Surface Metal-Insulator Transition in the Hubbard Model

PHYSICAL REVIEW B VOLUME 59, NUMBER 4 15 JANUARY 1999-II

Surface metal-insulator transition in the Hubbard model

M. Potthoff and W. Nolting Institut fu¨r Physik, Humboldt-Universita¨t zu Berlin, D-10115 Berlin, Germany ͑Received 8 July 1998; revised manuscript received 9 September 1998͒ The correlation-driven metal-insulator ͑Mott͒ transition at a solid surface is studied within the Hubbard model for a semi-infinite lattice by means of the dynamical mean-field theory. The transition takes place at a unique critical strength of the interaction. Depending on the surface geometry, the interaction strength, and the wave vector, we find one-electron excitations in the coherent part of the surface-projected metallic spectrum, which are confined to two dimensions. ͓S0163-1829͑99͒04904-8͔

I. INTRODUCTION considers the Mott transition at Tϭ0, but concentrates on a new aspect of the problem: the solid surface. Surface effects The correlation-driven transition from a paramagnetic are of interest for the following different reasons: metal to a paramagnetic insulator is a fundamental problem ͑i͒ Taking into account the surface comes closer to the in condensed-matter physics. The Mott transition is of inter- ͑inverse͒ photoemission experiment, which determines the est since strong electron correlations lead to low-energy elec- spectral function. The information depth in low-energy electronic properties that are qualitatively different from those tron spectroscopy is usually limited to a few surface layers predicted by band theory. only.19 Theoretical attempts started with the early ideas of Mott,1 ͑ii͒ The breakdown of translational symmetry due to the Hubbard2 and Brinkman and Rice3 ͑for an overview cf. Ref. surface introduces a layer dependence in the physical quan- 4͒. In recent years a more detailed understanding has been tities, which is worth studying. The layer dependence of the achieved by studying the half-filled Hubbard model in the quasiparticle weight is calculated within the DMFT and also limit of high spatial dimensions d.5 Here the electronic self- within the conventional second-order perturbation theory energy is a local quantity, and a self-consistent mapping onto around the Hartree-Fock solution ͑SOPT-HF͒.20 The pertur- the single-impurity Anderson model becomes possible.6,7 bative treatment applies to the weak-coupling regime, nonlo- The impurity model can be solved numerically7–11 or, for cal terms can be studied here. The ͑nonperturbative͒ DMFT weak or strong coupling, by using perturbative accounts for the local ͑temporal͒ fluctuations exactly, but approaches.6,12 This constitutes the dynamical mean-field neglects all spatial correlations. Within both approaches it is theory ͑DMFT͒, which is exact for dϭϱ but remains a pow- found that the presence of the surface gives rise to strong erful method also at finite d.13,14 oscillations depending on U and geometry. Extensive DMFT studies for the dϭϱ Bethe lattice with a ͑iii͒ One may ask whether or not the critical interaction semielliptic free density of states have established the fol- strength is modified at the surface, analogously to a ͑possi- lowing scenario for the Mott transition:14 For strong interac- bly͒ enhanced Curie temperature at the surface of a tion U the two high-energy charge excitation peaks are well ferromagnet.21 For uniform model parameters we find a separated by an insulating gap in the one-electron spectrum unique critical interaction Uc2 where the whole system un- similar to the Hubbard-III approach.2 For decreasing U the dergoes the transition. This excludes a surface phase differ- insulating solution ceases to exist at a critical value Uc1 .On ent from the bulk. the other hand, for small U the system is a metallic Fermi ͑iv͒ The low-energy electronic structure is of particular liquid with a quasiparticle band at the Fermi energy. For interest