- Email: [email protected]
Optical spectra and exciton Mott transition in correlated electron–hole systems
ARTICLE IN PRESS
Journal of Luminescence 128 (2008) 1022–1024 www.elsevier.com/locate/jlumin
Optical spectra and exciton Mott transition in correlated electron–hole systems Tetsuo Ogawaa,b,, Yuh Tomioa,b a
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan b CREST, JST, Toyonaka, Osaka 560-0043, Japan Available online 19 November 2007
Abstract Optical properties in the insulating states of three-dimensional electron–hole systems are studied using a two-band Hubbard model with both repulsion U and attraction U 0 . For two types of insulators, the Mott–Hubbard type and the biexciton type at half-filling, in particular, interband absorption spectra are calculated analytically within the two-site dynamical mean-field theory and ladder approximation. We show that an excitonic peak structure appears due to an ‘‘exciton’’ in the sea of correlated fermions when U 0 is larger than a certain critical value. r 2007 Elsevier B.V. All rights reserved. PACS: 71.10.Fd; 71.30.þh; 71.35.Cc Keywords: Interband optical absorption; Exciton Mott transition; Electron–hole systems; Two-band Hubbard model; Coulomb correlation; Dynamical mean-field theory
Coulomb correlation plays important roles in phase transitions and cooperative optical responses of electron–hole (e–h) systems realized, e.g., in strongly photoexcited semiconductors [1]. The e–h systems exhibit various remarkable properties depending on carrier density, temperature and dimensionality, and therefore have been investigated extensively both experimentally and theoretically. In particular, one of typical and interesting quantum phenomena is the exciton Mott transition (EMT) between an insulating exciton/biexciton gas phase and a metallic e–h plasma/liquid phase [2]. Not only to describe the EMT but also to understand optical responses of the system, many-body Coulomb correlation effects should be taken properly into account continuously from weak to strong coupling. However, most studies are still based on onebody approximations. In particular, there are no theoretical studies on optical responses around the EMT.
Corresponding author. Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan Tel.: +81 6 6850 5350; fax: +81 6 6850 5351. E-mail address: [email protected] (T. Ogawa).
0022-2313/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2007.11.068
The main aim of this study is to clarify analytically the EMT and its optical responses paying careful attention to effects of Coulomb correlation. To this end, we employ the simplest model, the two-band Hubbard model, and solve it with the use of the dynamical mean-field theory (DMFT) [3]. The DMFT requires only the locality of the self-energy, and can take full account of local correlations. The resulting DMFT becomes exact in the limit of infinite spatial dimensions, hence it is a good approximation for three-dimensional systems. Here we assume quasi-thermalequilibrium neglecting the e–h interband recombination. We start with the two-band Hubbard model: XX X a H¼ ta d ay ma nais is d js isa
hijis a¼e;h
þU
X ia
nai" nai#
U
0
X
neis nhis0 ,
ð1Þ
iss0
hy where d ey is (d is ) denotes the creation operator of a conduction-band electron (valence-band hole) with spin s a at the ith site, nais d ay is d is , ta and ma are the transfer integral between the neighboring sites and the chemical potential, respectively. The on-site Coulomb interaction of the e–e (h–h) repulsion and that of the e–h attraction are
ARTICLE IN PRESS T. Ogawa, Y. Tomio / Journal of Luminescence 128 (2008) 1022–1024
