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Si quantum wells
Superlattices and Microstructures, Vol. 23, No. 1, 1998
Evidence for a metal–insulator transition at B = 0 in Si/SiGe/Si quantum wells M. D’iorio, D. Brown, J. Lam, D. Stewart†, S. Deblois‡, H. Lafontaine National Research Council of Canada, IMS, Ottawa, ON, Canada K1A 0R6
(Received 15 July 1996) The temperature dependence of the resistivity of gated Si-SiGe quantum-well structures has revealed a metal–insulator transition as a function of carrier density at zero magnetic field. Although early scaling theories have argued against the existence of a metal–insulator transition at zero temperature in infinite two-dimensional systems, it is now clear experimentally that such a transition can occur in systems with short-range scatterers. We have studied the magneto-transport properties of holes confined in strained p-type Si-Si0.87 Ge0.13 -Si quantum wells grown by ultra-high-vacuum chemical-vapor deposition. In the temperature range 25 mK–4.2 K, there is a transition from an insulating phase at low carrier densities to a metallic phase at high carrier densities with a transition boundary near 3.3 × 1011 cm−2 . Evidence for a Coulomb gap is presented in the insulating phase. Key words: semiconductor quantum wells, metal insulator transition.
1. Introduction The study of metal–insulator transitions in two-dimensional systems has received a lot of attention recently with the discovery of re-entrant transitions from insulating to quantum Hall liquid states in the quantum [1] and extreme quantum limits [2]. In addition, in very high mobility Si-MOSFETs, there is now strong evidence for a true metal–insulator transition at zero magnetic field and low temperature as the density is driven into the dilute regime [3]. The temperature dependence of the resistivity in both metallic and insulating phases varies 1 as ρ(T ) = ρ0 exp(T0 /T ) 2 which is typical of a Coulomb gap [4] which opens in the density of states near the Fermi energy due to the Coulomb interactions between localized electrons. The zero-field metal–insulator transition bears the hallmarks of a true phase transformation as it scales along a single physical parameter. The resistivity as a function of temperature over a wide range of densities (7.12−13.7×1010 cm−2 ) can be made to overlap by scaling along the temperature axis [5]. When the resistivity is represented in terms of a ratio T /T0 where T0 depends only on the density, the data collapses on two separate curves, a metallic and an insulating one. The existence of a zero-field metal–insulator transition has now been verified in several material systems including superconducting thin films [6] and disordered GaAs-AlGaAs heterostructures [7] but runs counter to conventional wisdom which states that, at zero magnetic field, all states in a two-dimensional system are localized at low temperature in an infinite sample [8]. However, a model disorder potential with random .
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† Present address: Department of Applied Physics, Stanford University, U.S.A. ‡ Present address: D´epartement de physique, Universit´e Laval, Canada.
0749–6036/98/010055 + 07 $25.00/0
sm960298
56
Superlattices and Microstructures, Vol. 23, No. 1, 1998 3500 3000
ρxx(kΩ/ )
2500 ns = 1.77 × 1011 cm–2
2000 1500 1000 500 0
ns = 4.28 × 1011 cm–2 0
2
4 6 Magnetic field (T)
8
Fig. 1. Longitudinal resistivity of the two-dimensional hole gas as a function of the magnetic field for various carrier densities. The minimum at 6 T for the higher carrier densities corresponds to filling factor ν = 2. The measurements were taken at 25 mK in sample CVD32 K.
delta-type scatterers does allow for a transition to a metallic state above a mobility edge [9]. In this paper we describe the onset of a metal–insulator transition at zero magnetic field in p-type Si-SiGe-Si quantum wells. .
