- Email: [email protected]
Linear and nonlinear optics of Sr2CuO2Cl2
Physica B 312–313 (2002) 909–912
Linear and nonlinear optics of Sr2 CuO2Cl2 b,c . J.S. Dodgea,b,c,*, A.B. Schumacherb,c, R. Lovenich , M.A. Carnahanb,c, R.A. Kaindlb,c, L.L. Millerd, D.S. Chemlab,c a
Physics Department, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 b Department of Physics, University of California at Berkeley, Berkeley, CA 94720, USA c Materials Sciences Division, E.O. Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA d Ames Laboratory and Department of Physics, Iowa State University, Ames, IA 50011, USA
Abstract We present evidence from both linear and nonlinear optical spectroscopy for the existence of distinct exciton types at the charge-transfer gap of Sr2 CuO2 Cl2 ; which are linked by strong, phonon-mediated coupling. We show how several basic optical properties of this material depend on these excitations. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Charge-transfer insulators; Optical spectroscopy; Nonlinear optical spectroscopy
In an antiferromagnetic insulator, two charge carriers can travel without disturbing the antiferromagnetic order, and consequently can move more freely than one. The formation of Cooper pairs by such magnetic interactions is one of the dominant themes of research in high-temperature superconductivity, and similar interactions may participate in the binding of electron–hole pairs into excitons. Recent results from resonant inelastic X-ray scattering on insulating Ca2 CuO2 Cl2 indicates that the bandwidth of these particle–hole excitations is several times larger than the single-particle bandwidth measured by photoemission spectroscopy [1,2]. In a conventional picture of weakly interacting quasiparticles, this result is difficult to explain, and suggests that particle–hole excitations in correlated insulators possess properties that are distinct from those in band insulators. Optical spectroscopy is a natural tool for studying these particle–hole properties, and although a considerable body of experimental work exists on the insulating cuprates, our picture of their high-energy electronic structure remains incomplete [3]. For example, a variety *Corresponding author. Physics Department, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6. Tel.: +1-604-291-4736; fax: +1-604-0291-3592. E-mail address: [email protected] (J.S. Dodge).
of optically forbidden excitations are expected to exist in the insulating cuprates, yet some of them remain unobserved. Theoretically, no clear minimal model has emerged for treating any one of the various particle–hole excitations, much less their interactions with phonons, magnons, or other excitons. We have performed detailed optical measurements, in both the linear and nonlinear optical regime, on insulating Sr2 CuO2 Cl2 ; and find that optically forbidden states play important, previously unrecognized roles in determining their optical properties. Here we summarize the basic results, presented in more detail elsewhere [4,5]. All of the measurements discussed below were performed on single crystals of Sr2 CuO2 Cl2 [6], cleaved to a thickness ct100 nm: These thin samples facilitated measurements in a transmission geometry, which in turn enabled direct measurements of the linear absorption spectrum and simplified the calculation of wð3Þ from the optical third harmonic generation (THG) spectrum. The optical absorption spectrum of Sr2 CuO2 Cl2 is shown in Fig. 1, at three different temperatures [4]. The overall character of the absorption spectrum, together with its temperature dependence, is qualitatively similar to that measured by the MIT group in La2 CuO4 [7]. Both materials exhibit a broad absorption peak near 2 eV; which both shifts down in energy and broadens further as the temperature is raised. The temperature
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 5 2 7 - 7
J.S. Dodge et al. / Physica B 312–313 (2002) 909–912
Fig. 1. Absorption spectrum of Sr2 CuO2 Cl2 at three different temperatures: squares T ¼ 15 K; closed circles T ¼ 250 K; and open circles T ¼ 350 K: Solid lines are theoretical results from a model that includes strong phonon-mediated coupling between the optically allowed charge-transfer exciton and an optically forbidden excitation at lower energy [4].