close to the critical point. Here, but also well below increasing U the metallic solution of the DMFT equations Uc2 , the surface renormalization of the one-electron excita- coalesces with the insulating one at another critical interaction energies is shown to be sufficiently strong to generate a tion Uc2ϾUc1 .AsUapproaches Uc2 the effective mass surface mode. The mode is essentially confined to two di- 3 diverges as in the Brinkman-Rice solution. Between Uc1 mensions and shows up with a reduced dispersion. The gen- and Uc2 there is a region where both solutions coexist. For erating mechanism for this type of surface excitation is ex- TÞ0 the insulator is stable below Uc2 , and the transition is clusively due to electron correlations. A simple criterion for of first order.14 For Tϭ0 entropic effects favor the metallic their existence is derived analytically. solution. The transition is thus expected14–17 to take place at Uc2 ͑second-order transition͒ or at least close to Uc2 . Although some interesting questions are yet unsolved II. MODEL AND GREEN FUNCTION ͑e.g., concerning the concept of a preformed gap for 18 We study the Hubbard model at half-filling and Tϭ0: -Uc2 ͒, one can state that there is a comparatively deۋU tailed understanding of the Mott transition in infinite dimensions. For finite dimensions one has to assume that the mean- U field picture provided by the DMFT remains valid † Hϭ͚ tijci␴cj␴ϩ ͚ ni␴niϪ␴ , ͑1͒ qualitatively. This is the basis for the present study, which ij␴ 2 i␴

0163-1829/99/59͑4͒/2549͑7͒/$15.00PRB 59 2549 ©1999 The American Physical Society 2550 M. POTTHOFF AND W. NOLTING PRB 59 with i and j running over the sites of a semi-infinite dϭ3 lattice, which is chosen here to be the simple cubic lattice cut at a low-index lattice plane. The uniform nearest-neighbor hopping t͗ij͘ϵϪt with tϭ1 fixes the energy scale. To focus on the Mott transition from a paramagnetic metal to a paramagnetic insulator, we restrict ourselves to the spin- symmetric solutions of the DMFT equations as usual.4 Thereby, we set aside the fact that on a bipartite lattice the half-filled Hubbard model is actually in an antiferromagnetic phase below a finite Neél temperature. We concentrate on the metallic spectrum in the vicinity of the Fermi energy. This is governed by the low-energy expansion of the self-energy:22

U ⌺ ͑E͒ϭ␦ ϩ␤ Eϩi␥ E2ϩ . ͑2͒ ij ij 2 ij ij •••

With the help of the ͑symmetric͒ matrices ␤ϭ(␤ij) and Z Ϫ1 ϭ(1Ϫ␤) the one-electron Green function Gij(E) FIG. 1. Layer dependence of the on-site ͑left͒ and the off-site † ϭ͗͗ci␴ ;cj␴͘͘ can be written in the form: ͑right͒ linear coefficient in the low-energy expansion ͑2͒ of the self-energy at different surfaces of the s.c. lattice as obtained within 1 SOPT-HF. Second-nearest neighbors i and j within the same layer. G͑E͒ϭZ1/2 Z1/2, ͑3͒ Within SOPT-HF ␤ ϳU2ϫconst; we set Uϭ1 in the figure. Note 1/2 1/2 ij E1ϪZ TZ the different scales. Energy units such that tϭ1. where Gϭ(Gij) and Tϭ(tij). Only the term linear in E is ϱ ϱ ϱ ͑0͒ ͑0͒ ͑0͒ taken into account. In particular, we consider an energy ␳ij ͑x͒␳ij ͑y͒␳ij ͑z͒ 2 ␤ ϭϪ2U2 dx dy dz , range where damping effects (ϰi␥ijE ) are unimportant. ij ͵ ͵ ͵ 2 This coherent part of the spectrum is exclusively deter- 0 0 0 ͑xϩyϩz͒ ͑4͒ mined by the so-called quasi-particle weight matrix Z Ϫ1 ϭ(zij) ͑mass-enhancement matrix Z ). The usual bulk quasi-particle weight z(q) is obtained from z by three- if the HF ͑on- or off-site͒ density of states is symmetric, ij (0) (0) dimensional Fourier transformation. For the semi-infinite ␳ij (ϪE)ϭ␳ij (E), and ␤ijϭ0 if it is antisymmetric. Fig- system only lateral translational symmetry can be exploited. ure 1 shows the local term ␤iϭ j to exhibit a weak layer Considering the system to be built up by NЌ layers parallel dependence for the sc͑100͒ surface. Only for the topmost ϱ), there is a collection of NЌ one- surface layer a significant difference to the