2. Samples Pseudomorphic growth of Si-Si0.87 Ge0.13 -Si quantum wells was achieved by ultra-high-vacuum chemicalvapor deposition (UHV-CVD) on a n − Si (100) substrate. The symmetrically doped structures consisted of a ˚ i-Si spacer on both sides of ˚ i-Si buffer, a 500 A ˚ Si layer doped with boron (1 × 1018 cm−3 ), a 300A 500 A ˚ ˚ ˚ a 65 A Si0.87 Ge0.13 quantum well, and a 430 A (CVD32 J) or 395 A (CVD32 K) Si layer doped with boron (1 × 1018 cm−3 ). The samples were patterned as Hall bar structures 2000 µm long by 100 µm wide. Ohmic ˚ Au–Ti Schottky contacts were made using Al evaporation and annealing below the eutectic point. A 1700 A gate was deposited on top of the samples. The carrier density and mobility were 2.55 × 1011 cm−2 and 1100 cm2 V−1 s−1 respectively with zero bias on the gate. The mobility and carrier density increase with the ˚ quantum well: µ = 12 000 cm2 V−1 s−1 at n s = 3.5 × 1011 cm−2 . SiGe quantum well width, e.g. for a 200 A The range of densities used for these measurements was 1.77–4.28 × 1011 cm−2 . The temperature of the sample was varied from 25 mK to 4.2 K using an Oxford top-loading dilution refrigerator and DC transport measurements were carried out using four-terminal techniques with a high impedance differential electrometer.
3. Results and discussion The magneto-transport properties of the two-dimensional holes confined in the SiGe quantum well were measured by conventional DC transport. The longitudinal resistivity ρx x is shown in Fig. 1 for carrier densities in the range 1.77–4.28 × 1011 cm−2 . As the density decreases, the resistivity increases at zero magnetic field and the Shubnikov–de Haas oscillations become very large at half-filled Landau levels. We will focus only on the zero-field longitudinal resistivity. The metal–insulator transition which is re-entrant with the quantum
Superlattices and Microstructures, Vol. 23, No. 1, 1998
57
25 mK 20
ρxx(kΩ/ )
18
100 16
1.3 K
ns of 3.3 × 1011 cm–2 2.8 3.0 3.2 3.4
3.6
3.8
4.0
10 1.5
2.0
2.5
3.0
3.5
4.0
4.5
ns (×1011 cm–2) Fig. 2. The dependence of the zero magnetic field longitudinal resistivity on carrier density for different temperatures. The intersection of the traces shown in the inset indicates the critical carrier density for the metal–insulator transition.
Hall effect has been studied elsewhere [10, 11]. By plotting the carrier density dependence of the resistivity at different temperatures as illustrated in Fig. 2, it is clear that there exists a crossover of all the traces at a single critical carrier density n c = 3.3 × 1011 cm−2 . The temperature dependence of the zero-field resistivity was studied in two regimes: 1.2–4.2 K and 25 mK–1.3 K. In Fig. 3A, there is evidence of a transition from a metallic behavior (ρx x decreases with decreasing temperature) at high densities to an insulating behavior (ρx x increases with decreasing temperature) at low densities in CVD32 J. This is much more pronounced in Fig. 3B in the temperature range 25 mK–1.3 K where the resistivity scale is 25 times larger. These measurements were taken in CVD32 K. There is also a sharp dip in the resistivity around 120 mK which develops slightly above the critical carrier density and becomes very strong in the dilute density regime. The origin of this drop is not clear yet. In Fig. 3C, the crossover from an insulating to a metallic temperature dependence is illustrated in the carrier density range 2.90–4.28 × 1011 cm−2 . A fit to the temperature dependence above 400 mK shows a transition from an activated behavior ρx x ∝ exp(T0 /T ) around 2 × 1011 cm−2 to a characteristic Coulomb 1 gap behavior ρx x ∝ exp(T0 /T ) 2 near 1.8 × 1011 cm−2 . This is illustrated in Fig. 4. The data is consistent with the occurrence of a phase transition from a metallic to an insulating phase near a critical density n c = 3.3 × 1011 cm−2 . In the insulating phase the temperature dependence of the resistivity points to the opening of a Coulomb gap. This attests in part to the quality of the samples and to the importance of the Coulomb interaction. The hallmarks of a phase transition are the scaling parameter and universal critical exponents. The extraction of a critical exponent ν such that the zero magnetic field resistivity scales with a single parameter T0 which approaches zero at the critical density and varies as |n c − n s |ν where ν = 1.6 in the metallic (n s > n c ) and insulating (n s < n c ) phases will require more effort. .
