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Ultraslow electron spin dynamics in the fractional quantum Hall regime
Physica B 256±258 (1998) 121±124
Ultraslow electron spin dynamics in the fractional quantum Hall regime N.N. Kuzma
a,*
, P. Khandelwal a, S.E. Barrett a, L.N. Pfeier b, K.W. West a b
b
Department of Physics, Yale University, New Haven, CT 06520, USA Lucent Technologies Bell Laboratories, Murray Hill, NJ 07974, USA
Abstract Recent optically pumped nuclear magnetic resonance (OPNMR) measurements in two dierent electron-doped multiple quantum well samples near the fractional quantum Hall ground state m 13 are reviewed. Below 0.5 K, the spectra provide evidence that the spin-reversed charged excitations of this ground state are localized over the NMR time scale of 40 ls. By varying NMR pulse parameters, the electron spin temperature (as measured by the Knight shift) could be driven above the lattice temperature. For Tlattice < 0:5 K, these non-equilibrium measurements imply that the value of the electron spin-lattice relaxation time s1s lies between 100 ls and 500 ms at m 13. Ó 1998 Elsevier Science B.V. All rights reserved. Keywords: Fractional quantum Hall eect; NMR; GaAs
Optically pumped nuclear magnetic resonance (OPNMR) has been recently used to study two-dimensional electron systems (2DES) exhibiting the fractional quantum Hall eect (FQHE) [1,2]. We review evidence for ultraslow electron spin dynamics near the most studied FQHE ground state m 13 (where m nhc=Btot cos h) in two dierent multiple quantum well samples, `40 W' and `10 W' (n40 W 6:69 1010 cmÿ2 and n10 W 7:75 1010 cmÿ2 ), described elsewhere in this volume [3]. A series of 71 Ga OPNMR emission spectra are shown in Fig. 1 (solid lines). All spectra exhibit two peaks: one from nuclei in the Al0:1 Ga0:9 As barriers at zero frequency shift, and the other due to nuclei in the GaAs quantum wells [1,2,4], which is shifted to lower frequencies by the Fermi contact
* Corresponding author. Fax: 1 203 432 6175; e-mail: [email protected]
hyper®ne coupling to the spins of 2DES [5±7]. We de®ne the peak-to-peak splitting to be the Knight shift KS . The spectra at m 13 are well described by a simple model explained elsewhere (Fig. 1, dashed lines) [2,3]. The central assumption of this model is that the electron spins are delocalized along the well, such that hSz m; T i appears spatially homogeneous, when averaged over the NMR time scale ( 40 ls). In this limit, the delocalization of the low density 2DES produces a `motional narrowing' of the well peak. However, as Fig. 1(A) demonstrates, low-temperature measurements at m 0.267 show a crossover to more complicated line shapes. Although the spectra are in reasonable agreement with our model above 1 K, the width of the well peak increases dramatically as the temperature is lowered to T 0.45 K and then decreases upon further lowering to T 0.31 K. Fig. 1(B) shows that the extra broadening of the well resonance disappears
0921-4526/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 5 4 7 - X
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N.N. Kuzma et al. / Physica B 256±258 (1998) 121±124
Fig. 1. 71 Ga OPNMR spectra (solid lines) of sample 10 W, taken in Btot 12 T at: (A) m 0:267 (h 0 ), 1.49 K P T P 0:31 K; and (B) 0:267 6 m 6 0:325 0 6 h 6 34:9 ), T 0.46 K. Frequency shift is relative to fo 155.93 MHz. The extreme full width athalf maximum of the well peak is shown; the dashed lines are ®ts referred to in text. Adapted from [2].
as the sample is tilted from h10 W 0 (m 0.267) to 36.8 (m 13), despite a 10% increase in the dipolar broadening. Furthermore, there is a striking correspondence between the decrease in the linewidth (Fig. 2, open symbols) and the increase in KS (®lled symbols) as m ! 13 : The observed broadening of the well line shape beyond the `motionally narrowed' limit implies that the time-averaged product of j/j2 P becomes spatially inhomogeneous in the plane of 2DES, where j/j2 is the local electron density, and P is the local electron spin polarization. Neither the formation of a Wigner crystal nor irregularities in the charge density along the well are consistent with all the aspects of our data [2]. In contrast, the
Fig. 2. The temperature dependence of the Knight shift (®lled symbols) and the linewidth (open symbols) in sample 10 W for ®lling factors: (A) 0.267 6 m 6 13 (adapted from [2]), and (B) m P 13. Lines are to guide the eye.
