arXiv:cond-mat/0102136v1 [cond-mat.supr-con] 7 Feb 2001 nanal mt adi nrae,eetostr gradu- turn electrons increased, electrons is physical band of empty number nearly the a is As in holes 1. and Figure fre- electrons in the depicted of in conductivity difference dependent qualitative transport quency The electron coherent. while largely incoherent is largely is transport une oe r ev n lcrn r light, conse- are a electrons As disrup- and heavy disruption. large are no holes a or quence, little cause cause Holes propagate, electrons they while when environment holes. electronic their are in tion electrons they are metal when the in and carriers the when different tally from arises that models. Hamiltonian fundamental more effective these energy low with hopping the correlated or with is needed) model [8,9,6]. is Hubbard freedom generalized of band The degrees one degrees bosonic and than electronic electronic (more coupled only [7] These with freedom formulated of [5,6]. be can models models polaron termed en- asymmetric generically be high may electron-hole which also freedom, of describe degrees that in- ergy is Hamiltonians part by other essential The corporated the asymmetry. model of electron-hole part Hubbard of contains physics generalized [4] hopping The con- correlated be with matter. asymmetry to charge condensed need fundamental of Hamiltonians the contain new Hub- that Thus, the sidered as such model. to solids bard used in commonly phenomena Hamiltonians many-body by study captured not is respec- riers band the of bottom the level Fermi near the charge and tively. when the top metal the mean a near we in is electrons bands energy and state in holes solid carriers By contemporary in [3]. ignored science be to continues fact eageta h rnpr feetiiyi fundamen- is electricity of transport the that argue We car- hole and electron between difference physical The fundamental This [1,2]. electrons like not are Holes tmcobtl,adpooeasml aitna odescri to Hamiltonian simple a between propose interaction and that Coulomb orbitals, and the coherent atomic from is arises electrons asymmetry of tempera this transport high hig state and yield conductivity normal former state the the normal low : yield different latter fundamentally are electrons that bands asserts full conde theory of The asymmetry superconductivity. electron-hole hole fundamental the of tation n rma noeetbcgon,i pisa ilto of car violation superconducting as of optics reduction see in mass is effective background, this indicating incoherent photoemission an In from . ing transition ’undressing’ an is prily noeetosadbcmn oembl ntes the in mobile more becoming and electrons dresse into whereby physics, (partially) this from follows perconductivity hteiso n pia xeiet niaeta h tr the that indicate experiments optical and Photoemission uecnutvt rmHl Undressing Hole from Superconductivity eateto hsc,Uiest fClfri,SnDiego San California, of University Physics, of Department aJla A92093-0319 CA Jolla, La and Fbur ,2020) 7, (February hole , .E Hirsch E. J. 1 adhas band aiaincondition malization ilb ozr whenever nonzero be will n nyfrnnitrcigprilsis particles non-interacting for only and atce ihseta ucingvnby given function spectral with particle, eetpr fteseta ucin(ehv neglected have (we function of spectral dependence the possible of part herent n nrae iei nry arn feeto carriers state, electron superconductivity superconducting of drive pairing the cannot energy, heav- in kinetic becoming increased incoherent carriers and more to and lead electron ier of instead pairing ki- would Since and [10]. carriers state, lowered normal be will coherent the energy in more netic than and superconducting lighter character- the becoming in be carriers the full. will by less superconductivity ized is consequence, band the a where As thereby situation pairing, a by that is locally way electrons mimicking other into more turn The turn can and carriers undress. holes lighter hole gradually decreased, become they is as and physical coherent band electrons, of full number into nearly the gradually as a Conversely, in schemati- 2. electrons figure depicted due in as ’dressed’, cally interactions, increasingly electron-electron become they to incoherent. more because and is heavier This become and holes, into ally where oteFrisrae so h form the of is momentum surface, Fermi for the system, to many-body a in function with igeeeto na mt adi non-interacting a is band empty an in electron single A oecrir nrs ypiig turning pairing, by undress carriers hole d nnal mt ad n oe nnearly in holes and bands empty nearly in sachrn uspril ekemerg- peak quasiparticle coherent a as n ǫ ir.W rps htti samanifes- a is this that propose We riers. niint uecnutvt ncuprates in superconductivity to ansition k z uesprodciiy hsi because is This superconductivity. ture sdmte ecie yteter of theory the by described matter nsed odciiyadnra eas the metals, normal and conductivity h prodcigstate. uperconducting fhlsi noeet eepanhow explain We incoherent. is holes of ei.A it. be h adeeg.Tesnl atcespectral particle single The energy. band the steqaiatcewih,and weight, quasiparticle the