bulk value isۋto the surface (NЌ dimensional Fermi-‘‘surfaces’’ in the two-dimensional sur- found. Surface effects become stronger for the sc͑110͒ and face Brillouin zone ͑SBZ͒. The ␮th Fermi surface is given are most pronounced for the sc͑111͒ surface where a strong by ⑀k␮ϭ0, where ⑀k␮ (␮ϭ1,...,NЌ) are the eigenvalues oscillation of ␤ii is noticed. This can be understood as fol- 1/2 1/2 of the renormalized hopping matrix Z TZ at a two- lows: According to Eq. ͑4͒ the dominant contribution to ␤ii dimensional wave vector k of the SBZ. The corresponding comes from the HF density of states density near Eϭ0. For 23 eigenvectors uk␮ yield the discontinuous drops of the mo- the sc͑111͒ surface, however, the Eϭ0 value is known to mentum distribution function along the direction k in the oscillate between zero ͑for the subsurface and all even lay- .ϱ the de- ers͒ and a constant value ͑for the topmost and all odd layers͒ۋ SBZ via: ␦n(kϭk␮)ϭNϪ1u† Z u . For N F Ќ k␮• • k␮ Ќ 23 crease of n(k) is continuous, the quasiparticle weight matrix This anomalous behavior is found to dominate the shape of Ϫ2 Z, however, has a well-defined meaning according to Eq. the density of states within an energy range ⌬EϳtL , ͑3͒. where L denotes the distance to the surface. Passing from the ϱ, the bulk HF density of states isۋsurface to the volume, L recovered outside an infinitesimally small energy range III. PERTURBATIONAL APPROACH around Eϭ0. For the coefficient ␤ii this implies a damped Z shall be calculated at a mean-field level assuming that layer-by-layer oscillation with the maximum absolute value spin and charge fluctuations are reasonably local. This im- for the topmost layer. plies a local self-energy ⌺ijϷ␦ij⌺i , which needs justifica- For the discussion of the nonlocal terms ␤iÞ j we have to tion for dϽϱ and in particular for the dϭ3 semi-infinite consider second-nearest neighbors since electron-hole sym- (0) lattice. Some insight can be gained by conventional second- metry requires ␳ij (E) to be antisymmetric, and thus ␤ij order perturbation theory around the Hartree-Fock solution. ϭ0 for nearest neighbors i, j on a bipartite lattice and at SOPT-HF is capable of accounting for the complete nonlo- half-filling. We also find surface-induced oscillations in the cality of the self-energy in the weak-coupling regime also for case iÞ j as can be seen in Fig. 1. More important, however, the case of reduced translational symmetry.20 the absolute values are smaller by about two orders of mag- We have calculated the linear coefficient ␤ij for the dif- nitude compared with the local terms. ferent low-index surfaces. The low-energy expansion of the Let us consider the quasiparticle weight matrix. Decom- SOPT-HF self-energy yields posing the linear coefficient into local and nonlocal parts PRB 59 SURFACE METAL-INSULATOR TRANSITION IN THE . . . 2551

equation is solved on the semi-infinite lattice to get the on- site Green functions G␣(E). For this purpose we make use of lateral translational symmetry, which restricts the numeri- cal calculations to inversion of NЌϫNЌ matrices on a wave- vector mesh in the two-dimensional SBZ. Corresponding to each layer ␣, an effective impurity problem is set up. From the on-site Green function and the self-energy, the ␣th SIAM is specified by fixing the respective free impurity Green (0) function Gimp(E) via the DMFT self-consistency condition:

͑0͒ Ϫ1 Ϫ1 Gimp͑E͒ϭ͓G␣͑E͒ ϩ⌺␣͑E͔͒ . ͑6͒ The crucial step is the solution of the (␣th) SIAM to get the self-energy ⌺␣(E), which is required for the next cycle. Ap- plying DMFT to a film geometry implies a self-consistent treatment of NЌ impurity problems that are coupled indi- rectly via the respective baths of conduction electrons. For finite temperatures the impurity problems can be solved by employing the quantum Monte Carlo method7–9 using the Hirsch-Fye algorithm. For Tϭ0 the exact- diagonalization ͑ED͒ approach of Caffarel and Krauth10,11 FIG. 2. U dependence of the layer-dependent quasiparticle may be applied and will be chosen for the present study. ED Ϫ1 weight z␣ϭ(1Ϫ␤␣) within SOPT-HF. Results for different sur- is able to yield the essentially exact solution of the mean- faces. ‘‘1’’ stands for the topmost surface layer. Dashed lines: bulk field equations in a parameter range where the errors intro- s.c. lattice. duced by the finite system size are unimportant. For the Mott problem the relevant low-energy scale is set by the width of Ϫ1 ␤ijϭ␦ij␤ijϩ(1Ϫ␦ij)␤ij , and expanding Zϭ(1Ϫ␤) in the quasiparticle peak in the metallic solution. It has to be powers of the nonlocal one, yields expected that there are non-negligible finite-size effects

jÞi when this energy scale becomes comparatively small. We are 1 1 1 thus limited to interactions strengths that are not too close to ⌬ziiϭ͚ ␤ij ␤ji ͑5͒ j 1Ϫ␤ii 1Ϫ␤jj 1Ϫ␤ii Uc2 and cannot access the very critical regime. The algorithm proceeds as follows: The DMFT self- for the lowest-order nonlocal correction ⌬zii of the local consistency condition yields a bath Green function 2 element zii . The correction is of the order (␤iÞ j) and can (0) Gimp(iEn). The parameters of a SIAM with ns sites, the thus be neglected in the weak-coupling limit. conduction-band energies ⑀ , and the hybridization strengths Figure 2 shows the local elements of the quasiparticle k Vk (kϭ2,...,ns), are obtained by minimizing the follow- weight matrix ziiϷ1/(1Ϫ␤ii) within SOPT-HF as a function ing cost function: of U. For weak coupling we observe the quadratic depen- 2 dence on the interaction strength: 1Ϫz␣(U)ϰU . Depending 1 nmax on geometry, there is a considerable layer dependence even 2 ͑0͒ Ϫ1 ͑0͒ Ϫ1 ␹ ϭ ͚ ͉Gimp͑iEn͒ ϪGn ͑iEn͒ ͉, ͑7͒ for weak interaction. For strong U, SOPT-HF still predicts a nmaxϩ1 nϭ0 s Fermi-liquid state with z␣Ͼ0. Here, as expected, the pertur- (0) 2 Ϫ1 where G (iEnϪ␮)ϭ͓iEnϪ⑀dϪ͚kV /(iEnϪ⑀k)͔ is the bational approach breaks down. ns k free (Uϭ0) Green function of the ns-site SIAM. The fit of (0) Ϫ1 IV. DYNAMICAL MEAN-FIELD THEORY Gimp(iEn) is performed on the imaginary axis iEnϭi(2n ϩ1)␲/˜␤ with a fictitious inverse temperature ˜␤, which in- We now turn to the dynamical mean-field theory to con- troduces a low-energy cutoff. Lanczo`’s technique24 is used to sider the intermediate- and strong-coupling regime. DMFT is calculate the zero-temperature impurity Green function a nonperturbative method. However, we have to assume that G (iE ). The local self-energy of the ␣th layer is then a local self-energy functional is a reasonable approximation imp n obtained via ⌺ (iE )ϭG(0) (iE )Ϫ1ϪG (iE )Ϫ1.At for dϭ3 dimensions. From the perturbational results, this is ␣ n imp n imp n half-filling electron-hole symmetry can be enforced by ϭ well justified in the weak-coupling regime, for larger U the ⑀k Ϫ and V2ϭV2 with kϩk ϭn ϩ2. assumption may be questioned. In any case, we expect ⑀kЈ k kЈ Ј s DMFT to be a good starting point. We have generalized the mean-field equations to the case V. RESULTS AND DISCUSSION of film geometries. A film consisting of a sufficiently large Routinely, the calculations have been performed for n but finite number of layers NЌ is considered to simulate the s actual surface. Apart from mirror symmetry at the film cen- ϭ8 sites. For interaction strengths well below Uc2 this has ter, the layers are treated as being inequivalent. DMFT must turned out to be sufficient for convergence. The results are thus be applied in the following way: We start with a guess independent of the high-energy cutoff nmax . A small low- ˜ 3 for the ͑local͒ self-energy ⌺␣(E) for each layer ͑translational energy cutoff (␤Wϳ10 ) is necessary to obtain a converged iE)͔Im ⌺␣(iEϭ0). Although the)ץ/ץsymmetry is assumed within the layers͒. Second, the Dyson value for 1Ϫ1/z␣ϭ͓ 2552 M. POTTHOFF AND W. NOLTING PRB 59

FIG. 3. Bulk quasiparticle weight as a function of U. Results for nsϭ8 and nsϭ10.