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4. Conclusion The zero-field metal–insulator transition in Si-SiGe-Si quantum wells has similar characteristics to that observed in high-mobility Si-MOSFETs. This is understandable as both material systems have large effective
58
Superlattices and Microstructures, Vol. 23, No. 1, 1998 16 15
2.4 × 1011 cm–2
ns = 2.4–2.9 × 1011 cm–2
14
ρxx(kΩ/ )
13 12 11 10 9
2.9 × 1011 cm–2
8 7
Sample 32 J 2
1
3
4
Temperature (K)
Increasing ns
400
ρxx(kΩ/ )
300
ns = 1.77–4.28 × 1011 cm–2
200 100 0 Increasing ns
ρxx(kΩ/ )
20.00 18.75
ns = 2.9–4.28 × 1011 cm–2
17.50 16.25 15.00 13.75 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Temperature (K) Fig. 3. A, The temperature dependence of the zero-field longitudinal resistivity in the temperature range 1.2–4.2 K and carrier density range 2.4–2.9×1011 cm−2 for sample CVD32 J. B, in the temperature range 25 mK–1.3 K and carrier density range 1.77–4.28×1011 cm−2 for sample CVD32 K. C, in the temperature range 25 mK–1.3 K and carrier density range 2.9–4.28 × 1011 cm−2 for CVD32 K.
Superlattices and Microstructures, Vol. 23, No. 1, 1998
59
400 ns = 1.77 × 1011 cm–2 ns = 1.91 × 1011 cm–2
ρxx(kΩ/ )
300
200
(1/T)1/2 (1/T)
100
0.4
0.6
0.8 1.0 Temperature (K)
1.2
1.4
1
Fig. 4. Fit to the temperature dependence of the zero-field resistivity: ρx x = ρ0 exp(T0 /T ) 2 for n s = 1.77 × 1011 cm−2 and ρx x = ρ0 exp(T0 /T ) for n s = 1.91 × 1011 cm−2 .
masses, large Coulomb interaction compared to the Fermi energy and similar Wigner-Seitz to Bohr radius ratios. In both cases, the transport to quantum lifetime ratios are close to unity meaning that the carriers undergo mostly large-angle or short-range scattering. This is consistent with Azbel’s theory for which the metal–insulator transition is not forbidden in two-dimensional systems with short-range scatterers. Acknowledgements—We wish to thank John Stapledon and Yan Feng for their assistance with the preparation of the samples. This work was supported in part by a Natural Sciences and Engineering Research Council (NSERC) operating grant.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
}V. M. Pudalov, M. D’Iorio, S. V. Kravchenko, and J. W. Campbell, Phys. Rev. Lett. 70, 1866 (1993). }S.-T. Chui (ed.) Physics of the Electron Solid World Scientific, Singapore (1995), for a review. }A. L. Efros and B. I. Shklovskii, J. Phys. C8, L49 (1975). }S. V. Kravchenko, W. E. Mason, G. E. Bowker, J. E. Furneaux, V. M. Pudalov, and M. D’Iorio, Phys. Rev. B51, 7038 (1995). }W. Mason, S. V. Kravchenko, G. E. Bowker, and J. E. Furneaux, Phys. Rev. B52, 7857 (1995). }Y. Liu, K. A. McGreer, B. Nease, D. B. Haviland, G. Martinez, J. W. Halley, and A. M. Goldman, Phys. Rev. Lett. 67, 2068 (1991). }H. W. Jiang, C. E. Johnson, K. L. Wang, and S. T. Hannahs, Phys. Rev. Lett. 71, 1439 (1993). }E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979). }M. Ya. Azbel, Phys. Rev. B45 4208 (1992). }M. D’Iorio, D. Stewart, S. Deblois, D. Brown, and J.-P. No¨el, Surface Sci. 361/362, 937 (1996). }S. I. Dorozhkin, C. J. Emeleus, T. E. Whall, and G. Landwehr, Phys. Rev. B52, 11638 (1995). .