Knight shift and thus the total spin polarization drops monotonically below m 13, allowing the local spin polarization P ~ R0 ) to be spatially inhomogeneous (Fig. 3). This suggests that localization of spin-reversed regions is responsible for the behavior shown in Figs. 1 and 2. The time scale of this localization may be inferred from the simple model in which, after every jump time sJ , the local electron polarization at each nuclear site instantaneously assumes values of either 1 or ÿ0:15, with probability p or 1 ÿ p ), respectively [2]. Fig. 4 shows how the simulated spectra depend upon the value of sJ , for the case p 0.85, in qualitative agreement with the corresponding data in Fig. 1(A). When sJ is very fast, all nuclei see the same time-averaged local polarization, equal to the total polarization. At the other extreme (sJ ! 1), the motion is fro-
N.N. Kuzma et al. / Physica B 256±258 (1998) 121±124
Fig. 3. The evolution of the well linewidth and Knight shift as the temperature of sample 10 W is varied from 1.5 K (lower left corner) to 0.3 K (upper right corner), for ®lling factors below 13. Same data as in Fig. 2(A).
Fig. 4. Simulated OPNMR spectra using the model described in the text. KSint is set to 12 kHz for P 1. The barrier is suppressed (ab 0) for clarity. Adapted from [2].
zen out, and the single resonance splits into two lines. In the intermediate motion regime, the FWHM of the well peak goes through a maximum when sJ 40 ls.
123
Based upon this simple model, the peaks in the FWHM at Tloc 0:5 K (Fig. 2(A)) re¯ect the localization of reversed spins, such that they fail to cover the sample uniformly over 40 ls. The self-similar curves in Fig. 2(A) suggest that Tloc is not a strong function of the ®lling factor (or the density of reversed spins) for m < 13. Note also that below 0.5 K, the measured KS m < 13) increases toward KS m 13), as seen in the model. However, even down to T 0.3 K, the spectra do not appear to match the frozen limit of our simulation. Surprisingly, as m is varied below 13, the trends in the KS and FWHM data (Fig. 2(A) and 3) continue smoothly through m 27 without interruption. High-®eld magnetotransport measurements on samples taken from the same wafer as 10 W show much more structure, with well-developed minima in qxx at m 13 ; 25 ; 27, and 15 at T 0:3 K [8,9]. Additional measurements of the linewidth for m > 13 in sample 10 W are consistent with the above picture (Fig. 2(B)). The observed spectra contain more information than our simple simulation has revealed. A more sophisticated model might explain the non-trivial behavior shown in Fig. 3 and should probably include: (i) a detailed structure for the reversed spin regions present below m 13, (ii) the 2D dynamics of these reversed spins, and (iii) the eects of thermally excited spin ¯ips, since Tsat is not that much greater than Tloc . All of the above results were obtained using a weak rf pulse long after optical pumping to probe the equilibrium properties of the 2DES. The nonequilibrium properties of the 2DES can also be studied by varying these parameters, with a number of remarkable results at m 13. The rf tipping pulse for the NMR experiment is produced by a coil wrapped around the sample (Fig. 5(C), inset), which generates a linearly polar! ized (perpendicular to z0 ) magnetic ®eld of amplitude 2H1 at fo 155:93 MHz. The equilibrium value of KS (T) is independent of the rf pulse parameters for weak H1 (that is, H1 5 Gauss, spulse 20 ls). However, if stronger pulses are used at T < 0:5 K, the measured KS drops sharply below the equilibrium value, even though the lattice temperature is unaected by the pulses. The equilibrium measurements [1] of KS (T) (Fig. 5 (A)