film geometry implies a high-dimensional parameter space ͓NЌ(nsϪ1)/2, i.e., Ϸ100 parameters to be determined self- consistently͔ we always obtained a stabilized metallic or insulating solution. A moderate number of layers NЌ in the film is sufficient to simulate the semi-infinite system. We used NЌϭ11,15,21 layers to investigate the ͑100͒, ͑110͒, and ͑111͒ surfaces, respectively. The convergence has been checked by comparing the results from calculations for different N . Ќ Ϫ1 FIG. 4. U and layer dependence of z␣ϭ(1Ϫ␤␣) in the vicin- Uc2 . Due to the finite number ofۋDifficulties arise for U sites considered, the energy resolution of the conduction ity of the surface ͑bulk: dashed lines͒. ED calculation for 8 sites band cannot be better than ⌬EϷ␩W/n , where W is the free ͑filled/open circles: nsϭ10). NЌϭ11,15,21 for the sc͑100͒,sc͑110͒, s sc͑111͒ surface, respectively. Insets: z in the critical regime. Thin bandwidth and ␩ is a constant, which accounts for the fact ␣ ͑solid and dashed͒ lines: SOPT-HF results for the sc͑100͒ surface that the conduction-band energies ⑀ are not equally spaced k and UϽ8 ͑top panel͒. and depend on U. ⌬E can be estimated by comparing the results for different n . Figure 3 shows the bulk quasiparticle s sc͑100͒ surface there are no significant differences between weight z as a function of U as obtained from calculations for the DMFT and the SOPT result for z up to UϷ2Ϫ3 ͑see n ϭ8 and n ϭ10 sites. We notice that there is a good agree- ␣ s s top panel in Fig. 4͒. The same holds for the sc͑110͒ surface ment between both results for U smaller than ϳ14.5 or z ͑not shown͒. Contrary, perturbation theory breaks down at a larger than ϳ0.01. For zϽ0.01 the energy scale set by the much weaker interaction strength for the sc͑111͒ surface as it width of the quasiparticle peak ϷzW can no longer be re- is obvious comparing the results in Figs. 2 and 3. solved, i.e., ⌬EϷ0.12ϭW/100. For nsϭ8 and Uϭ15 this For low and intermediate U, z␣ has an oscillating layer ⌬E is also the typical energetic distance between the dependence while it is monotonous close to the transition conduction-band level at the Fermi energy ⑀kϭ0, and the ͑Fig. 4, insets͒. The strongest surface effects are seen for the level with lowest absolute energy ⑀k Þ0 in the self- Ј open sc͑111͒ surface while for the sc͑100͒ surface z␣Ϸzbulk consistent solution. The error that is introduced by the low- except for ␣ϭ1. This trend is related to the decrease of the energy cutoff ˜ is found to be of minor importance com- (100) (110) (111) ␤ surface-coordination numbers: nS ϭ5, nS ϭ4, nS pared with the error due to finite-size effects. This is ensured ϭ3, to be compared with the bulk value nBϭ6. by choosing ˜␤ such that ␲˜␤Ϫ1(Ϸ0.03) is smaller than ⌬E. For all surfaces and layers there is a unique and common From the zero of z(U) in Fig. 3 we can estimate the critical interaction Uc2 where z␣(U) approaches zero. The critical interaction: Uc2Ϸ16tϭ1.33W for nsϭ8 and Uc2 value of Uc2 is the same as the bulk critical interaction Ϸ15tϭ1.25W for nsϭ10. Obviously, a precise determina- strength ͑found for nsϭ8), i.e., Uc2 is determined by the tion is not possible. The results may be compared with the bulk system. In all cases the quasiparticle weight of the top- value Uc2Ϸ1.34W, which has been obtained for the Bethe most layer zS is significantly reduced with respect to the bulk 2 lattice with infinite connectivity using ED at a finite but low value z. This is plausible since the variance ⌬ϭnSt of the 25 temperature ␤tϭ100. For interaction strengths UϾUc1 free (Uϭ0) surface density of states is smaller due to the ϭ11.5t we also obtain an insulating solution. Contrary to reduced