dependence of the absorption peak energy, Epeak ðTÞ; and the line width, gðTÞ; each may be fit to Bose–Einstein occupation functions, with a single oscillator energy Eosc E50 meV; characteristic of a bond-bending optical phonon [4,7]. Furthermore, the absorption tail at low energies fits an exponential, Urbach form [4]. Most importantly, the Urbach behavior in Sr2 CuO2 Cl2 persists down to 10 K; the lowest temperature measured, with a very broad width. The Urbach tail has been studied extensively in semiconductors, where it appears at relatively high temperatures, and may be calculated using conventional Green’s function techniques [8,9]. Applying similar techniques to the absorption spectrum of Sr2 CuO2 Cl2 has enabled us to identify characteristics that distinguish it from more conventional semiconductors [4]. We assume that the absorption peak is due to an exciton, and calculate the phonon-mediated coupling of this exciton to a continuum in a cumulant expansion. The optical absorption processes in the Urbach tail are associated with phonon emission to states below the bare exciton energy; at high temperatures, these states are generated by phonon-mediated renormalization of the exciton. At low temperatures, however, the phonons freeze out, the renormalization disappears, and the absorption peak narrows considerably, no matter how strong the exciton–phonon coupling, a result which clearly contradicts the experimental observation. Therefore, to reproduce the experimentally observed, lowtemperature line width, we find it necessary to assume that there are additional, optically forbidden states that couple to the optically allowed exciton via phonons. The
10
4
10
3
esu)
2.2
-13
1.8 2.0 Energy (eV)
10
102 1
(3)
1.6
|χ xxxx| of Sr2CuO2Cl2 (10
0.0
Quartz
0.5
/ I 3ω
1.0
Sr CuO2Cl2
α (105 cm-1)
1.5
solid lines in Fig. 1 show the results of our model after including this assumption, and show very good agreement with the experimental line shape and temperature dependence. Thus, the behavior of the optical absorption peak in the cuprates can be understood as a consequence of strong exciton–phonon coupling, together with the existence of nearby, optically forbidden states. To test this hypothesis further, however, we have exhausted the information provided by optical absorption. To study optically forbidden excitations with optics, one must turn to nonlinear optical techniques. The most familiar technique is spontaneous Raman scattering, which was among the first methods used to observe and characterize the ligand-field (dd) excitations in the cuprates [10,11]. While Raman scattering is linear in both the incoming and outgoing electric fields, it is actually associated with the third-order nonlinear optical susceptibility, wð3Þ ; since the polarization of the medium also participates [12]. We have employed a related technique, THG spectroscopy [5], which is also a wð3Þ process but possesses several characteristics that distinguish it from Raman. The outgoing, frequencytripled photons are emitted in a coherent beam of light, making their detection more straightforward than in Raman spectroscopy, and eliminating the issue of background luminescence that can be problematic in Raman [13]. THG is a parametric process; that is, the medium is left in the ground state after producing the third harmonic photon. Moreover, in our experiments on Sr2 CuO2 Cl2 ; resonance-induced phenomena are extremely unlikely, since the incoming photons are well below the energy of any electronic excitations. These facts greatly simplify the measurement and interpretation of the THG spectrum.
I3ω2
910
0.6
0.8
1.0 1.2 1.4 Energy (eV)
1.6
Fig. 2. Third harmonic generation in Sr2 CuO2 Cl2 [5]. The left axis shows the Ið3oÞ relative to quartz, which is proportional to ð3Þ 2 jwxxxx j : The right scale indicates absolute values for Sr2 CuO2 Cl2 : The dashed line is a guide to the eye.
J.S. Dodge et al. / Physica B 312–313 (2002) 909–912
3 ð3Þ 2 Px ð3oÞ ¼ wð3Þ xxxx Ex þ 3wxxyy Ex Ey ; ð3Þ 2 Ey3 þ 3wð3Þ Py ð3oÞ ¼ wxxxx xxyy Ex Ey :
ð1Þ
