Superconductivity from Hole Undressing J. E. Hirsch Department of Physics, University of California, San Diego La Jolla, CA 92093-0319 (February 7, 2020) Photoemission and optical experiments indicate that the transition to superconductivity in cuprates is an ’undressing’ transition . In photoemission this is seen as a coherent quasiparticle peak emerg- ing from an incoherent background, in optics as violation of the Ferrell-Glover-Tinkham sum rule indicating effective mass reduction of superconducting carriers. We propose that this is a manifes- tation of the fundamental electron-hole asymmetry of condensed matter described by the theory of hole superconductivity. The theory asserts that electrons in nearly empty bands and holes in nearly full bands are fundamentally different : the former yield high conductivity and normal metals, the latter yield low normal state conductivity and high temperature superconductivity. This is because the normal state transport of electrons is coherent and that of holes is incoherent. We explain how this asymmetry arises from the Coulomb interaction between electrons in atoms and the nature of atomic orbitals, and propose a simple Hamiltonian to describe it. A universal mechanism for su- perconductivity follows from this physics, whereby dressed hole carriers undress by pairing, turning (partially) into electrons and becoming more mobile in the superconducting state. Holes are not like electrons [1,2]. This fundamental ally into holes, and become heavier and more incoherent. fact continues to be ignored in contemporary solid state This is because they become increasingly ’dressed’, due science [3]. By holes and electrons we mean the charge to electron-electron interactions, as depicted schemati- carriers in energy bands in a metal when the Fermi level cally in figure 2. Conversely, as the number of physical is near the top and near the bottom of the band respec- electrons in a nearly full band is decreased, holes turn tively. gradually into electrons, and become lighter and more The physical difference between electron and hole car- coherent as they gradually undress. The other way that riers is not captured by Hamiltonians commonly used to hole carriers can turn into electrons is by pairing, thereby study many-body phenomena in solids such as the Hub- mimicking locally a situation where the band is less full. bard model. Thus, new Hamiltonians need to be con- As a consequence, superconductivity will be character- sidered that contain the fundamental charge asymmetry ized by the carriers becoming lighter and more coherent of condensed matter. The generalized Hubbard model in the superconducting than in the normal state, and ki- with correlated hopping [4] contains part of the essential netic energy will be lowered [10]. Since pairing of electron physics of electron-hole asymmetry. The other part is in- carriers would instead lead to carriers becoming heav- corporated by Hamiltonians that describe also high en- ier and more incoherent in the superconducting state, ergy degrees of freedom, which may be generically termed and increased kinetic energy, pairing of electron carriers electron-hole asymmetric polaron models [5,6]. These cannot drive superconductivity. models can be formulated with only electronic degrees A single electron in an empty band is a non-interacting of freedom [7] (more than one band is needed) or with particle, with spectral function given by coupled electronic and bosonic degrees of freedom [8,9,6]. The generalized Hubbard model with correlated hopping A(k,ω)= δ(ω ǫk) (1) − is the low energy effective Hamiltonian that arises from these more fundamental models. with ǫk the band energy. The single particle spectral We argue that the transport of electricity is fundamen- function in a many-body system, for momentum k close tally different when the carriers in the metal are electrons to the Fermi surface, is of the form arXiv:cond-mat/0102136v1 [cond-mat.supr-con] 7 Feb 2001 and when they are holes. Holes cause a large disrup- A(k,ω)= zδ(ω ǫ )+ A′(k,ω) (2) tion in their electronic environment when they propagate, − k while electrons cause little or no disruption. As a conse- where z is the quasiparticle weight, and A′ is the inco- quence, holes are heavy and electrons are light, and, hole herent part of the spectral function (we have neglected transport is largely incoherent while electron transport possible dependence of z on k). One has in general z < 1, is largely coherent. The qualitative difference in the fre- and only for non-interacting particles is z = 1. The nor- quency dependent conductivity of electrons and holes is malization condition A(k,ω)dω = 1 ensures that A′ depicted in Figure 1. As the number of physical electrons R will be nonzero whenever