expressed and U 0 , respectively. We use ra0 ðeÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi by U 2 2 2 4ta e =ð2pta Þ as a typical density of states (DOS) of three-dimensional systems. In the DMFT, the original lattice model is mapped onto an effective single impurity model. Here we use the two-site DMFT [4], where the quantum impurity model is represented simply by a few parameters. This makes analytical calculations possible, hence this is suitable for the present purpose. In the following, we restrict ourselves to P the case of te ¼ th ð tÞ and half-filling ðn s hnais i ¼ 1Þ. Quantity 2t is taken as unity. When both electron and hole bands are half-filled, two types of insulating states have been shown to exist at zero temperature under the assumption of no e–h pair condensation: (i) the Mott–Hubbard (MH) insulator for U4U 0 and (ii) the biexciton-like (BX) insulator for UoU 0 [5]. The phase diagram in the U 0 2U plane is shown in Fig. 1. The second-order transitions among the metallic and insulating states occur on the solid curves. In these insulating states, the interacting DOS has a gap at the Fermi level (ma ¼ U=2 U 0 ), which is given analytically by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ra ðeÞ ¼ ðe2 o2 =4Þðo2þ =4 e2 Þ=ð2pt2 jejÞ, where o ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2t þ 4t2 þ D2 with D ¼ U ðD ¼ 2U 0 UÞ for the MH (BX) insulator. Now we evaluate the optical response function wðoÞ in the MH and BX insulating states. A ladder approximation is applied [6] and leads wðoÞ ¼ w0 ðoÞ=½1 þ U 0 w0 ðoÞ, where w0 ðoÞ is the response function without the vertex part (i.e., the imaginary part of w0 ðoÞ represents the joint DOS), pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w0 ðoÞ ’ ½o2 8t2 D2 i sgnðoÞ ðo2 o2 Þðo2þ o2 Þ= ð16t2 DÞ for Db2t. Thus, the interband absorption spectrum including excitonic effects (through the vertex part) is obtained as
1023
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ob ¼ o2 ð8t2 D=U 0 Þð1 U 0 =Dc Þ2 , I b ¼ ð4pt
2
ð3Þ
D=ob Þ½1=D2c
þ Re w0 ðob Þ, ð4Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with Dc ¼ 4tD= 4t2 þ D2 . This is the analytical expression of the interband absorption spectrum of a half-filled e–h system. Fig. 2 shows the interband absorption spectra at U ¼ 5 against the (normalized) photon energy o. The dotted curves denote the spectra without excitonic effects, but fully reflect the band renormalization due to the Coulomb correlation. With increasing U 0 , the spectral weight is shifted toward the lower edge of the continuum band. The pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi spectrum exhibits the edge singularity of 1= o o around U 0 ’ 2, and finally an isolated peak emerges at o ¼ ob for U 0 \2 (see also the inset of Fig. 3). This clearly means the formation of an ‘‘exciton’’ in the background of correlated MH/BX insulator. The exact condition for appearance of the isolated peak is given by U 0 Dc . The region satisfying this condition is displayed in Fig. 1. The corresponding boundaries (dashed lines) can be expressed pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi as U ¼ 2tU 0 = 16t2 U 0 2 ðU ¼ 2U 0 2tU 0 = 16t2 U 0 2 Þ for the MH (BX) insulating states. In the limit of D ¼ 1, one obtains Dc ¼ 4t, ob ¼ D U 0 =4 4t2 =U 0 , and I b ¼ ðp=4Þ½1 ð4t=U 0 Þ2 YðU 0 4tÞ. These are similar to those of the band insulator (low-density limit): a free exciton can be formed for U 0 X2t and the exciton peak intensity I 0b ¼ p½1 ð2t=U 0 Þ2 YðU 0 2tÞ. However, while there is no optical gain in the band insulator, the MH and BX insulators have gain for oo0 (not shown in Fig. 2). Fig. 3 shows the exciton binding energy in the MH/BX insulator, E b o ob as a function of U 0 , together with
Im wðoÞ ¼ sgnðoÞ½I b dðjoj ob Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ðo2 o2 Þðo2þ o2 Þ=ðo2 o2b Þ, ð2Þ 2U 0
U=5
U′=1
U′=4
U′=2
U′=5
U′=3
U′=5.5
0.5
5
e 0
h -Im χ(ω)
4
U/(2t)
(II) Mott-Hubbard insulator 3
0.5
0
2
e
(I) Metal
h
0.5
1 (III) Biexciton-like insulator 0 3
0 0
1
2
3 U ′ /(2t)
4
5
Fig. 1. Phase diagram in the U 0 2U plane at half-filling ðn ¼ 1Þ. The interband absorption spectra exhibits an isolated ‘‘excitonic’’ peak in the shaded region.
4
5 6 ω/(2t)
7
3
4
5 6 ω/(2t)
7
Fig. 2. Interband absorption spectra Im wðoÞ at U ¼ 5 for several values of U 0 . Origin of the horizontal axis corresponds to me þ mh . The dotted curves denote the absorption spectra without excitonic effects Im w0 ðoÞ, corresponding to the joint DOS.
