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Crossover from Coulomb glass to Fermi glass in Si:P
ARTICLE IN PRESS
Physica B 359–361 (2005) 1469–1471 www.elsevier.com/locate/physb
Crossover from Coulomb glass to Fermi glass in Si:P Marco Heringa,b,, Marc Schefflera, Martin Dressela, Hilbert v. Lo¨hneysenb a
1. Physikalisches Institut, Universita¨t Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany b Physikalisches Institut, Universita¨t Karlsruhe, 76128 Karlsruhe, Germany
Abstract Electronic transport in highly doped but still insulating Si:P at low temperatures is dominated by localized states. Because these states are due to the randomly distributed phosphorus atoms in the silicon crystal, the whole system is disordered and thus Si:P is a perfect model system for so-called electron glasses. We have studied the frequencydependent conductivity of this system in the THz frequency range. In agreement with previous experimental results and the theoretical predictions by Efros and Shklovskii with increasing energy, we have found a crossover from (interacting) Coulomb glass to (noninteracting) Fermi glass behavior. Additionally, we have observed a strong influence of the Coulomb gap which was not taken into account by previous workers. r 2005 Published by Elsevier B.V. Keywords: Coulomb glass; Fermi glass; AC conductivity
1. Introduction At low excitation energies and temperatures, hopping processes between the disordered, localized states dominate the charge carrier transport in Si:P, in particular variable range hopping (VRH) as predicted by Mott [1] in 1968. Mott’s theoretical model for the conduction process in the VRH regime describes the so-called Fermi glasses where correlation effects between the localized electronic states are neglected, whereas taking into Corresponding author. 1. Physikalisches Institut, Universi-
ta¨t Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany. Tel.: +49 711 685 4893; fax: +49 711 685 4886. E-mail address: [email protected] (M. Hering). 0921-4526/$ - see front matter r 2005 Published by Elsevier B.V. doi:10.1016/j.physb.2005.01.457
account Coulomb interactions between the states leads to the Coulomb glass model. For both cases, the theoretical predictions for the DC conductivity have been validated experimentally [2]. A transition was found from a temperature dependence corresponding to that suggested by Mott for a Fermi glass to one as predicted by Efros and Shklovskii [3] for the Coulomb glass at lower temperatures. For their derivation of the DC conductivity, Efros and Shklovskii (ES) took the Coulomb gap into account that opens in the density of states (DOS) near the Fermi edge due to Coulomb interactions. The transition from one behavior to the other shows that at higher excitation energies, the effects of Coulomb interaction are negligible.
ARTICLE IN PRESS M. Hering et al. / Physica B 359– 361 (2005) 1469–1471
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Also for the AC conductivity at T ¼ 0 K there are different predictions. In contrast to Mott’s derivation [4] for the Fermi glass, ES [5] introduced the Coulomb potential Uðro Þ ¼ e2 =1 ro between the two states directly involved in a hopping process and separated by a mean distance ro ¼ x lnð2I 0 =_oÞ: Here x is the localization length and I 0 is the prefactor of the overlap integral. Its value is of the order of the binding energy of the electronic state [6] and is commonly taken to be the Bohr energy of the dopant (for Si:P: I 0 45 meV). Thus ES showed that there is also a transition in frequency space from the Coulomb glass at lower energies, with approximately linearfrequency dependence, to the Fermi glass with quadratic behavior s1 ðoÞ ¼ ae2 N 20 _oxr4o ½_o þ Uðro Þ.
(1)
Fig. 1. Frequency-dependent conductivity of a Si:P sample with n ¼ 1:6 1018 cm 3 at T ¼ 1:8 K: (Experimental error bars correspond to symbol size.) The solid lines are fits with linear and quadratic behavior, respectively. The dotted line is a fit with the frequency dependence as predicted by ES (1).
The factor a is a constant of order one and N 0 is the noninteracting DOS close to the Fermi level which is assumed to be constant. But for Coulomb glasses, with the DOS at the Fermi level affected by the Coulomb gap, this only holds for Uðro ÞbD; with D the width of the Coulomb gap. In the opposite case, mainly states within the gap participate in the hopping conduction and, as claimed by ES [5], instead of the sub-linear behavior of Eq. (1) at low energies _o5Uðro Þ: s1 ðoÞ ¼
ae4 2 N _ox4 ½lnð2I 0 =_oÞ3 . 1 0
(2)
The frequency dependence turns to a super-linear behavior for _o5Uðro Þ5D: s1 ðoÞ ¼ ae2 N 20 x4
_o . lnð2I 0 =_oÞ
(3)
2. Results We have studied the frequency-dependent conductivity of Si:P in a range between 30 and 3000 GHz using far-infrared and quasi-optical submillimeter [7] techniques. In qualitative agreement with the theory by ES, the measured frequency dependence at various phosphorus concentrations shows a crossover from approximately linear to quadratic behavior. But when it comes to a quantitative description Eq. (1) fails for
Fig. 2. Frequency-dependent conductivity in the Coulomb glass regime for samples with phosphorus concentrations n ¼ 2:29 1018 cm 3 and n ¼ 2:57 1018 cm 3 at T ¼ 1:8 K: The fits (solid) correspond to a super-linear behavior following Eq. (3) with I 0 ¼ 45 meV; strictly linear behavior is plotted for comparison (dotted). At higher frequencies the transition to quadratic behavior is apparent.