is h erl-lvrTnhmsmrule sum Ferrell-Glover-Tinkham the < z lcrn naosadtentr of nature the and atoms in electrons A ( ,ω k, and 1 universal A = ) ( ,ω k, A zδ ′ R ,tu sqaiaieydifferent qualitatively is thus 0, 6= z = ) ( A ω on ( ,ω k, < z − ehns o su- for mechanism δ k ( ǫ .Oehsi general in has One ). ω k ) + ) .Asnl oei full a in hole single A 1. dω − . ǫ k A nue that ensures 1 = (1) ) ′ ( ,ω k, z (2) ) A .Tenor- The 1. = ′ steinco- the is k < z close A 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,−σ (14) vant ions, O=, are highly negatively charged, correspond- ing to a small Z. Moreover the entire CuO planes are so that the hole quasiparticle spectral weight is highly negatively charged, giving rise to highly ’floppy’ −g2 orbitals thus creating the most favorable environment for zh = e (15) hole pairing. This also provides a rationale for the advan- tage of the two-dimensional structure. Since the negative and holes become heavily dressed if g is large, while elec- charge needs to be compensated for charge neutrality, one trons remain undressed. The quasiparticle weight for way to do it is to arrange the excess negative charge in general electronic band filling n is given by the conducting two-dimensional planes where the pairing n 2 z(n) = [1 + (e−g /2 1)]2 (16) occurs and the compensating positive charge outside the 2 − main conducting structures. It is possible that highly negatively charged conducting substructures could also which interpolates between 1 and zh and quantifies the be created in three-dimensional structures, which would magnitude of coherent response of the system for given be even more advantageous for high temperature super- band filling. The relation Eq. (12) can be written as conductivity since, everything else being equal, higher † −g2/2 † coordination strongly enhances superconductivity in this c = [1 + (e 1)ni, ˜−σ]˜c +˜ni,−σ incoherent part iσ − iσ × model [13]. (17) Superconductivity occurs in this theory from the en- hanced hopping amplitude when two hole carriers pair. where the ’incoherent part’ describes the processes where ′ In that case the hopping is th = √zht and the difference bosons are created when the electron is created at the ∆t = t′ t drives the transition to superconductivity, as h − h site. This represents the second term in Eq. (6), where discussed in detail elsewhere [13]. The superconducting the ion ends up in an excited state when the second elec- condensation energy is kinetic [10], and the quasiparti- tron is created. The full Hamiltonian to be studied is cle weight is larger in the superconducting than in the then normal state [6]. † To study quantitatively the physics described above H = X Hi X tij (c cjσ + h.c.)+ X Vij ninj (18) − iσ requires information on excitation energies and matrix i elements of electronic states in multi-electron atoms, and is a difficult many-body problem. It is useful to first un- with Hi given by Eq. (11). The low energy effective derstand throughly the novel physical phenomena that Hamiltonian for quasiparticles that results, using Eq. emerge, for which we can use a variety of model Hamilto- (14), is nians that contain the essential physics [6–9]. As perhaps σ † the simplest realization, consider the site Hamiltonian Heff = X tij (˜c c˜jσ + h.c.)+ X Vij n˜in˜j (19a) − iσ † † Hi = ω0ai ai + gω0(ai + ai)ni↑ni↓ (11) σ 2 t = t [1 (1 √z )(˜n − +˜n − )+(1 √z ) n˜ − n˜ − ] where ai is a local boson operator and niσ is an electron ij ij − − h i, σ j, σ − h i, σ j, σ number operator. This is a special case of the generalized (19b) Holstein model discussed in Ref. 9, with fully non-linear coupling. Using a generalized Lang-Firsov transforma- which in particular describes the superconducting state † tion the following relation between electron particle (ciσ) [13]. However to understand the fundamental processes † and quasiparticle (˜ciσ) operators results [6]: of spectral weight transfer that occur it is necessary to use the full Hamiltonian Eq. (17) [5,6] . This Hamilto- † † † g(ai −ai)˜ni,−σ nian predicts that spectral weight will be transfered from ciσ = e c˜iσ (12) the high energy scale determined by ω0, which represents so that for an empty site, creating an electron particle is an electronic excitation energy scale of the multielectron the same as creating a quasiparticle atom, down to low (intraband) energies, both when the system goes superconducting and when it is doped with † † ciσ =˜ciσ (13a) holes, both in the one-particle spectral function (photoe- mission [14]) and in the two-particle spectral function and the quasiparticle weight is 1, while instead if the site (optical conductivity [15]). is already occupied by an electron of opposite spin, We believe that the physics described by these mod- els represents a new paradigm in many-body physics. † † † g(ai −ai) ciσ = e c˜iσ. (13b) When the system goes superconducting, hole quasipar- ticles ’undress’ and become more like bare particles. In Taking the ground state expectation value of the expo- the conventional Fermi liquid approach quasiparticles are nential in Eq. (12) yields fixed objects, that interact weakly with one another and

