Particlehole Asymmetry in Doped Mott Insulators

At half filling (one electron per lattice site) there are no holes, and thus no conduction in the large U Hubbard model of (2.5) at T = 0. This is the Mott insulating state whose low-energy physics is described by the antiferromagnetic S = 1/2 Heisenberg Hamiltonian on a square lattice. It is by now very well established that the undoped parent compounds of the high Tc superconductors are antiferromagnetically ordered Mott insulators (actually charge transfer insulators, but that distinction is not important for our present purposes).

The question then arises: Is the strong Hubbard U that leads to Mott insulating behavior at half filling also relevant for the doped materials? Or, could the screening in the doped materials effectively lead to a weakly correlated system? We show here that the answer is unequivocal: the strong Hubbard U leads to a characteristic doping-dependent particle-hole asymmetry [17-22] in the tunneling spectra of the doped Mott insulators. Theoretically, this is a rigorous consequence of Gutzwiller projection, requiring no assumptions about the nature of the ground state. Experimentally, this characteristic p-h asymmetry has been observed [23-26] by STM experiments on several hole-doped cuprates including Bi2212 and oxychlo-rides. This establishes unambiguously that correlations are important in the doped materials.

We discuss the asymmetry between the electron and hole density of states (DOS) in the lightly doped Mott insulators in a very general setting that does not depend on the details of the Hamiltonian, e.g., presence of disorder, presence of longer range interaction or longer range hopping etc., as long as the local Hubbard repulsion U is the largest energy scale in the model. Our results do not make any assumptions about the broken symmetry (if any) in the projected state and are valid even in the presence of quenched disorder as we do not assume translational invariance.

We focus, for the sake of simplicity, on the case of T = 0 and small but finite hole doping x = (1 — n) ^ 1. The STM spectrum at a location r and a bias V is proportional to the local density of states (LDOS) N(r,a = eV). (The question of the STM tunneling matrix element, which can itself be a function of r, is discussed below.) The LDOS is given by N(r, a) = A(r, r, a), where the spectral function A(r, r0, a) = — (1/^)ImG(r, r0,a + ir) is related to the imaginary part of the retarded (real space) Green's function of the electrons. It has the spectral decomposition

where | 0) is the ground state with energy E0 and | m) are excited states with energy Em and an implicit sum over o is assumed. To look at the asymmetry we want to derive sum rules for the frequency integrals of A(r, r0, a). The first sum rule is that f1 daA(r, r0, a) = 21r,r, which is just a statement of probability conservation and the factor of 2 comes from the spin sum. We can also obtain f.

where n(r) is the local electron density and x(r) is the corresponding local hole doping. (At finite temperatures, the integration would be cut off by a Fermi function in the integrand. In the T = 0 case discussed here, we get a hard cutoff at the chemical potential a = 0). This sum rule just implies that the probability of extracting an electron from the system at point r is proportional to the local density. The above two sum rules also lead to an obvious third one, f1 daN(r, a) = 1 C x(r). However, a nontrivial and more interesting sum rule is obtained if the positive frequency integral is cutoff at a scale Q, where t,/,... ^ Q ^ U. Physically, this means that we are restricting attention to the "lower Hubbard band" in which there is no double occupancy. Thus, the states | m) that contribute to the sum rule all lie in the projected low-energy subspace, and are of the form |m) = e_i5P|0m) with |0m) forming a complete basis in the full Hilbert space; i.e., J2m |^m)(0m| = 1. Using the form of the low-energy states and the spectral decomposition of the DOS, we find that pQ

/ daN(r, a) = V{0|PcrffP|0m)(0m|P4P|0) = (0|P¿wP<4P|0), (2.8)

j0 m where ct is the canonically transformed form of the creation operator and we have used the completeness of |0m) to write the final form. We have

where the first term in (2.9) simply ensures that the creation of the electron does not create a double occupancy, while the second term represents a process where the double occupancy formed by the creation operator is relaxed through a hopping process. Using the above form for the creation operator, we finally arrive at the sum rule for low-energy electron injection:

Here, t,/,... ^ ^ ^ U and (K(r)) is the local kinetic energy of the electrons at r. The first term in (2.10) simply says that one can inject an electron into any of the x empty sites, with two for spin degeneracy. The second term gives an order xt/U correction since the injected electron can create a temporary double occupancy and then hop off to a neighboring empty site. Note that while the probability to add a low-energy electron (2.10) is obtained to order t/ U, the corresponding result (2.7) to extract an electron is exact to all orders in t/U.

Thus, we conclude that the integrated weight for extracting an electron (2.7) is much larger that the integrated low-energy weight to add an electron (2.10) in a doped Mott insulator, and this asymmetry increases with underdoping.

The spectral weight asymmetry in doped Mott insulators can be understood by the following argument [18-20], which also shows why these systems are completely different from doped band insulators; see Fig. 2.3. Upon doping a band insulator with a density x of holes, the chemical potential just moves into the

BAND INSULATOR MOTT INSULATOR

HALF g FILLING Q

HOLE DOPED

bii o

Fig. 2.3 Schematic figure showing the difference between doped band insulators and doped Mott insulators, based on [18-20]. Upon doping a band insulator, the chemical potential moves with the bands remaining "rigid." However, there is a transfer of spectral weight across the Mott-Hubbard gap upon doping a Mott insulator, with the creation of new low-energy states

x valence band, with the bands remaining "rigid." The integrated spectral weight for electron extraction is then (1 — x), whereas that for addition at low energies is x. In marked contrast, the upper and lower Hubbard bands are not "rigid." While they each have spectral weight of 1 at half-filling, upon hole doping, there is a transfer of spectral weight x from the upper to the lower Hubbard band. Thus, in the doped system the spectral weight for electron extraction is still (1 — x) ((2.7) with translation invariance), while that for addition at low energies is now 2x (large-U limit of (2.10)). This transfer of spectral weight from high energies to the low-energy subspace with doping is a hallmark of doped Mott insulators [18-20,27].

As already noted, the p-h asymmetry and its doping dependence are in excellent qualitative agreement with all available STM data on cuprates [23-26], where the measured conductance g(r; V) = [d//dV](r) shows strong asymmetry between positive and negative bias V. The conductance is related to the LDOS via g(r, V) = M(r)N(r;! = eV) where M(r) is a spatially varying tunneling matrix element. It was proposed in [21,22] that local ratios of V > 0 and V < 0 conductances would lead to a cancellation of the unknown matrix elements. This has certainly proved very useful in the analysis of STM data [28] and the extracted LDOS ratios are very similar from one material to another, even though the matrix elements are very different. However, it has not been possible to quantitatively test the ratio of sum rules (2.10) to (2.7) and extract the local charge density. The difficulty stems from a choice of a suitable (negative) lower cutoff in (2.7) for experimental data.

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