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Evidence for a metal–insulator transition at B = 0 in Si/SiGe/Si quantum wells M. D’iorio, D. Brown, J. Lam, D. Stewart†, S. Deblois‡, H. Lafontaine National Research Council of Canada, IMS, Ottawa, ON, Canada K1A 0R6
(Received 15 July 1996) The temperature dependence of the resistivity of gated Si-SiGe quantum-well structures has revealed a metal–insulator transition as a function of carrier density at zero magnetic field. Although early scaling theories have argued against the existence of a metal–insulator transition at zero temperature in infinite two-dimensional systems, it is now clear experimentally that such a transition can occur in systems with short-range scatterers. We have studied the magneto-transport properties of holes confined in strained p-type Si-Si0.87 Ge0.13 -Si quantum wells grown by ultra-high-vacuum chemical-vapor deposition. In the temperature range 25 mK–4.2 K, there is a transition from an insulating phase at low carrier densities to a metallic phase at high carrier densities with a transition boundary near 3.3 × 1011 cm−2 . Evidence for a Coulomb gap is presented in the insulating phase. Key words: semiconductor quantum wells, metal insulator transition.
1. Introduction The study of metal–insulator transitions in two-dimensional systems has received a lot of attention recently with the discovery of re-entrant transitions from insulating to quantum Hall liquid states in the quantum [1] and extreme quantum limits [2]. In addition, in very high mobility Si-MOSFETs, there is now strong evidence for a true metal–insulator transition at zero magnetic field and low temperature as the density is driven into the dilute regime [3]. The temperature dependence of the resistivity in both metallic and insulating phases varies 1 as ρ(T ) = ρ0 exp(T0 /T ) 2 which is typical of a Coulomb gap [4] which opens in the density of states near the Fermi energy due to the Coulomb interactions between localized electrons. The zero-field metal–insulator transition bears the hallmarks of a true phase transformation as it scales along a single physical parameter. The resistivity as a function of temperature over a wide range of densities (7.12−13.7×1010 cm−2 ) can be made to overlap by scaling along the temperature axis [5]. When the resistivity is represented in terms of a ratio T /T0 where T0 depends only on the density, the data collapses on two separate curves, a metallic and an insulating one. The existence of a zero-field metal–insulator transition has now been verified in several material systems including superconducting thin films [6] and disordered GaAs-AlGaAs heterostructures [7] but runs counter to conventional wisdom which states that, at zero magnetic field, all states in a two-dimensional system are localized at low temperature in an infinite sample [8]. However, a model disorder potential with random .
.
.
.
.
.
.
.
† Present address: Department of Applied Physics, Stanford University, U.S.A. ‡ Present address: D´epartement de physique, Universit´e Laval, Canada.
0749–6036/98/010055 + 07 $25.00/0
sm960298
56
Superlattices and Microstructures, Vol. 23, No. 1, 1998 3500 3000
ρxx(kΩ/ )
2500 ns = 1.77 × 1011 cm–2
2000 1500 1000 500 0
ns = 4.28 × 1011 cm–2 0
2
4 6 Magnetic field (T)
8
Fig. 1. Longitudinal resistivity of the two-dimensional hole gas as a function of the magnetic field for various carrier densities. The minimum at 6 T for the higher carrier densities corresponds to filling factor ν = 2. The measurements were taken at 25 mK in sample CVD32 K.
delta-type scatterers does allow for a transition to a metallic state above a mobility edge [9]. In this paper we describe the onset of a metal–insulator transition at zero magnetic field in p-type Si-SiGe-Si quantum wells. .