3 develop special correlations when a transition to a col- [11] J.E. Hirsch, Physica B 199& 200, 366 (1994). lective state occurs, but do not change their intrinsic [12] J.E. Hirsch, Phys. Rev. B 48, 3327 (1993); 3340 (1993). nature. Here instead quasiparticles change their most [13] J.E. Hirsch and F. Marsiglio, Phys. Rev. B 39, 11515 fundamental properties, their weight and their effective (1989); Physica C 162-164, 591 (1989); F. Marsiglio and mass, as the ordered state develops. The energy that J.E. Hirsch, Phys. Rev. B 41, 6435 (1990). drives the transition to superconductivity originates in [14] H. Ding et al, cond-mat/0006143 (2000); D.L. Feng et al, the very large energy renormalization that is involved in Science 289, 277 (2000). 43 going from a description based on bare, strongly interact- [15] S. Uchida et al, Phys. Rev. B , 7942 (1991); I. Fugol et 86 ing particles, to one based on dressed, weakly interact- al, Sol.St.Comm. , 385 (1993); D. Basov et al, Science 283, 49 (1999). ing quasiparticles. The theory of hole superconductivity 159 proposes that holes, by undressing and turning into elec- [16] F. Marsiglio and J.E. Hirsch, Physica C , 157 (1989). [17] J.E. Hirsch, Phys. Rev. B 58, 8727 (1998); 59, 11962 trons, manage to take advantage of this rich source of (1999). untapped energy. [18] J. E. Hirsch, cond-mat/0012517 (2000), to be published What are measurable consequences of the fundamen- in Phys.Lett.A. tal charge asymmetry on which this theory is based? We [19] J. E. Hirsch, Physica C 158, 326 (1989); Phys.Lett. have shown that it leads to universal asymmetry in NIS A138, 83 (1989); Physica C 341-348, 213 (2000). tunneling [16], with larger conductance predicted for a negatively biased sample, and to the prediction of posi- tive thermoelectric power for NIS and SIS tunnel junc- FIG. 1. Qualitative difference tions [17]. There exists some experimental evidence for between frequency-dependent conductivity when the conduc- the former, while the latter has not been experimentally tion band is almost empty (electron carriers) and when it is tested. Furthermore, we have recently proposed that it almost full(hole carriers) (schematic). Holes have a large ef- should lead to negatively charged vortices in the mixed fective mass, resulting in a small value of σ1(ω = 0), and give state of type II superconductors, and more generally to rise to a large incoherent contribution to σ1(ω) extending to a tendency for superconductors to expel negative charge frequencies well beyond the band energies. Electrons give rise from the bulk [18], leading to higher negative charge den- to Drude-like conductivity with large σ1(ω = 0) due to their sity and superfluid density near the surface, as shown small effective mass. schematically in figure 6. Because the principles on which the theory is based are very general, we expect that if FIG. 2. When the Fermi level is near the bottom of the theory is valid it should apply to all superconducting a band, carriers are undressed, light, coherent, and elec- materials [19]. tron-like. As electrons are added to the band and the Fermi level rises, carriers become increasingly dressed, heavier, inco- herent, and hole-like. The thickness of the ǫ versus k line in- dicates qualitatively the strength of the quasiparticle weight.