Consequently, by controlling the polarization of the input field E; we can determine the tensorial properties of wð3Þ : We used pffiffiffi an input polarization of the form E ¼ # E0 ðx# þ eiD yÞ= 2; where we varied the retardance D to produce curves of the type shown in the inset of Fig. 3. ia ð3Þ ib Introducing the notation wð3Þ xxxx ¼ kxx e ; wxxyy ¼ kxy e ; and d ¼ a b; we plot in Fig. 3 the relative amplitude kxx =kxy and the relative phase d as a function of fundamental photon frequency. If the only contribution to the nonlinear susceptibility were from the threephoton resonance to the charge-transfer gap, the relative amplitude and phase would be frequency independent, with the amplitude ratio kxx =kxy ¼ 3 and relative phase d ¼ 0: The clear structure in these quantities near _oB1 eV is unambiguous evidence for the participation in the THG susceptibility of a two-photon resonance to a state with a1g symmetry at 2 eV [5]. Candidates for this state include a ligand-field exciton, into the Cuð3d3z2 r2 Þ orbital, and a charge-transfer exciton, with s-wave symmetry. Both of these excitations have proved difficult to observe with other means, indicating that nonlinear optical spectroscopy could play an important future role in identifying and characterizing even-parity excitations in correlated insulators. In summary, our study of Sr2 CuO2 Cl2 indicates that even-parity excitations may play an important role in determining basic linear and nonlinear optical properties of correlated insulators. We have introduced powerful new theoretical and experimental tools for understanding these properties. While limited to q ¼ 0 excitations,
5
(a)
(b)
I3ω (norm)
∆ (rad) 0.0 1.5 3.0 4.5
6
3
-0.4
2
-0.6
1
-0.8
0
0.8 1.0 1.2 1.4 1.6 Energy (eV)
0.8 1.0 1.2 1.4 1.6 Energy (eV)
δ (rad)
4 κxx/κxy
The THG spectrum of Sr2 CuO2 Cl2 is shown in Fig. 2 [5]. These measurements were taken in transmission at room temperature, with both the incoming and outgoing photons polarized along the [1 0 0] axis. We used a femtosecond-pulsed, 250 kHz regenerative amplifier system together with parametric amplification as a tunable laser source. To obtain accurate measurements ð3Þ of the THG susceptibility jwxxxx j at several different wavelengths, we referenced all of our measurements to crystalline quartz. The data show a broad resonance at _o0 B0:7 eV: We can exclude the participation of a onephoton resonance, because this would imply a strong, optically allowed excitation at _oB0:7 eV; which is not observed. Clearly, at 3_oB2:1 eV; a three-photon resonance to the charge-transfer gap is expected. However, we cannot exclude the participation of a two-photon resonance to an optically forbidden state; moreover, we expect such processes to occur. To address this issue, we studied the spectrum of the THG tensor, through polarization-sensitive experiments [5]. For the point group symmetry 4=mmm of Sr2 CuO2 Cl2 ; the third harmonic polarization is given by
911
-1.0
Fig. 3. (a) Relative phase d and (b) relative magnitude r of ð3Þ wð3Þ xxxx and wxxyy of Sr2 CuO2 Cl2 ; derived from polarization analysis of the THG spectrum. The dashed lines are guides to the eye. The inset shows an example of the raw data used to derive the relative quantities shown in the two main panels, for _o ¼ 0:89 eV: As the retardance D is varied, Ix ð3oÞ (filled circles) and Iy ð3oÞ (open circles) vary sinusoidally. The phase shift between the two curves is a measure of d; and r is obtained from the ratio of the maximum and minimum intensities [5].
these techniques complement momentum-resolved experimental probes [1], both by providing better energy resolution and by measuring the optical matrix elements between states. In the future, the results of these studies may allow us to determine the role of strong correlations in setting the binding energies of these excitations [14], and help us to understand the recent observation of large nonlinear optical susceptibilities in strongly correlated, one-dimensional materials [15].
Acknowledgements This work was supported by the US Department of Energy under Contracts No. DE-AC03-76SF00098 and W-7405-Eng-82. Fellowships from the German National Merit Foundation (ABS), and the Deutsche Forschungsgemeinschaft (RAK) are gratefully acknowledged.
References [1] [2] [3] [4] [5]
M.Z. Hasan, et al., Science 288 (2000) 1811. B.O. Wells, et al., Phys. Rev. Lett. 74 (1996) 964. M.A. Kastner, et al., Rev. Mod. Phys. 70 (1998) 897. . R. Lovenich, et al., Phys. Rev. B 63 (2001) 235104. A.B. Schumacher, et al., preprint, cond-mat/0105502.
912 [6] [7] [8] [9]
J.S. Dodge et al. / Physica B 312–313 (2002) 909–912
L.L. Miller, et al., Phys. Rev. B 41 (1990) 1921. J.P. Falck, et al., Phys. Rev. Lett. 69 (1992) 1109. M.V. Kurik, Phys. Stat. Sol. A 8 (1971) 9. J.G. Liebler, S. Schmitt-Rink, H. Haug, J. Lum. 24 (1985) 1. [10] R. Liu, et al., Phys. Rev. Lett. 71 (1993) 3709. [11] P. Kuiper, et al., Phys. Rev. Lett. 80 (1998) 5204.
[12] R.W. Boyd, Nonlinear Optics, Academic Press, Inc., San Diego, 1992. [13] D. Salamon, et al., Phys. Rev. B 51 (1995) 6617. [14] E. Hanamura, N.T. Dan, Y. Tanabe, Phys. Rev. B 62 (2000) 7033. [15] H. Kishida, et al., Nature 405 (2000) 929.
