured in the insulating YBCO single crystals. This energy is much lower than the charge-transfer gap, which is about 1 eV [4], so that the bound state is rather shallow. The size of the bound-state wave function, f im (ρ) ∝ exp(−ρ/a im ), is comparable to the lattice constant, a im = 2 ǫ/4me 2 ≈ 0.4 nm. Excess oxygen ions in YBCO6.5 give rise to a potential V ( − → ρ ), Fig.(1), periodic in a plane − → ρ ≡ {x, y}. It is "gentle" [24] in the outlined sense due to the high polar- izability, ǫ ≫ 1 of perovskites, so that the valence-band states of the parent insulator are mainly involved. Hence the impurity-band wave function, ψ(r) can be expanded in a complete set of the orthogonal valence-band Wannier functions, w(r), which account for most of the correlations [11]. These functions are atomic-like with their extension, a 0 smaller than the lattice constant, a. Thus where the envelope function F (m) satisfies the following differential equation [25] [ Here E(p) is the LDU+U dispersion of the 2D valence band [4]. The eigenstates of Eq.(2) can be expanded as since the potential V ( − → ρ ) is periodic. Here k is the quasimomentum in the reduced 2D Brillouin zone, (−π/2a < k x < π/2a, −π/a < k y < π/a), f ( − → ρ − n) are the orthogonal impurity-band Wannier orbitals built of the boundstate wave function f im ( − → ρ ), and n are 2D position vectors of excess-oxygen ions N im or their projections in the plane, Fig.(1). Then the impurity-band dispersion is found as where are the impurity-band hopping integrals, which are positive for the attractive V ( − → ρ ). It should be noted that the impurity-band dispersion, Eq.(4) holds for the whole Brillouin zone as long as we do not expand the valenceband energy operator E(−i − → ∇) to any finite order in − → ∇. The envelope function Eq.(2) works correctly to any power in k as long as the extension of the valence-band Wannier orbitals is much less than the extension of the impurity-band orbitals, a 0 ≪ a im ≈ a.
A momentum-distribution map of the impurity-band spectral function at the Fermi surface, A(k, 0) ∝ 1/[(E e im (k) − µ) 2 + γ 2 ] is shown in Fig.(3) with the inverse quasiparticle life-time, γ = 0.04t about the Dingle temperature measured in MQO experiments [3]. It hardly resembles the observed ARPES map [9], which is not surprising because the impurity-band ARPES matrix element, M (K) strongly depends on the photoelectron momentum K [17]. Calculating M (K) ∝ dr exp(iK · r)ψ(r) with the impurity-band wave function Eq.(1) one obtains M (K) ∝ f K , where f k is the Fourier component of the impurity-band Wannier orbital, f ( − → ρ ), introduced in Eq.(3). The extension of this orbital is comparable to the lattice constant, which explains the strong dependence of the matrix element on the in-plane photoelectron momentum K .
Approximating f ( − → ρ ) as the 2D Coulomb bound state f im (ρ) yields and the ARPES momentum-distribution map of the Fermi surface, I(K, 0) ∝ M (K) 2 A(K, 0), is . (8) A very close resemblance between theoretical Eq.(8) and experimental ARPES maps is shown in Fig.(4). The theory reproduces the locations, relative intensities and the half-moon shape of the ARPES spots in YBCO6.5 reconciling puzzling ARPES and MQO data. In fact, the small electron pockets observed in MQO experiments [3] are seen in ARPES [9], partially shadowed by the matrix element. The ARPES quasiparticle life-time turns out much shorter than in MQO, presumably due to an instrumental broadening and a high level of impurity scattering off deposited potassium atoms on cleaved YBCO [9]. In the electron-doped cuprates, one expects an impurity band somewhat below the conduction band, if these compounds are doped charge-transfer insulators as the hole-doped ones. Then the outlined scenario with reversed holes and electrons also accounts for the small hole pockets observed in MQO of the electron-doped cuprates [7].