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Photoemission: Low energy and high energy scales
J. Phys. Chem.
Solid.vVol. 56, No. 12, pp. 1735-1736, 1995 Cocwrieht @ 1995 Elsevier Science Ltd Printed-in Great Britain. All rights reserved 0022-3697195$9.50 + 0.00
PHOTOEMISSION: LOW ENERGY AND HIGH ENERGY SCALES K. MATH0 Centre de Recherches sur les Trits Basses Tempkratures, associk B 1’UniversitC Joseph Fourier, C.N.R.S., BP 166, 38042 Grenoble-Cidex 9, France
Abstract-The electron removal spectrum and momentum distribution of a many-body system are calculated, combining low energy FL phenomenology and high energy Pad& approximants. The influence of non-FL
power laws at low energy is also studied.
1. INTRODUCTION In the ongoing efforts to interprete low energy ARPES signals of strongly correlated materials [1] via the electron removal spectrum of a many-body system, a k-independent self-energy Z(w) has so far been employed. Considerable success in modeling the lineshapes and dispersions is somewhat attenuated by our uncertainty about the nature of the background, which is anomalously strong in the cuprates [2]. Since the background is dispersionless, it may be possible to explain it by X(w). On surfaces of constant band structure eigenvalue Ek, the Green function is given by:
G(k, w) = (w - E.(w) -
Q-’ = (G(~F, w)-’ - Ek)-‘.(1)
It is sufficient to model G&, w). Assuming a decomposition into a singular part and a background, G&, w) = G,y(o) + Gh(w), the signal away from k = kF is nevertheless not additive. As shown [3], the lineshape is profoundly influenced by an interference term. Fermi liquid (FL) models are obtained with G,V(w) = Q* /co, a simple pole of residue Q*. The background is then defined as the regular part of the corresponding Laurent expansion of G(kF, w). The quasiparticles (QP) have weight Z(kF) = Q*, dispersion E: = Q*Ek and damping rk = Q*$JGh(O) = &/A*. This defines the energy scale A*, beyond which the QP are overdamped. In discussions with J. Allen and his collaborators [2], two objectives for possible extensions of this phenomenology have been envisaged: (i) define a singular part that allows for non-FL power laws [4], (ii) obtain a physical picture of the background, its weight and structure beyond A*, possibly in relation to microscopic models. A brief description of progress on these two points is given here.
Fig. 1. Momentum distribution as function of bandstructure eigenvalue Ek. The energy scale is A-’ = tlGh(O). Inserts: same scales; see discussion section. The influence of non-FL exponents c( is shown for Q* = 0.3.
2. LOW ENERGY POWER LAW To model power law behavior [4], we define a self-energy function:
olln(-(A*/(U)‘) 2 (1 + (w/A*)*)
1)
(2)
with 0 I (x < 1 and A*, the previous energy scale of the FL. ThesingularpartofG(kF, w) isG,V(w) = Q*/ (w -C,(w)), with Q* the previous weight, which loses the meaning of a residue, as soon as o( # 0. Details leading to this particular ‘Ansatz” and other possible choices will be discussed elsewhere [S]. Essential requirements are: tr& (w ) - 0 for large w, and the correct analytical properties along the cuts I#w > 0 and Xw < 0, with w = 0 as branching point. In the limit ti < 1 (misprinted in [3] as o( - l), eqn (2) reduces to a MFL model
WI.
1735
1736
K. MATH0
Binding
energy
w/A’
Fig. 2. Momentum resolved spectra for a particle hole symmetric Z(w). It is shown, how a QP resonance just emerges from the background, for the given bandstructure input lk. Scales A-’ = clGb(0), A* = Q*A, Q* = l/6. Influence of (r: Values as in Fig. 1
3. HIGH ENERGY
SUMRULES
The high energy behavior can be obtained to a desired accuracy by terminating the continued fraction expansion G(~F, w) = I/ (w - wi - S;/(w
- wj - S;/. . .,)
(3)
at sufficiently high level, giving rise to a Pade approximant. Each real coefficient with index n in the expansion involves moments of the spectral function A(&, w) up to order n. In practice, this becomes tedious beyond the level n = 3 [7]. We have used the levels n = 1 and n = 3, introducing also a phenomenological imaginary part !Jw, # 0 to take into account the energy independent damping of the background excitations. The low energy singular part is incorporated by introducing one more phenomenological stage, consisting of complex energies S,,, 1 and w,,+r. These are uniquely determined by the conditions: (i) Z(0) = 0; (ii) the singularity has real, positive weight 0 < Q* < 1. A non-FL power law is incorporated by replacing w by R = w - Ef(w), but only in the last stage, if all moments to order n are to be conserved. The influence of the power law exponent a on the momentum distribution n(k) is shown in Fig. 1, and on the k-resolved spectrum in Fig. 2.