Linear and nonlinear optics of Sr2 CuO2Cl2 b,c . J.S. Dodgea,b,c,*, A.B. Schumacherb,c, R. Lovenich , M.A. Carnahanb,c, R.A. Kaindlb,c, L.L. Millerd, D.S. Chemlab,c a
Physics Department, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 b Department of Physics, University of California at Berkeley, Berkeley, CA 94720, USA c Materials Sciences Division, E.O. Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA d Ames Laboratory and Department of Physics, Iowa State University, Ames, IA 50011, USA
Abstract We present evidence from both linear and nonlinear optical spectroscopy for the existence of distinct exciton types at the charge-transfer gap of Sr2 CuO2 Cl2 ; which are linked by strong, phonon-mediated coupling. We show how several basic optical properties of this material depend on these excitations. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Charge-transfer insulators; Optical spectroscopy; Nonlinear optical spectroscopy
In an antiferromagnetic insulator, two charge carriers can travel without disturbing the antiferromagnetic order, and consequently can move more freely than one. The formation of Cooper pairs by such magnetic interactions is one of the dominant themes of research in high-temperature superconductivity, and similar interactions may participate in the binding of electron–hole pairs into excitons. Recent results from resonant inelastic X-ray scattering on insulating Ca2 CuO2 Cl2 indicates that the bandwidth of these particle–hole excitations is several times larger than the single-particle bandwidth measured by photoemission spectroscopy [1,2]. In a conventional picture of weakly interacting quasiparticles, this result is difficult to explain, and suggests that particle–hole excitations in correlated insulators possess properties that are distinct from those in band insulators. Optical spectroscopy is a natural tool for studying these particle–hole properties, and although a considerable body of experimental work exists on the insulating cuprates, our picture of their high-energy electronic structure remains incomplete [3]. For example, a variety *Corresponding author. Physics Department, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6. Tel.: +1-604-291-4736; fax: +1-604-0291-3592. E-mail address: [email protected] (J.S. Dodge).
of optically forbidden excitations are expected to exist in the insulating cuprates, yet some of them remain unobserved. Theoretically, no clear minimal model has emerged for treating any one of the various particle–hole excitations, much less their interactions with phonons, magnons, or other excitons. We have performed detailed optical measurements, in both the linear and nonlinear optical regime, on insulating Sr2 CuO2 Cl2 ; and find that optically forbidden states play important, previously unrecognized roles in determining their optical properties. Here we summarize the basic results, presented in more detail elsewhere [4,5]. All of the measurements discussed below were performed on single crystals of Sr2 CuO2 Cl2 [6], cleaved to a thickness ct100 nm: These thin samples facilitated measurements in a transmission geometry, which in turn enabled direct measurements of the linear absorption spectrum and simplified the calculation of wð3Þ from the optical third harmonic generation (THG) spectrum. The optical absorption spectrum of Sr2 CuO2 Cl2 is shown in Fig. 1, at three different temperatures [4]. The overall character of the absorption spectrum, together with its temperature dependence, is qualitatively similar to that measured by the MIT group in La2 CuO4 [7]. Both materials exhibit a broad absorption peak near 2 eV; which both shifts down in energy and broadens further as the temperature is raised. The temperature
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 5 2 7 - 7
J.S. Dodge et al. / Physica B 312–313 (2002) 909–912
Fig. 1. Absorption spectrum of Sr2 CuO2 Cl2 at three different temperatures: squares T ¼ 15 K; closed circles T ¼ 250 K; and open circles T ¼ 350 K: Solid lines are theoretical results from a model that includes strong phonon-mediated coupling between the optically allowed charge-transfer exciton and an optically forbidden excitation at lower energy [4].