ARTICLE IN PRESS T. Ogawa, Y. Tomio / Journal of Luminescence 128 (2008) 1022–1024
1024
1 8 U=5 ω+
Eb/(2t)
ω
6
U=6
U=5
ω-
4
ωb
0.5
MH
BX
2 0
1
free exciton
2
3 U′
4
5
6 U=∞
0 0
1
2
3 U′/(2t)
4
5
6
Fig. 3. The binding energy E b as a function of U 0 for U ¼ 5, 6, 1 and n ¼ 0 limit (free exciton). The inset shows positions of the peak ðob Þ and continuum (shaded area) as a function of U 0 for U ¼ 5.
the free exciton binding energy E 0b ¼ ðU 0 2tÞ2 =U 0 (dotted curve). We find the binding energy E b is suppressed by the repulsion U, and E b ¼ ðU 0 4tÞ2 =ð4U 0 Þ in the U ¼ 1 limit.
Our present study on optical spectra implies that there are two different insulating mechanism even in the MH insulating state: one is due to only U and the other due to both U and U 0 . This difference cannot be distinguished by the DOS ðra Þ. As seen in the optical spectra, when an additional e–h pair is created, they are unbound (bound) for U 0 oDc 2 ðU 0 XDc Þ. This fact predicts that at na1 the insulating phase remains as an exciton gas phase for U 0 \2, while the metallic phase is extended to the region U 0 t2 regardless of U. Indeed, this prediction agrees with the phase diagram at na1 [6,7]. Optical spectra at na1 will be discussed and reported elsewhere. The computation in this work has been done using the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. References [1] H. Haug, S.W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, World Scientific, Singapore, 2004. [2] J. Shah, M. Combescot, A.H. Dayem, Phys. Rev. Lett. 38 (1977) 1497. [3] A. Georges, G. Kotliar, W. Krauth, M.J. Rozenberg, Rev. Mod. Phys. 68 (1996) 13. [4] M. Potthoff, Phys. Rev. B 64 (2001) 165114. [5] Y. Tomio, T. Ogawa, J. Lumin. 112 (2005) 220. [6] T. Ogawa, Y. Tomio, K. Asano, J. Phys.: Conf. Series 21 (2005) 112. [7] Y. Tomio, T. Ogawa, AIP Conf. Proc. 850 (2006) 1313.
Journal of Luminescence 128 (2008) 1022–1024 www.elsevier.com/locate/jlumin
Optical spectra and exciton Mott transition in correlated electron–hole systems Tetsuo Ogawaa,b,, Yuh Tomioa,b a
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan b CREST, JST, Toyonaka, Osaka 560-0043, Japan Available online 19 November 2007
Abstract Optical properties in the insulating states of three-dimensional electron–hole systems are studied using a two-band Hubbard model with both repulsion U and attraction U 0 . For two types of insulators, the Mott–Hubbard type and the biexciton type at half-filling, in particular, interband absorption spectra are calculated analytically within the two-site dynamical mean-field theory and ladder approximation. We show that an excitonic peak structure appears due to an ‘‘exciton’’ in the sea of correlated fermions when U 0 is larger than a certain critical value. r 2007 Elsevier B.V. All rights reserved. PACS: 71.10.Fd; 71.30.þh; 71.35.Cc Keywords: Interband optical absorption; Exciton Mott transition; Electron–hole systems; Two-band Hubbard model; Coulomb correlation; Dynamical mean-field theory
Coulomb correlation plays important roles in phase transitions and cooperative optical responses of electron–hole (e–h) systems realized, e.g., in strongly photoexcited semiconductors [1]. The e–h systems exhibit various remarkable properties depending on carrier density, temperature and dimensionality, and therefore have been investigated extensively both experimentally and theoretically. In particular, one of typical and interesting quantum phenomena is the exciton Mott transition (EMT) between an insulating exciton/biexciton gas phase and a metallic e–h plasma/liquid phase [2]. Not only to describe the EMT but also to understand optical responses of the system, many-body Coulomb correlation effects should be taken properly into account continuously from weak to strong coupling. However, most studies are still based on onebody approximations. In particular, there are no theoretical studies on optical responses around the EMT.
Corresponding author. Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan Tel.: +81 6 6850 5350; fax: +81 6 6850 5351. E-mail address: [email protected] (T. Ogawa).