surface-coordination number nS , and thus U/ͱ⌬ is Uc2 , the value found for Uc1 is almost independent of ns . larger at the surface, which enhances correlation effects. The This is plausible since there is no small energy scale in the reduced surface coordination number, therefore, tends to case of the insulator. drive the surface to an insulating phase at an interaction We now return to our main subject of interest: the layer strength lower than the bulk critical interaction Uc2 (z and interaction dependence of the quasiparticle weight at the Ͼ0, zSϭ0). A real surface transition, however, is not surfaces of the s.c. lattice. Figure 4 shows the results ob- found; zS remains to be nonzero up to Uc2 . tained for nsϭ8. At weak coupling we notice a quadratic U This is interpreted as follows: For UϽUc2 the bulk qua- dependence consistent with the perturbation theory. For the siparticle band has a finite dispersion. The corresponding PRB 59 SURFACE METAL-INSULATOR TRANSITION IN THE . . . 2553 low-energy excitations are thus extended over the entire lattice. Due to hopping processes between the very surface and the bulk, they will have a finite weight also in the top layer. Therefore, below Uc2 the bulk excitations will to some ex- tent induce a quasiparticle peak with a nonzero weight zS Ͼ0 in the topmost layer. To test this interpretation, we de- coupled the top layer from the rest of the system by switch- ing off the hopping between the top and the subsurface layer t12ϭ0. For the sc͑111͒ surface this implies nSϭ0. The top- layer self-energy is thus given by the ͑insulating͒ atomic- 2 limit expression ⌺␣ϭ1(E)ϭU/2ϩU /4E, while for the other layers we get the same layer-dependent ͑metallic͒ self- energy as before, but shifted by one layer. Taking this as the starting point for the DMFT self-consistency cycle where t12ϭt again, we observe that a finite quasiparticle weight in the top layer zSϾ0 is generated immediately and that the cycle converges to the same solution as before. There is no Mott localization at the surface of the metallic Fermi liquid. However, the tendency of the system to reduce the surface quasiparticle weight ͑enhance the surface effective mass͒ results in a partial ͑wave-vector-dependent͒ local- 2 ization of quasiparticles at the surface: Let us consider the FIG. 5. Upper left: z(U), zS(U) and the ratio r ϭzS /z for the coherent part of the surface electronic structure. To make the sc͑100͒ surface. Upper right: (2Ϫr2)/(1Ϫr2) as a function of r2. Lower right: Ratio between parallel and perpendicular dispersion calculations more transparent, we assume that z␣ϭz except for the top layer (z ϭz ). This is well justified for the along high-symmetry directions in the two-dimensional SBZ. At k ␣ϭ1 S 2 2 sc͑100͒ and, except for the very critical regime, also for the points with (2Ϫr )/(1Ϫr )Ͻ͉⑀ʈ(k)/⑀Ќ(k)͉ surface excitations are sc͑110͒ surface ͑Fig. 4͒. Furthermore, we define the ratio split off the bulk continuum. Lower left: Coherent part of the qua- r2ϭz /z. Two-dimensional Fourier transformation of the siparticle spectrum projected onto the SBZ calculated from Eqs. ͑8͒ S and ͑9͒. Filled area: bulk continuum. Solid line: surface excitation renormalized hopping matrix T¯ ϭͱZTͱZ yields a tridiago- 2 for r ϭ0.25 corresponding to UϭUc2 . ¯ nal matrix T␣␣Ј(k) in the layer indices ␣,␣Јϭ1,...,NЌ. ¯ A surface excitation is characterized by an energy that For ␣,␣Ју2 the nonzero elements are given by T␣␣(k) ¯ ͑for a given wave vector k of the SBZ͒ falls outside the ϭz⑀ʈ(k) and T␣␣Ϯ1(k)ϭz⑀Ќ(k). For the sc͑100͒ surface 26 continuum of bulk excitations described by G0 . A condition the parallel and perpendicular dispersions read: ⑀ʈ(k) for the