2. Samples Pseudomorphic growth of Si-Si0.87 Ge0.13 -Si quantum wells was achieved by ultra-high-vacuum chemicalvapor deposition (UHV-CVD) on a n − Si (100) substrate. The symmetrically doped structures consisted of a ˚ i-Si spacer on both sides of ˚ i-Si buffer, a 500 A ˚ Si layer doped with boron (1 × 1018 cm−3 ), a 300A 500 A ˚ ˚ ˚ a 65 A Si0.87 Ge0.13 quantum well, and a 430 A (CVD32 J) or 395 A (CVD32 K) Si layer doped with boron (1 × 1018 cm−3 ). The samples were patterned as Hall bar structures 2000 µm long by 100 µm wide. Ohmic ˚ Au–Ti Schottky contacts were made using Al evaporation and annealing below the eutectic point. A 1700 A gate was deposited on top of the samples. The carrier density and mobility were 2.55 × 1011 cm−2 and 1100 cm2 V−1 s−1 respectively with zero bias on the gate. The mobility and carrier density increase with the ˚ quantum well: µ = 12 000 cm2 V−1 s−1 at n s = 3.5 × 1011 cm−2 . SiGe quantum well width, e.g. for a 200 A The range of densities used for these measurements was 1.77–4.28 × 1011 cm−2 . The temperature of the sample was varied from 25 mK to 4.2 K using an Oxford top-loading dilution refrigerator and DC transport measurements were carried out using four-terminal techniques with a high impedance differential electrometer.
3. Results and discussion The magneto-transport properties of the two-dimensional holes confined in the SiGe quantum well were measured by conventional DC transport. The longitudinal resistivity ρx x is shown in Fig. 1 for carrier densities in the range 1.77–4.28 × 1011 cm−2 . As the density decreases, the resistivity increases at zero magnetic field and the Shubnikov–de Haas oscillations become very large at half-filled Landau levels. We will focus only on the zero-field longitudinal resistivity. The metal–insulator transition which is re-entrant with the quantum
Superlattices and Microstructures, Vol. 23, No. 1, 1998
57
25 mK 20
ρxx(kΩ/ )
18
100 16
1.3 K
ns of 3.3 × 1011 cm–2 2.8 3.0 3.2 3.4
3.6
3.8
4.0
10 1.5
2.0
2.5
3.0
3.5
4.0
4.5
ns (×1011 cm–2) Fig. 2. The dependence of the zero magnetic field longitudinal resistivity on carrier density for different temperatures. The intersection of the traces shown in the inset indicates the critical carrier density for the metal–insulator transition.
Hall effect has been studied elsewhere [10, 11]. By plotting the carrier density dependence of the resistivity at different temperatures as illustrated in Fig. 2, it is clear that there exists a crossover of all the traces at a single critical carrier density n c = 3.3 × 1011 cm−2 . The temperature dependence of the zero-field resistivity was studied in two regimes: 1.2–4.2 K and 25 mK–1.3 K. In Fig. 3A, there is evidence of a transition from a metallic behavior (ρx x decreases with decreasing temperature) at high densities to an insulating behavior (ρx x increases with decreasing temperature) at low densities in CVD32 J. This is much more pronounced in Fig. 3B in the temperature range 25 mK–1.3 K where the resistivity scale is 25 times larger. These measurements were taken in CVD32 K. There is also a sharp dip in the resistivity around 120 mK which develops slightly above the critical carrier density and becomes very strong in the dilute density regime. The origin of this drop is not clear yet. In Fig. 3C, the crossover from an insulating to a metallic temperature dependence is illustrated in the carrier density range 2.90–4.28 × 1011 cm−2 . A fit to the temperature dependence above 400 mK shows a transition from an activated behavior ρx x ∝ exp(T0 /T ) around 2 × 1011 cm−2 to a characteristic Coulomb 1 gap behavior ρx x ∝ exp(T0 /T ) 2 near 1.8 × 1011 cm−2 . This is illustrated in Fig. 4. The data is consistent with the occurrence of a phase transition from a metallic to an insulating phase near a critical density n c = 3.3 × 1011 cm−2 . In the insulating phase the temperature dependence of the resistivity points to the opening of a Coulomb gap. This attests in part to the quality of the samples and to the importance of the Coulomb interaction. The hallmarks of a phase transition are the scaling parameter and universal critical exponents. The extraction of a critical exponent ν such that the zero magnetic field resistivity scales with a single parameter T0 which approaches zero at the critical density and varies as |n c − n s |ν where ν = 1.6 in the metallic (n s > n c ) and insulating (n s < n c ) phases will require more effort. .