0022-2313/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2007.11.068
The main aim of this study is to clarify analytically the EMT and its optical responses paying careful attention to effects of Coulomb correlation. To this end, we employ the simplest model, the two-band Hubbard model, and solve it with the use of the dynamical mean-field theory (DMFT) [3]. The DMFT requires only the locality of the self-energy, and can take full account of local correlations. The resulting DMFT becomes exact in the limit of infinite spatial dimensions, hence it is a good approximation for three-dimensional systems. Here we assume quasi-thermalequilibrium neglecting the e–h interband recombination. We start with the two-band Hubbard model: XX X a H¼ ta d ay ma nais is d js isa
hijis a¼e;h
þU
X ia
nai" nai#
U
0
X
neis nhis0 ,
ð1Þ
iss0
hy where d ey is (d is ) denotes the creation operator of a conduction-band electron (valence-band hole) with spin s a at the ith site, nais d ay is d is , ta and ma are the transfer integral between the neighboring sites and the chemical potential, respectively. The on-site Coulomb interaction of the e–e (h–h) repulsion and that of the e–h attraction are
ARTICLE IN PRESS T. Ogawa, Y. Tomio / Journal of Luminescence 128 (2008) 1022–1024
expressed and U 0 , respectively. We use ra0 ðeÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi by U 2 2 2 4ta e =ð2pta Þ as a typical density of states (DOS) of three-dimensional systems. In the DMFT, the original lattice model is mapped onto an effective single impurity model. Here we use the two-site DMFT [4], where the quantum impurity model is represented simply by a few parameters. This makes analytical calculations possible, hence this is suitable for the present purpose. In the following, we restrict ourselves to P the case of te ¼ th ð tÞ and half-filling ðn s hnais i ¼ 1Þ. Quantity 2t is taken as unity. When both electron and hole bands are half-filled, two types of insulating states have been shown to exist at zero temperature under the assumption of no e–h pair condensation: (i) the Mott–Hubbard (MH) insulator for U4U 0 and (ii) the biexciton-like (BX) insulator for UoU 0 [5]. The phase diagram in the U 0 2U plane is shown in Fig. 1. The second-order transitions among the metallic and insulating states occur on the solid curves. In these insulating states, the interacting DOS has a gap at the Fermi level (ma ¼ U=2 U 0 ), which is given analytically by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ra ðeÞ ¼ ðe2 o2 =4Þðo2þ =4 e2 Þ=ð2pt2 jejÞ, where o ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2t þ 4t2 þ D2 with D ¼ U ðD ¼ 2U 0 UÞ for the MH (BX) insulator. Now we evaluate the optical response function wðoÞ in the MH and BX insulating states. A ladder approximation is applied [6] and leads wðoÞ ¼ w0 ðoÞ=½1 þ U 0 w0 ðoÞ, where w0 ðoÞ is the response function without the vertex part (i.e., the imaginary part of w0 ðoÞ represents the joint DOS), pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w0 ðoÞ ’ ½o2 8t2 D2 i sgnðoÞ ðo2 o2 Þðo2þ o2 Þ= ð16t2 DÞ for Db2t. Thus, the interband absorption spectrum including excitonic effects (through the vertex part) is obtained as
1023
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ob ¼ o2 ð8t2 D=U 0 Þð1 U 0 =Dc Þ2 , I b ¼ ð4pt
2
ð3Þ
D=ob Þ½1=D2c