4. DISCUSSION Figure 1 shows n(k) for a hole doped Hubbard model (X = 0.3), assuming Q* = X. Self-consistency is obtained on the Pade level n = 3 [5], with input CIws = 2t, U = 40t and a d = 2 (tight binding) square lattice. A softened step of amplitude Q* is the characteristic of the non-FL power laws.
The lower insert shows the same result on an expanded scale, stressing the characteristic asymmetry of doped Hubbard models with large U. The correct sign of the dynamical weight transfer [8] is obtained, irrespective of the low energy power law. The upper insert shows, that previously used self-energies FL-TAYLOR [2] and MFL [2,6] cause the total spectral weight to decay with ek. This feature, traced to tlC(w) growing without limitation at large w, suppresses the background. A proper limitation of Z:(w) is generated by the Padt approximants. Figure 2 shows spectra for a particle hole symmetric C( w ) with Q* = l/6 and background generated on the PadC level n = 1. The case (x = 0 was used for linefits in TiTer [1,9]. The chosen values for the bandstructure input correspond to a QP resonance just emerging, for 1~: I/A* = lek l/A I 1. The influence of o( is still weak. The non-FL power law influences the lineshapes and dispersions more strongly in the regime lek]/A << 1, not shown here. Similar spectra for the doped Hubbard model will be discussed elsewhere [5].
REFERENCES 1. Allen J. W. et al. this conference, 1995. 2. Liu L.-Z. et aI., J Phys. Chem. Sol. 52, 1473 (1991). 3. Matho K., Physica B 199,200, 382 (1994). 4. Anderson F! W., Phys. Rev. B 42, 2624 (1990). 5. Matho K., to be published. 6. Varma C. M. et al, Phys Rev. Lett. 63, 1996 (1989). 7. Nolting W. and Borgiel W., Phys. Rev. 39, 6962 (1989). 8. Meinders M. B. J. et al., Phys. Rev. B 48, 3916 (1993). 9. Harm S. et al., J Electron Spectroscopy 68, 111(1994).
Solid.vVol. 56, No. 12, pp. 1735-1736, 1995 Cocwrieht @ 1995 Elsevier Science Ltd Printed-in Great Britain. All rights reserved 0022-3697195$9.50 + 0.00
PHOTOEMISSION: LOW ENERGY AND HIGH ENERGY SCALES K. MATH0 Centre de Recherches sur les Trits Basses Tempkratures, associk B 1’UniversitC Joseph Fourier, C.N.R.S., BP 166, 38042 Grenoble-Cidex 9, France
Abstract-The electron removal spectrum and momentum distribution of a many-body system are calculated, combining low energy FL phenomenology and high energy Pad& approximants. The influence of non-FL
power laws at low energy is also studied.
1. INTRODUCTION In the ongoing efforts to interprete low energy ARPES signals of strongly correlated materials [1] via the electron removal spectrum of a many-body system, a k-independent self-energy Z(w) has so far been employed. Considerable success in modeling the lineshapes and dispersions is somewhat attenuated by our uncertainty about the nature of the background, which is anomalously strong in the cuprates [2]. Since the background is dispersionless, it may be possible to explain it by X(w). On surfaces of constant band structure eigenvalue Ek, the Green function is given by:
G(k, w) = (w - E.(w) -
Q-’ = (G(~F, w)-’ - Ek)-‘.(1)
It is sufficient to model G&, w). Assuming a decomposition into a singular part and a background, G&, w) = G,y(o) + Gh(w), the signal away from k = kF is nevertheless not additive. As shown [3], the lineshape is profoundly influenced by an interference term. Fermi liquid (FL) models are obtained with G,V(w) = Q* /co, a simple pole of residue Q*. The background is then defined as the regular part of the corresponding Laurent expansion of G(kF, w). The quasiparticles (QP) have weight Z(kF) = Q*, dispersion E: = Q*Ek and damping rk = Q*$JGh(O) = &/A*. This defines the energy scale A*, beyond which the QP are overdamped. In discussions with J. Allen and his collaborators [2], two objectives for possible extensions of this phenomenology have been envisaged: (i) define a singular part that allows for non-FL power laws [4], (ii) obtain a physical picture of the background, its weight and structure beyond A*, possibly in relation to microscopic models. A brief description of progress on these two points is given here.