existence of surface excitations is, therefore, given by ϭ2t͓cos(kxa)ϩcos(kya)͔ and ⑀Ќ(k)ϭt. At the very surface ͉ESϪa͉Ͼ2b where ES is the excitation energy obtained ¯ 2 ¯ Ϫ1 we have: T11(k)ϭr z⑀ʈ(k) and T12/21(k)ϭrz⑀Ќ(k). The from the perturbed surface Green function: G(k,EϭES) 2 specific surface renormalization (r Ͻ1) leads to two ͑local- ϭ0. With G0(k,E)ϭϮz/b at the band edges EϭaϮ2b this ization͒ effects relevant on the energy scale zW: a dimin- is equivalent to ished parallel dispersion in the top layer and a tendency to 2 2 decouple the top-layer from the bulk spectrum. 0Ͼ2b Ϯb͑aϪa0͒Ϫb0 , ͑10͒ As a consequence, surface excitations split off the con- 2 2 tinuum of bulk excitations in the SBZ: For tridiagonal where a0ϵr z⑀ʈ(k)ϭr a and b0ϵrz⑀Ќ(k)ϭrb. Equation ¯ ͑10͒ generally tells us whether or not a discrete eigenvalue is T␣␣Ј(k) the k-resolved surface Green function is readily calculated;24 we have split off the eigenvalue continuum of a semi-infinite tridiagonal matrix with modified ‘‘surface’’ parameters. For the present case it immediately leads to the following simple 1 2 2 criterion for the existence of a surface excitation: G0͑k,E͒ϭz ͓EϪaϯͱ͑EϪa͒ Ϫ4b ͔ 2b2 2 2Ϫr ⑀ʈ͑k͒ Ͻ . ͑11͒ ͓for ϮRe͑EϪa͒Ͼ0͔, ͑8͒ 2 1Ϫr ͯ⑀Ќ͑k͒ͯ with a z (k), b z (k) for the unperturbed surface ϵ ⑀ʈ ϵ ⑀Ќ Hereby, the ratio r2ϭz /z is related to the ratio between the 2ϭ 2ϭ S (r 1). Thus, for r 1 the coherent part of the surface free dispersions. spectral density ϪIm G ϩ ϩ 0(k,E i0 )/␲ is semielliptical for Figure 5 visualizes the criterion for the case of the sc͑100͒ each k with band edges at EϭaϮ2b. The surface Green surface. At small U, the renormalization factor r2 is too close function G for the perturbed surface (r2Ͻ1) can be ex- 24 to 1 to generate surface excitations. These appear above U pressed in terms of G0 as: Ϸ11t and split off the bulk continuum at the ¯⌫ and M¯ high- symmetry points in the SBZ. The minimum interaction zS G͑k,E͒ϭ . ͑9͒ strength is well below Uc2 . Finite-size effects are of no im- 2 2 2 Uc2 surface excitations are found in aۋEϪr aϪr b G0͑k,E͒ portance here. For U 2554 M. POTTHOFF AND W. NOLTING PRB 59 fairly extended region around ¯⌫ and M¯ . In the insulating At the surface electron-correlation effects are found to be phase the whole coherent spectrum disappears. The sc͑110͒ significantly more pronounced compared with the bulk. The surface is even more favorable for surface excitations. Equa- top-layer quasiparticle weight is considerably reduced for all surfaces and interaction strengths. Surface correlation effects tion ͑11͒ always holds along the kyϭ␲/ͱ2a direction where 26 are the stronger the larger is the reduction of the surface ⑀Ќ(k)ϭ0. For the sc͑111͒ surface z␣ deviates from z also in some subsurface layers. Calculating the coherent surface coordination number. For the comparatively open sc͑111͒ surface there is a pronounced layer dependence of the qua- spectrum from the DMFT results for z␣(U) shows that surface excitations do not exist for any U. siparticle weight z␣ for weak and intermediate couplings. As As can be seen in Fig. 5, the surface excitation splits off U approaches the critical region, the layer dependence be- the bulk continuum away from the Fermi energy. Due to comes monotonous in all cases. quasiparticle damping the surface peak in the spectrum will The reduced-surface coordination number tends to drive 2 the system to a phase with an insulating surface on top of a aquire a finite width which is of the order ⌬EϷz ͉␥ ͉E S S S metallic bulk. Eventually, however, the low-energy bulk ex- where ␥ is the damping coefficient at the surface defined in S citations induce a finite-spectral weight at the Fermi energy