FIG. 3. An electron created in the empty 1s orbital of the [1] For a complete list of references see ion (a) produces no disruption in another degree of freedom; http://physics.ucsd.edu/∼jorge/hole.html. an electron created in the orbital that is already occupied by [2] J. E. Hirsch, Phys.Lett.A134, 451 (1989); another electron (b) will creat a disruption of that degree of Mat.Res.Soc.Symp.Proc.156, 349 (1989). freedom, and the resulting state can be any of the excited [3] The origin of the pervasive belief that electrons and holes states of the doubly occupied ion. Similarly a hole created in are equivalent dates back to W. Heisenberg, Ann. der the doubly occupied ion (an electro destroyed) will leave the Physik 10, 888 (1931). remaining electron in any of the possible excited states of the [4] D.K. Campbell, J.T. Gammel and E.Y. Loh, Phys. Rev. singly occupied ion. B 42, 475 (1990) and references therein. [5] J. E. Hirsch, Physica C 201, 347 (1992); in ”Polarons and Bipolarons in high-Tc Superconductors and Related Ma- FIG. 4. Qualitative difference of single particle spectral terials”, ed. by E.K.H. Salje, A.S. Alexandrov and W.Y. functions for particles near the Fermi level for a nearly empty Liang, Cambridge University Press, Cambridge, 1995, p. band (electrons) and a nearly full band (holes). For the nearly 234 . full band the quasiparticle spectral weight is small and a large [6] J.E. Hirsch, Phys.Rev. B 62, 14487, 14498 (2000). incoherent contribution exists. For the nearly empty band the [7] J.E. Hirsch, Phys. Rev. B 43, 11400 (1991). spectral function is entirely coherent. [8] J.E. Hirsch and S. Tang, Sol.St.Comm. 69, 987 (1989); Phys. Rev. B 40, 2179 (1989). [9] J.E. Hirsch, Phys. Rev. B 47, 5351 (1993) . [10] J. E. Hirsch, Physica C 199, 305 (1992); J.E. Hirsch and F. Marsiglio, Phys. Rev. B 45, 4807 (1992); Physica C 331, 150 (2000); Phys. Rev. B 62, 15131 (2000).

4 FIG. 5. Hopping processes giving rise to conduction when the band is almost empty (a) and when it is almost full (b). In (b) the ’diagonal hopping processes’, where the ions make ground state to ground state transitions, involve a rearrange- ment of the electrons that are not hopping. In addition, the hopping process may lead to the ion making a transition to an excited state.

FIG. 6. Schematic picture of a spherical superconducting body. Negative charge is expelled from the bulk to the surface.

5 σ ω σ ω 1( ) 1( )

almost empty band almost full band

ω ω

Figure 1

electronic energy band carriers

Figure 2

l

(a) (b)

Figure 3 A(k,ω) A(k,ω)

z=1

single electron single hole spectral function spectral function

zh<1 1-zh

ω ω

Figure 4

(a) single electron hopping

(b) single hole hopping

Figure 5

Figure 6