z < 1. A single hole in a full in a nearly empty band is increased, electrons turn gradu- band has z < 1 and A′ = 0, thus is qualitatively different 6 1 from a single electron in an empty band (Eq. (1)) for from ϕ1s to the self-consistent stateϕ ¯1s; in addition, the which z = 1 and A′ = 0. end result of this process may be any of the excited states The difference between a single electron in an empty of the doubly occupied ion, as given by Eq. (6). Similarly band and a single hole in a full band is simplest to un- the spectral function for creating a hole (destroying an derstand in a tight binding formulation. Consider as the electron) in the doubly occupied ion is simplest case the band formed by overlap of 1s orbitals (0) ′ in a lattice of hydrogen-like ions, with nuclear charge Z. A2(ω)= zhδ(ω + (ǫ1s ǫ¯1s )) + A2(ω) (8a) The process of creating an electron in the empty 1s or- − bital (figure 3a) ′ l † 2 (l) A2(ω)= X < c¯↑ > δ(ω + (ǫ1s ǫ¯1s(0) )) (8b) >= ϕ (r)= Ce−Zr/a0 (3) ↑↓ | |↓ | − |↑ 1s l=06 (a0 =Bohr radius) does not affect any other degree of Eqs. (4) and (7), (8) show the fundamental differ- freedom. Hence the single site spectral function is given ence between the spectral functions for creating electrons by in empty bands and creating or destroying holes in full bands. The qualitative structure does not depend on the A (ω)= δ(ω ǫ ) (4) 0 − 1s Hartree approximation, and illustrates the general fact that the hole spectral function will have a large inco- (with ǫ = 13.6eV ). Consider instead the process of 1s herent contribution and a small quasiparticle weight, as creating an electron− of spin when an electron of spin shown schematically in Figure 4. This is due to the phys- already exists in the 1s orbital↑ (figure 3b). If the sec- ical fact that the spacing between atomic energy levels is ond↓ electron is created in the 1s orbital also, a state of smaller than the Coulomb repulsion between electrons in very high energy will result, due to the large Coulomb a level, thus when an electron is created in an already repulsion between two electrons in that orbital (17ZeV ). occupied level the wavefunctions will expand into other Instead, consider the ground state of the two-electron ion atomic levels. This physics cannot be captured with in the Hartree approximation models with a single orbital per site [11]. This effect 0 then leads to the fundamental difference in the transport >=ϕ ¯1s(r1)¯ϕ1s(r2) (5a) | ↑↓ properties when the band is almost empty and when it is almost full. For electrons in nearly empty bands the hop- −Zr/a¯ 0 ϕ¯1s(r)= Ce¯ (5b) ping between atoms t is unrenormalized, while for holes in a full band it is reduced by the quasiparticle weight Z¯ = Z 5/16 (5c) t = z t (9) − h h To obtain the lowest energy state we want to create the due to the disruption caused in the other electron in the second electron in the expanded orbitalϕ ¯1s, with creation orbital during the hopping process, as depicted schemat- † operatorc ¯↑. We have then ically in figure 5. This leads to an enhanced effective mass for holes, hence a larger dc resistivity, and to a large † 0 0 † l l † c¯ >= >< c¯ > + X >< c¯ > incoherent contribution to the frequency dependent con- ↑|↓ | ↑↓ ↑↓ | ↑|↓ | ↑↓ ↑↓ | ↑|↓ l=06 ductivity from hopping processes where the electrons end up in excited states. (6) We can find the quasiparticle weight for the hole ex- plicitely in this Hartree approximation where l> are a complete set of excited states of the | ↑↓ doubly occupied ion, with energiesǫ ¯(l). The single parti- 5 3 1s 2 (1 16Z ) zh = < ϕ1s ϕ¯1s > = − (10) cle spectral function for this process is | | | (1 5 )6 − 32Z A (ω)= z δ(ω (¯ǫ(0) ǫ )) + A′ (ω) (7a) 1 h − 1s − 1s 1 which approaches unity as the nuclear charge Z increases, and becomes small as Z 0.3125. Even though obtained in the Hartree approximation,→ the qualitative effect will z = < 0 c¯† > 2 (7b) h | ↑↓ | ↑|↓ | be generally true (and even larger when one goes beyond the Hartree approximation [12]). When the effective nu- ′ l † 2 (l) clear charge Z is large zh 1, hence hole quasiparticles A1(ω)= X < c¯ > δ(ω (¯ǫ1s ǫ1s)) (7c) → ↑↓ | ↑|↓ | − − become coherent and light, and resemble electron quasi- l=06 particles. Instead, when the effective nuclear charge is hence it has now an incoherent part, and a quasiparticle small, zh 0 and holes become very heavy and incoher- ent. This→ is the regime most favorable for high temper- weight zh < 1. This is because when the electron is created the pre-existing electron will change↑ its state, ature superconductivity. Note how this is in qualitative ↓ 2 g(a†−a )˜n −(g2/2)˜n agreement with the situation in cuprates, where the rele- <e i i i,−σ >= e i,−σ (14) vant ions, O=, are highly negatively charged, correspond- ing to a small Z.