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N.N. Kuzma et al. / Physica B 256±258 (1998) 121±124
The non-equilibrium spectra remain motionally narrowed, and appear indistinguishable from the corresponding equilibrium spectra measured at a higher lattice temperature. Thus, the electron spin system achieves internal equilibrium prior to our measurement, justifying our use of Tspin . However, our measurement also shows that Tspin remains greater than T long after the rf pulse is turned o, which implies that the electron spin-lattice relaxation time s1s > 100 ls, for T < 0:5 K at m 13. The fact, that the measured Knight shifts [2] are essentially independent of sD after the ®rst 0.5 s, places an upper bound on s1s . Thus we ®nd 100 ls < s1s < 500 ms, for T < 0:5 K at m 13. While this value of s1s is at least a factor of 1000 longer than recent measurements of the transverse relaxation time s2 in bulk GaAs [10], it is consistent with a previous theoretical prediction [11] which assumed conditions rather similar to our experiment. Fig. 5. Top: T KS calibration curves based on the equilibrium KS (T) data for: (A) sample 40 W and (B) sample 10 W. Error bars for KS are shown. Bottom: The dependence of the eective spin temperature on the rf pulse length (H1 7 Gauss) for (C) sample 40 W and (D) sample 10 W. The intercept of the straight line ®t was constrained to be the lattice temperature: T 0:31 K (®lled circles 10 W and 40 W), T 0:42K (open circles 40 W), and T 0:44 K (open circles 10 W). The inset shows the top (along z0 ) and the front (along the rotation axis) views of the grooved sapphire platform holding a sample in a 5-turn rf coil. Adapted from [2].
and (B)), can be used to convert the measured Knight shift into an eective electron spin temperature Tspin ; Tspin rises linearly above the lattice temperature T as the duration of the rf pulse spulse increases, for H1 7G (Fig. 5(C) and (D)). The increase of Tspin drops o sharply with increasing lattice temperature and is not observable for T > 0:5 K. Furthermore, the apparent heating depends strongly on the rf ®eld strength and scales as H1g 2 < g < 5, which rules out nuclear spins and ohmic heating as the heat sources. Rather, these data provide evidence for a direct coupling between the rf pulse and the spins in the 2DES. The exact mechanism is not known, since the pulse frequency is well below the electron spin resonance at 74 GHz. Impurities in the bulk or edge states may be involved.
Acknowledgements We thank S.M. Girvin, D.A. Huse, A.H. MacDonald, N. Read, S. Sachdev, and S.L. Sondhi for helpful discussions. We also thank K.E. Gibble, R.L. Willett, and K.W. Zilm for experimental assistance. This work was supported by NSF CAREER Grant #DMR-9501925. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
P. Khandelwal et al., Phys. Rev. Lett. 81 (1998) 673. N.N. Kuzma et al., Science 281 (1998) 686. P. Khandelwal et al., Physica B, submitted. S.E. Barrett et al., Phys. Rev. Lett. 74 (1995) 5112. C.P. Slichter, Principles of Magnetic Resonance, 3rd ed., Springer, New York, 1990. D. Heiman, The Spectroscopy of Semiconductors, in: D.G. Seiler, C.L. Littler (Eds.), Ch. 1, Academic Press, San Diego, 1992. J. Winter, Magnetic Resonance in Metals, Oxford University Press, London, 1971. L.N. Pfeier et al., Appl. Phys. Lett. 61 (10) (1992) 1211. R.L. Willett, private communication. J. Kikkawa, D. Awschalom, Phys. Rev. Lett. 80 (1998) 4313. D.M. Frenkel, Phys. Rev. B 43 (1991) 14228.