ARTICLE IN PRESS M. Hering et al. / Physica B 359– 361 (2005) 1469–1471
all samples. As can be seen exemplarily in Fig. 1 for a sample with doping concentration n ¼ 1:6 1018 cm 3 ; the crossover from Coulomb to Fermi glass behavior is much sharper than theoretically predicted. Taking into account the logarithmic corrections one finds that the data sets of all samples at low energies are better described by Eq. (3). In Fig. 2 the super-linear fit is compared to the strictly linear fit for two different samples. This super-linear frequency dependence indicates that the Coulomb gap strongly influences the AC conductivity. Also the sharp crossover, that was observed in Si:B [8] as well, might be connected to the Coulomb gap as discussed by Lee and Stutzmann [8]. These interpretations are in contrast to results by Helgren et al. [9] who found the concentration-dependent crossover energy corresponding to the scaling behavior of the interaction energy as predicted by ES.
1471
References [1] N.F. Mott, J. Non-Crystal. Solids 1 (1968) 1. [2] M. Hornung, M. Iqbal, S. Waffenschmidt, H.v. Lo¨hneysen, Phys. Stat. Sol. B 218 (2000) 75. [3] B.I. Shklovskii, A.L. Efros, Electronic Properties of Doped Semiconductors, Springer, Berlin, 1984. [4] N.F. Mott, E.A. Davis, Electronic Processes in NonCrystalline Materials, Oxford University Press, Oxford, 1979. [5] A.L. Efros, B.I. Shklovskii, Sov. Phys. JETP 54 (1981) 218. [6] T.G. Castner, in: M. Pollak, B.I. Shklovskii (Eds.), Hopping Transport in Solids, North-Holland, Amsterdam, 1991. [7] G. Kozlov, A. Volkov, in: G. Gru¨ner (Ed.), Millimeter and Submillimeter Wave Spectroscopy of Solids, Springer, Heidelberg, 1998. [8] M. Lee, M.L. Stutzmann, Phys. Rev. Lett. 87 (2001) 56402. [9] E. Helgren, N.P. Armitage, G. Gru¨ner, Phys. Rev. Lett. 89 (2002) 246601.
Physica B 359–361 (2005) 1469–1471 www.elsevier.com/locate/physb
Crossover from Coulomb glass to Fermi glass in Si:P Marco Heringa,b,, Marc Schefflera, Martin Dressela, Hilbert v. Lo¨hneysenb a
1. Physikalisches Institut, Universita¨t Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany b Physikalisches Institut, Universita¨t Karlsruhe, 76128 Karlsruhe, Germany
Abstract Electronic transport in highly doped but still insulating Si:P at low temperatures is dominated by localized states. Because these states are due to the randomly distributed phosphorus atoms in the silicon crystal, the whole system is disordered and thus Si:P is a perfect model system for so-called electron glasses. We have studied the frequencydependent conductivity of this system in the THz frequency range. In agreement with previous experimental results and the theoretical predictions by Efros and Shklovskii with increasing energy, we have found a crossover from (interacting) Coulomb glass to (noninteracting) Fermi glass behavior. Additionally, we have observed a strong influence of the Coulomb gap which was not taken into account by previous workers. r 2005 Published by Elsevier B.V. Keywords: Coulomb glass; Fermi glass; AC conductivity
1. Introduction At low excitation energies and temperatures, hopping processes between the disordered, localized states dominate the charge carrier transport in Si:P, in particular variable range hopping (VRH) as predicted by Mott [1] in 1968. Mott’s theoretical model for the conduction process in the VRH regime describes the so-called Fermi glasses where correlation effects between the localized electronic states are neglected, whereas taking into Corresponding author. 1. Physikalisches Institut, Universi-
ta¨t Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany. Tel.: +49 711 685 4893; fax: +49 711 685 4886. E-mail address: [email protected] (M. Hering). 0921-4526/$ - see front matter r 2005 Published by Elsevier B.V. doi:10.1016/j.physb.2005.01.457
account Coulomb interactions between the states leads to the Coulomb glass model. For both cases, the theoretical predictions for the DC conductivity have been validated experimentally [2]. A transition was found from a temperature dependence corresponding to that suggested by Mott for a Fermi glass to one as predicted by Efros and Shklovskii [3] for the Coulomb glass at lower temperatures. For their derivation of the DC conductivity, Efros and Shklovskii (ES) took the Coulomb gap into account that opens in the density of states (DOS) near the Fermi edge due to Coulomb interactions. The transition from one behavior to the other shows that at higher excitation energies, the effects of Coulomb interaction are negligible.