Eq. ͑2͒. Since we have E Ϸz t for the position of the exci- S S in the top-layer density of states. Consequently, all layer- the (0ۋU (zۋtation, the width is ⌬EϷz3 ␥ t2. For U S͉ S͉ c2 dependent quasiparticle weights vanish at the same coupling E2 coefficient in the expansion of the self-energy diverges as 2 strength Uc2 ; there is a unique transition point. This ex- ␥ϳϪ1/z W as has been shown in Ref. 16 for the dϭϱ cludes a modified surface critical interaction, and thus the Bethe lattice. This result may be used to estimate ⌬EϽzSt. existence of a surface phase. Comparing ⌬E with the energetic distance between the sur- On the other hand, a different localization effect has been face peak and the edge of the bulk continuum, ͉ESϪEB͉ found: There are special regions in the surface Brillouin Ϸzt ͑Fig. 5͒, shows both quantities to be of the same order zone, where the low-energy excitations in the top layer can- -Uc2 . We conclude that the surface not couple to the bulk modes and are confined to twoۋof magnitude for U peak cannot be fully separated in energy from the bulk fea- dimensional lateral propagation. These surface excitations tures in the surface-projected spectrum. are of particular interest since their nature is rather different A complete energetic separation between bulk and surface compared with the well-known Tamm- and Shockley-type excitations is possible if ES crosses the Fermi energy as a surface states.28,29 Opposed to the latter they are caused by function of k. The dispersion of the surface mode can be electron correlations exclusively. Quite generally, as long as calculated from Eqs. ͑8͒ and ͑9͒ to be the concept of a highly renormalized Fermi liquid applies, the specific surface renormalization of the effective mass 1 k 2 k 2 2 2 ⑀ʈ͑ ͒ ⑀Ќ͑ ͒ tends to generate excitations split off the bulk continuum. ES͑k͒ϭ zr ⑀ʈ͑k͒Ϯzr Ϫ . ͑12͒ 2 ͱ 4 1Ϫr2 This mechanism is sufficiently general to survive also at finite temperatures and nonbipartite lattices where antiferro- Setting ES(k)ϭ0 yields ⑀Ќ(k)ϭ0. As mentioned above, this magnetic order is expected to be suppressed. High-resolution is fulfilled, e.g., along the kyϭ␲/ͱ2a direction for the photoemission from single-crystal samples of transition- sc͑110͒ surface. metal oxides should be appropriate to detect the excitation by suitably tuning the Mott transition. VI. CONCLUSION The present study has been restricted to surface geometries and uniform model parameters. It is an open question While the investigation of the surface electronic structure whether or not there is a surface phase if the hopping or the of a metal has a long tradition, there are only few attempts to interaction strength at the very surface is modified. Possibly, account for correlation effects beyond the Hartree-Fock ap- a metallic surface coexisting with a Mott-insulating bulk can proximation or the ͑ab initio͒ local-density approach ͑see be found for a strongly reduced surface U. Future investiga- Refs. 20 and 27 and references therein͒. The present paper tions may also concern thin-film geometries where the study has focussed on the Hubbard model as a standard model for of the thickness dependence of the critical interaction is of correlated itinerant electrons and on the Mott transition as a particular interest. genuine correlation effect. Adapting the dynamical mean- field theory for general film geometries and using the exact- ACKNOWLEDGMENTS diagonalization approach, we have investigated the specific surface properties of the Tϭ0 semi-infinite Hubbard model This work was supported by the Deutsche Forschungsge- at half-filling. meinschaft within the SFB 290.

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