dependence of the absorption peak energy, Epeak ðTÞ; and the line width, gðTÞ; each may be fit to Bose–Einstein occupation functions, with a single oscillator energy Eosc E50 meV; characteristic of a bond-bending optical phonon [4,7]. Furthermore, the absorption tail at low energies fits an exponential, Urbach form [4]. Most importantly, the Urbach behavior in Sr2 CuO2 Cl2 persists down to 10 K; the lowest temperature measured, with a very broad width. The Urbach tail has been studied extensively in semiconductors, where it appears at relatively high temperatures, and may be calculated using conventional Green’s function techniques [8,9]. Applying similar techniques to the absorption spectrum of Sr2 CuO2 Cl2 has enabled us to identify characteristics that distinguish it from more conventional semiconductors [4]. We assume that the absorption peak is due to an exciton, and calculate the phonon-mediated coupling of this exciton to a continuum in a cumulant expansion. The optical absorption processes in the Urbach tail are associated with phonon emission to states below the bare exciton energy; at high temperatures, these states are generated by phonon-mediated renormalization of the exciton. At low temperatures, however, the phonons freeze out, the renormalization disappears, and the absorption peak narrows considerably, no matter how strong the exciton–phonon coupling, a result which clearly contradicts the experimental observation. Therefore, to reproduce the experimentally observed, lowtemperature line width, we find it necessary to assume that there are additional, optically forbidden states that couple to the optically allowed exciton via phonons. The
10
4
10
3
esu)
2.2
-13
1.8 2.0 Energy (eV)
10
102 1
(3)
1.6
|χ xxxx| of Sr2CuO2Cl2 (10
0.0
Quartz
0.5
/ I 3ω
1.0
Sr CuO2Cl2
α (105 cm-1)
1.5
solid lines in Fig. 1 show the results of our model after including this assumption, and show very good agreement with the experimental line shape and temperature dependence. Thus, the behavior of the optical absorption peak in the cuprates can be understood as a consequence of strong exciton–phonon coupling, together with the existence of nearby, optically forbidden states. To test this hypothesis further, however, we have exhausted the information provided by optical absorption. To study optically forbidden excitations with optics, one must turn to nonlinear optical techniques. The most familiar technique is spontaneous Raman scattering, which was among the first methods used to observe and characterize the ligand-field (dd) excitations in the cuprates [10,11]. While Raman scattering is linear in both the incoming and outgoing electric fields, it is actually associated with the third-order nonlinear optical susceptibility, wð3Þ ; since the polarization of the medium also participates [12]. We have employed a related technique, THG spectroscopy [5], which is also a wð3Þ process but possesses several characteristics that distinguish it from Raman. The outgoing, frequencytripled photons are emitted in a coherent beam of light, making their detection more straightforward than in Raman spectroscopy, and eliminating the issue of background luminescence that can be problematic in Raman [13]. THG is a parametric process; that is, the medium is left in the ground state after producing the third harmonic photon. Moreover, in our experiments on Sr2 CuO2 Cl2 ; resonance-induced phenomena are extremely unlikely, since the incoming photons are well below the energy of any electronic excitations. These facts greatly simplify the measurement and interpretation of the THG spectrum.
I3ω2
910
0.6
0.8
1.0 1.2 1.4 Energy (eV)
1.6
Fig. 2. Third harmonic generation in Sr2 CuO2 Cl2 [5]. The left axis shows the Ið3oÞ relative to quartz, which is proportional to ð3Þ 2 jwxxxx j : The right scale indicates absolute values for Sr2 CuO2 Cl2 : The dashed line is a guide to the eye.
J.S. Dodge et al. / Physica B 312–313 (2002) 909–912
3 ð3Þ 2 Px ð3oÞ ¼ wð3Þ xxxx Ex þ 3wxxyy Ex Ey ; ð3Þ 2 Ey3 þ 3wð3Þ Py ð3oÞ ¼ wxxxx xxyy Ex Ey :
ð1Þ