þ Re w0 ðob Þ, ð4Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with Dc ¼ 4tD= 4t2 þ D2 . This is the analytical expression of the interband absorption spectrum of a half-filled e–h system. Fig. 2 shows the interband absorption spectra at U ¼ 5 against the (normalized) photon energy o. The dotted curves denote the spectra without excitonic effects, but fully reflect the band renormalization due to the Coulomb correlation. With increasing U 0 , the spectral weight is shifted toward the lower edge of the continuum band. The pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi spectrum exhibits the edge singularity of 1= o o around U 0 ’ 2, and finally an isolated peak emerges at o ¼ ob for U 0 \2 (see also the inset of Fig. 3). This clearly means the formation of an ‘‘exciton’’ in the background of correlated MH/BX insulator. The exact condition for appearance of the isolated peak is given by U 0 Dc . The region satisfying this condition is displayed in Fig. 1. The corresponding boundaries (dashed lines) can be expressed pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi as U ¼ 2tU 0 = 16t2 U 0 2 ðU ¼ 2U 0 2tU 0 = 16t2 U 0 2 Þ for the MH (BX) insulating states. In the limit of D ¼ 1, one obtains Dc ¼ 4t, ob ¼ D U 0 =4 4t2 =U 0 , and I b ¼ ðp=4Þ½1 ð4t=U 0 Þ2 YðU 0 4tÞ. These are similar to those of the band insulator (low-density limit): a free exciton can be formed for U 0 X2t and the exciton peak intensity I 0b ¼ p½1 ð2t=U 0 Þ2 YðU 0 2tÞ. However, while there is no optical gain in the band insulator, the MH and BX insulators have gain for oo0 (not shown in Fig. 2). Fig. 3 shows the exciton binding energy in the MH/BX insulator, E b o ob as a function of U 0 , together with
Im wðoÞ ¼ sgnðoÞ½I b dðjoj ob Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ðo2 o2 Þðo2þ o2 Þ=ðo2 o2b Þ, ð2Þ 2U 0
U=5
U′=1
U′=4
U′=2
U′=5
U′=3
U′=5.5
0.5
5
e 0
h -Im χ(ω)
4
U/(2t)
(II) Mott-Hubbard insulator 3
0.5
0
2
e
(I) Metal
h
0.5
1 (III) Biexciton-like insulator 0 3
0 0
1
2
3 U ′ /(2t)
4
5
Fig. 1. Phase diagram in the U 0 2U plane at half-filling ðn ¼ 1Þ. The interband absorption spectra exhibits an isolated ‘‘excitonic’’ peak in the shaded region.
4
5 6 ω/(2t)
7
3
4
5 6 ω/(2t)
7
Fig. 2. Interband absorption spectra Im wðoÞ at U ¼ 5 for several values of U 0 . Origin of the horizontal axis corresponds to me þ mh . The dotted curves denote the absorption spectra without excitonic effects Im w0 ðoÞ, corresponding to the joint DOS.
ARTICLE IN PRESS T. Ogawa, Y. Tomio / Journal of Luminescence 128 (2008) 1022–1024
1024
1 8 U=5 ω+
Eb/(2t)
ω
6
U=6
U=5
ω-
4
ωb
0.5
MH
BX
2 0
1
free exciton
2
3 U′
4
5
6 U=∞
0 0
1
2
3 U′/(2t)
4
5
6
Fig. 3. The binding energy E b as a function of U 0 for U ¼ 5, 6, 1 and n ¼ 0 limit (free exciton). The inset shows positions of the peak ðob Þ and continuum (shaded area) as a function of U 0 for U ¼ 5.
the free exciton binding energy E 0b ¼ ðU 0 2tÞ2 =U 0 (dotted curve). We find the binding energy E b is suppressed by the repulsion U, and E b ¼ ðU 0 4tÞ2 =ð4U 0 Þ in the U ¼ 1 limit.
Our present study on optical spectra implies that there are two different insulating mechanism even in the MH insulating state: one is due to only U and the other due to both U and U 0 . This difference cannot be distinguished by the DOS ðra Þ. As seen in the optical spectra, when an additional e–h pair is created, they are unbound (bound) for U 0 oDc 2 ðU 0 XDc Þ. This fact predicts that at na1 the insulating phase remains as an exciton gas phase for U 0 \2, while the metallic phase is extended to the region U 0 t2 regardless of U. Indeed, this prediction agrees with the phase diagram at na1 [6,7]. Optical spectra at na1 will be discussed and reported elsewhere. The computation in this work has been done using the facilities of the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. References [1] H. Haug, S.W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, World Scientific, Singapore, 2004. [2] J. Shah, M. Combescot, A.H. Dayem, Phys. Rev. Lett. 38 (1977) 1497. [3] A. Georges, G. Kotliar, W. Krauth, M.J. Rozenberg, Rev. Mod. Phys. 68 (1996) 13. [4] M. Potthoff, Phys. Rev. B 64 (2001) 165114. [5] Y. Tomio, T. Ogawa, J. Lumin. 112 (2005) 220. [6] T. Ogawa, Y. Tomio, K. Asano, J. Phys.: Conf. Series 21 (2005) 112. [7] Y. Tomio, T. Ogawa, AIP Conf. Proc. 850 (2006) 1313.