Fig. 1. Momentum distribution as function of bandstructure eigenvalue Ek. The energy scale is A-’ = tlGh(O). Inserts: same scales; see discussion section. The influence of non-FL exponents c( is shown for Q* = 0.3.
2. LOW ENERGY POWER LAW To model power law behavior [4], we define a self-energy function:
olln(-(A*/(U)‘) 2 (1 + (w/A*)*)
1)
(2)
with 0 I (x < 1 and A*, the previous energy scale of the FL. ThesingularpartofG(kF, w) isG,V(w) = Q*/ (w -C,(w)), with Q* the previous weight, which loses the meaning of a residue, as soon as o( # 0. Details leading to this particular ‘Ansatz” and other possible choices will be discussed elsewhere [S]. Essential requirements are: tr& (w ) - 0 for large w, and the correct analytical properties along the cuts I#w > 0 and Xw < 0, with w = 0 as branching point. In the limit ti < 1 (misprinted in [3] as o( - l), eqn (2) reduces to a MFL model
WI.
1735
1736
K. MATH0
Binding
energy
w/A’
Fig. 2. Momentum resolved spectra for a particle hole symmetric Z(w). It is shown, how a QP resonance just emerges from the background, for the given bandstructure input lk. Scales A-’ = clGb(0), A* = Q*A, Q* = l/6. Influence of (r: Values as in Fig. 1
3. HIGH ENERGY
SUMRULES
The high energy behavior can be obtained to a desired accuracy by terminating the continued fraction expansion G(~F, w) = I/ (w - wi - S;/(w
- wj - S;/. . .,)
(3)
at sufficiently high level, giving rise to a Pade approximant. Each real coefficient with index n in the expansion involves moments of the spectral function A(&, w) up to order n. In practice, this becomes tedious beyond the level n = 3 [7]. We have used the levels n = 1 and n = 3, introducing also a phenomenological imaginary part !Jw, # 0 to take into account the energy independent damping of the background excitations. The low energy singular part is incorporated by introducing one more phenomenological stage, consisting of complex energies S,,, 1 and w,,+r. These are uniquely determined by the conditions: (i) Z(0) = 0; (ii) the singularity has real, positive weight 0 < Q* < 1. A non-FL power law is incorporated by replacing w by R = w - Ef(w), but only in the last stage, if all moments to order n are to be conserved. The influence of the power law exponent a on the momentum distribution n(k) is shown in Fig. 1, and on the k-resolved spectrum in Fig. 2.
4. DISCUSSION Figure 1 shows n(k) for a hole doped Hubbard model (X = 0.3), assuming Q* = X. Self-consistency is obtained on the Pade level n = 3 [5], with input CIws = 2t, U = 40t and a d = 2 (tight binding) square lattice. A softened step of amplitude Q* is the characteristic of the non-FL power laws.
The lower insert shows the same result on an expanded scale, stressing the characteristic asymmetry of doped Hubbard models with large U. The correct sign of the dynamical weight transfer [8] is obtained, irrespective of the low energy power law. The upper insert shows, that previously used self-energies FL-TAYLOR [2] and MFL [2,6] cause the total spectral weight to decay with ek. This feature, traced to tlC(w) growing without limitation at large w, suppresses the background. A proper limitation of Z:(w) is generated by the Padt approximants. Figure 2 shows spectra for a particle hole symmetric C( w ) with Q* = l/6 and background generated on the PadC level n = 1. The case (x = 0 was used for linefits in TiTer [1,9]. The chosen values for the bandstructure input correspond to a QP resonance just emerging, for 1~: I/A* = lek l/A I 1. The influence of o( is still weak. The non-FL power law influences the lineshapes and dispersions more strongly in the regime lek]/A << 1, not shown here. Similar spectra for the doped Hubbard model will be discussed elsewhere [5].
REFERENCES 1. Allen J. W. et al. this conference, 1995. 2. Liu L.-Z. et aI., J Phys. Chem. Sol. 52, 1473 (1991). 3. Matho K., Physica B 199,200, 382 (1994). 4. Anderson F! W., Phys. Rev. B 42, 2624 (1990). 5. Matho K., to be published. 6. Varma C. M. et al, Phys Rev. Lett. 63, 1996 (1989). 7. Nolting W. and Borgiel W., Phys. Rev. 39, 6962 (1989). 8. Meinders M. B. J. et al., Phys. Rev. B 48, 3916 (1993). 9. Harm S. et al., J Electron Spectroscopy 68, 111(1994).