Consequently, by controlling the polarization of the input field E; we can determine the tensorial properties of wð3Þ : We used pffiffiffi an input polarization of the form E ¼ # E0 ðx# þ eiD yÞ= 2; where we varied the retardance D to produce curves of the type shown in the inset of Fig. 3. ia ð3Þ ib Introducing the notation wð3Þ xxxx ¼ kxx e ; wxxyy ¼ kxy e ; and d ¼ a b; we plot in Fig. 3 the relative amplitude kxx =kxy and the relative phase d as a function of fundamental photon frequency. If the only contribution to the nonlinear susceptibility were from the threephoton resonance to the charge-transfer gap, the relative amplitude and phase would be frequency independent, with the amplitude ratio kxx =kxy ¼ 3 and relative phase d ¼ 0: The clear structure in these quantities near _oB1 eV is unambiguous evidence for the participation in the THG susceptibility of a two-photon resonance to a state with a1g symmetry at 2 eV [5]. Candidates for this state include a ligand-field exciton, into the Cuð3d3z2 r2 Þ orbital, and a charge-transfer exciton, with s-wave symmetry. Both of these excitations have proved difficult to observe with other means, indicating that nonlinear optical spectroscopy could play an important future role in identifying and characterizing even-parity excitations in correlated insulators. In summary, our study of Sr2 CuO2 Cl2 indicates that even-parity excitations may play an important role in determining basic linear and nonlinear optical properties of correlated insulators. We have introduced powerful new theoretical and experimental tools for understanding these properties. While limited to q ¼ 0 excitations,
5
(a)
(b)
I3ω (norm)
∆ (rad) 0.0 1.5 3.0 4.5
6
3
-0.4
2
-0.6
1
-0.8
0
0.8 1.0 1.2 1.4 1.6 Energy (eV)
0.8 1.0 1.2 1.4 1.6 Energy (eV)
δ (rad)
4 κxx/κxy
The THG spectrum of Sr2 CuO2 Cl2 is shown in Fig. 2 [5]. These measurements were taken in transmission at room temperature, with both the incoming and outgoing photons polarized along the [1 0 0] axis. We used a femtosecond-pulsed, 250 kHz regenerative amplifier system together with parametric amplification as a tunable laser source. To obtain accurate measurements ð3Þ of the THG susceptibility jwxxxx j at several different wavelengths, we referenced all of our measurements to crystalline quartz. The data show a broad resonance at _o0 B0:7 eV: We can exclude the participation of a onephoton resonance, because this would imply a strong, optically allowed excitation at _oB0:7 eV; which is not observed. Clearly, at 3_oB2:1 eV; a three-photon resonance to the charge-transfer gap is expected. However, we cannot exclude the participation of a two-photon resonance to an optically forbidden state; moreover, we expect such processes to occur. To address this issue, we studied the spectrum of the THG tensor, through polarization-sensitive experiments [5]. For the point group symmetry 4=mmm of Sr2 CuO2 Cl2 ; the third harmonic polarization is given by
911
-1.0
Fig. 3. (a) Relative phase d and (b) relative magnitude r of ð3Þ wð3Þ xxxx and wxxyy of Sr2 CuO2 Cl2 ; derived from polarization analysis of the THG spectrum. The dashed lines are guides to the eye. The inset shows an example of the raw data used to derive the relative quantities shown in the two main panels, for _o ¼ 0:89 eV: As the retardance D is varied, Ix ð3oÞ (filled circles) and Iy ð3oÞ (open circles) vary sinusoidally. The phase shift between the two curves is a measure of d; and r is obtained from the ratio of the maximum and minimum intensities [5].
these techniques complement momentum-resolved experimental probes [1], both by providing better energy resolution and by measuring the optical matrix elements between states. In the future, the results of these studies may allow us to determine the role of strong correlations in setting the binding energies of these excitations [14], and help us to understand the recent observation of large nonlinear optical susceptibilities in strongly correlated, one-dimensional materials [15].
Acknowledgements This work was supported by the US Department of Energy under Contracts No. DE-AC03-76SF00098 and W-7405-Eng-82. Fellowships from the German National Merit Foundation (ABS), and the Deutsche Forschungsgemeinschaft (RAK) are gratefully acknowledged.
References [1] [2] [3] [4] [5]
M.Z. Hasan, et al., Science 288 (2000) 1811. B.O. Wells, et al., Phys. Rev. Lett. 74 (1996) 964. M.A. Kastner, et al., Rev. Mod. Phys. 70 (1998) 897. . R. Lovenich, et al., Phys. Rev. B 63 (2001) 235104. A.B. Schumacher, et al., preprint, cond-mat/0105502.
912 [6] [7] [8] [9]
J.S. Dodge et al. / Physica B 312–313 (2002) 909–912
L.L. Miller, et al., Phys. Rev. B 41 (1990) 1921. J.P. Falck, et al., Phys. Rev. Lett. 69 (1992) 1109. M.V. Kurik, Phys. Stat. Sol. A 8 (1971) 9. J.G. Liebler, S. Schmitt-Rink, H. Haug, J. Lum. 24 (1985) 1. [10] R. Liu, et al., Phys. Rev. Lett. 71 (1993) 3709. [11] P. Kuiper, et al., Phys. Rev. Lett. 80 (1998) 5204.
[12] R.W. Boyd, Nonlinear Optics, Academic Press, Inc., San Diego, 1992. [13] D. Salamon, et al., Phys. Rev. B 51 (1995) 6617. [14] E. Hanamura, N.T. Dan, Y. Tanabe, Phys. Rev. B 62 (2000) 7033. [15] H. Kishida, et al., Nature 405 (2000) 929.