.
4. Conclusion The zero-field metal–insulator transition in Si-SiGe-Si quantum wells has similar characteristics to that observed in high-mobility Si-MOSFETs. This is understandable as both material systems have large effective
58
Superlattices and Microstructures, Vol. 23, No. 1, 1998 16 15
2.4 × 1011 cm–2
ns = 2.4–2.9 × 1011 cm–2
14
ρxx(kΩ/ )
13 12 11 10 9
2.9 × 1011 cm–2
8 7
Sample 32 J 2
1
3
4
Temperature (K)
Increasing ns
400
ρxx(kΩ/ )
300
ns = 1.77–4.28 × 1011 cm–2
200 100 0 Increasing ns
ρxx(kΩ/ )
20.00 18.75
ns = 2.9–4.28 × 1011 cm–2
17.50 16.25 15.00 13.75 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Temperature (K) Fig. 3. A, The temperature dependence of the zero-field longitudinal resistivity in the temperature range 1.2–4.2 K and carrier density range 2.4–2.9×1011 cm−2 for sample CVD32 J. B, in the temperature range 25 mK–1.3 K and carrier density range 1.77–4.28×1011 cm−2 for sample CVD32 K. C, in the temperature range 25 mK–1.3 K and carrier density range 2.9–4.28 × 1011 cm−2 for CVD32 K.
Superlattices and Microstructures, Vol. 23, No. 1, 1998
59
400 ns = 1.77 × 1011 cm–2 ns = 1.91 × 1011 cm–2
ρxx(kΩ/ )
300
200
(1/T)1/2 (1/T)
100
0.4
0.6
0.8 1.0 Temperature (K)
1.2
1.4
1
Fig. 4. Fit to the temperature dependence of the zero-field resistivity: ρx x = ρ0 exp(T0 /T ) 2 for n s = 1.77 × 1011 cm−2 and ρx x = ρ0 exp(T0 /T ) for n s = 1.91 × 1011 cm−2 .
masses, large Coulomb interaction compared to the Fermi energy and similar Wigner-Seitz to Bohr radius ratios. In both cases, the transport to quantum lifetime ratios are close to unity meaning that the carriers undergo mostly large-angle or short-range scattering. This is consistent with Azbel’s theory for which the metal–insulator transition is not forbidden in two-dimensional systems with short-range scatterers. Acknowledgements—We wish to thank John Stapledon and Yan Feng for their assistance with the preparation of the samples. This work was supported in part by a Natural Sciences and Engineering Research Council (NSERC) operating grant.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
}V. M. Pudalov, M. D’Iorio, S. V. Kravchenko, and J. W. Campbell, Phys. Rev. Lett. 70, 1866 (1993). }S.-T. Chui (ed.) Physics of the Electron Solid World Scientific, Singapore (1995), for a review. }A. L. Efros and B. I. Shklovskii, J. Phys. C8, L49 (1975). }S. V. Kravchenko, W. E. Mason, G. E. Bowker, J. E. Furneaux, V. M. Pudalov, and M. D’Iorio, Phys. Rev. B51, 7038 (1995). }W. Mason, S. V. Kravchenko, G. E. Bowker, and J. E. Furneaux, Phys. Rev. B52, 7857 (1995). }Y. Liu, K. A. McGreer, B. Nease, D. B. Haviland, G. Martinez, J. W. Halley, and A. M. Goldman, Phys. Rev. Lett. 67, 2068 (1991). }H. W. Jiang, C. E. Johnson, K. L. Wang, and S. T. Hannahs, Phys. Rev. Lett. 71, 1439 (1993). }E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979). }M. Ya. Azbel, Phys. Rev. B45 4208 (1992). }M. D’Iorio, D. Stewart, S. Deblois, D. Brown, and J.-P. No¨el, Surface Sci. 361/362, 937 (1996). }S. I. Dorozhkin, C. J. Emeleus, T. E. Whall, and G. Landwehr, Phys. Rev. B52, 11638 (1995). .
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