ARTICLE IN PRESS M. Hering et al. / Physica B 359– 361 (2005) 1469–1471
1470
Also for the AC conductivity at T ¼ 0 K there are different predictions. In contrast to Mott’s derivation [4] for the Fermi glass, ES [5] introduced the Coulomb potential Uðro Þ ¼ e2 =1 ro between the two states directly involved in a hopping process and separated by a mean distance ro ¼ x lnð2I 0 =_oÞ: Here x is the localization length and I 0 is the prefactor of the overlap integral. Its value is of the order of the binding energy of the electronic state [6] and is commonly taken to be the Bohr energy of the dopant (for Si:P: I 0 45 meV). Thus ES showed that there is also a transition in frequency space from the Coulomb glass at lower energies, with approximately linearfrequency dependence, to the Fermi glass with quadratic behavior s1 ðoÞ ¼ ae2 N 20 _oxr4o ½_o þ Uðro Þ.
(1)
Fig. 1. Frequency-dependent conductivity of a Si:P sample with n ¼ 1:6 1018 cm 3 at T ¼ 1:8 K: (Experimental error bars correspond to symbol size.) The solid lines are fits with linear and quadratic behavior, respectively. The dotted line is a fit with the frequency dependence as predicted by ES (1).
The factor a is a constant of order one and N 0 is the noninteracting DOS close to the Fermi level which is assumed to be constant. But for Coulomb glasses, with the DOS at the Fermi level affected by the Coulomb gap, this only holds for Uðro ÞbD; with D the width of the Coulomb gap. In the opposite case, mainly states within the gap participate in the hopping conduction and, as claimed by ES [5], instead of the sub-linear behavior of Eq. (1) at low energies _o5Uðro Þ: s1 ðoÞ ¼
ae4 2 N _ox4 ½lnð2I 0 =_oÞ3 . 1 0
(2)
The frequency dependence turns to a super-linear behavior for _o5Uðro Þ5D: s1 ðoÞ ¼ ae2 N 20 x4
_o . lnð2I 0 =_oÞ
(3)
2. Results We have studied the frequency-dependent conductivity of Si:P in a range between 30 and 3000 GHz using far-infrared and quasi-optical submillimeter [7] techniques. In qualitative agreement with the theory by ES, the measured frequency dependence at various phosphorus concentrations shows a crossover from approximately linear to quadratic behavior. But when it comes to a quantitative description Eq. (1) fails for
Fig. 2. Frequency-dependent conductivity in the Coulomb glass regime for samples with phosphorus concentrations n ¼ 2:29 1018 cm 3 and n ¼ 2:57 1018 cm 3 at T ¼ 1:8 K: The fits (solid) correspond to a super-linear behavior following Eq. (3) with I 0 ¼ 45 meV; strictly linear behavior is plotted for comparison (dotted). At higher frequencies the transition to quadratic behavior is apparent.
ARTICLE IN PRESS M. Hering et al. / Physica B 359– 361 (2005) 1469–1471
all samples. As can be seen exemplarily in Fig. 1 for a sample with doping concentration n ¼ 1:6 1018 cm 3 ; the crossover from Coulomb to Fermi glass behavior is much sharper than theoretically predicted. Taking into account the logarithmic corrections one finds that the data sets of all samples at low energies are better described by Eq. (3). In Fig. 2 the super-linear fit is compared to the strictly linear fit for two different samples. This super-linear frequency dependence indicates that the Coulomb gap strongly influences the AC conductivity. Also the sharp crossover, that was observed in Si:B [8] as well, might be connected to the Coulomb gap as discussed by Lee and Stutzmann [8]. These interpretations are in contrast to results by Helgren et al. [9] who found the concentration-dependent crossover energy corresponding to the scaling behavior of the interaction energy as predicted by ES.
1471
References [1] N.F. Mott, J. Non-Crystal. Solids 1 (1968) 1. [2] M. Hornung, M. Iqbal, S. Waffenschmidt, H.v. Lo¨hneysen, Phys. Stat. Sol. B 218 (2000) 75. [3] B.I. Shklovskii, A.L. Efros, Electronic Properties of Doped Semiconductors, Springer, Berlin, 1984. [4] N.F. Mott, E.A. Davis, Electronic Processes in NonCrystalline Materials, Oxford University Press, Oxford, 1979. [5] A.L. Efros, B.I. Shklovskii, Sov. Phys. JETP 54 (1981) 218. [6] T.G. Castner, in: M. Pollak, B.I. Shklovskii (Eds.), Hopping Transport in Solids, North-Holland, Amsterdam, 1991. [7] G. Kozlov, A. Volkov, in: G. Gru¨ner (Ed.), Millimeter and Submillimeter Wave Spectroscopy of Solids, Springer, Heidelberg, 1998. [8] M. Lee, M.L. Stutzmann, Phys. Rev. Lett. 87 (2001) 56402. [9] E. Helgren, N.P. Armitage, G. Gru¨ner, Phys. Rev. Lett. 89 (2002) 246601.