Ultraslow electron spin dynamics in the fractional quantum Hall regime N.N. Kuzma
a,*
, P. Khandelwal a, S.E. Barrett a, L.N. Pfeier b, K.W. West a b
b
Department of Physics, Yale University, New Haven, CT 06520, USA Lucent Technologies Bell Laboratories, Murray Hill, NJ 07974, USA
Abstract Recent optically pumped nuclear magnetic resonance (OPNMR) measurements in two dierent electron-doped multiple quantum well samples near the fractional quantum Hall ground state m 13 are reviewed. Below 0.5 K, the spectra provide evidence that the spin-reversed charged excitations of this ground state are localized over the NMR time scale of 40 ls. By varying NMR pulse parameters, the electron spin temperature (as measured by the Knight shift) could be driven above the lattice temperature. For Tlattice < 0:5 K, these non-equilibrium measurements imply that the value of the electron spin-lattice relaxation time s1s lies between 100 ls and 500 ms at m 13. Ó 1998 Elsevier Science B.V. All rights reserved. Keywords: Fractional quantum Hall eect; NMR; GaAs
Optically pumped nuclear magnetic resonance (OPNMR) has been recently used to study two-dimensional electron systems (2DES) exhibiting the fractional quantum Hall eect (FQHE) [1,2]. We review evidence for ultraslow electron spin dynamics near the most studied FQHE ground state m 13 (where m nhc=Btot cos h) in two dierent multiple quantum well samples, `40 W' and `10 W' (n40 W 6:69 1010 cmÿ2 and n10 W 7:75 1010 cmÿ2 ), described elsewhere in this volume [3]. A series of 71 Ga OPNMR emission spectra are shown in Fig. 1 (solid lines). All spectra exhibit two peaks: one from nuclei in the Al0:1 Ga0:9 As barriers at zero frequency shift, and the other due to nuclei in the GaAs quantum wells [1,2,4], which is shifted to lower frequencies by the Fermi contact
* Corresponding author. Fax: 1 203 432 6175; e-mail: [email protected]
hyper®ne coupling to the spins of 2DES [5±7]. We de®ne the peak-to-peak splitting to be the Knight shift KS . The spectra at m 13 are well described by a simple model explained elsewhere (Fig. 1, dashed lines) [2,3]. The central assumption of this model is that the electron spins are delocalized along the well, such that hSz m; T i appears spatially homogeneous, when averaged over the NMR time scale ( 40 ls). In this limit, the delocalization of the low density 2DES produces a `motional narrowing' of the well peak. However, as Fig. 1(A) demonstrates, low-temperature measurements at m 0.267 show a crossover to more complicated line shapes. Although the spectra are in reasonable agreement with our model above 1 K, the width of the well peak increases dramatically as the temperature is lowered to T 0.45 K and then decreases upon further lowering to T 0.31 K. Fig. 1(B) shows that the extra broadening of the well resonance disappears
0921-4526/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 5 4 7 - X
122
N.N. Kuzma et al. / Physica B 256±258 (1998) 121±124
Fig. 1. 71 Ga OPNMR spectra (solid lines) of sample 10 W, taken in Btot 12 T at: (A) m 0:267 (h 0 ), 1.49 K P T P 0:31 K; and (B) 0:267 6 m 6 0:325 0 6 h 6 34:9 ), T 0.46 K. Frequency shift is relative to fo 155.93 MHz. The extreme full width athalf maximum of the well peak is shown; the dashed lines are ®ts referred to in text. Adapted from [2].
as the sample is tilted from h10 W 0 (m 0.267) to 36.8 (m 13), despite a 10% increase in the dipolar broadening. Furthermore, there is a striking correspondence between the decrease in the linewidth (Fig. 2, open symbols) and the increase in KS (®lled symbols) as m ! 13 : The observed broadening of the well line shape beyond the `motionally narrowed' limit implies that the time-averaged product of j/j2 P becomes spatially inhomogeneous in the plane of 2DES, where j/j2 is the local electron density, and P is the local electron spin polarization. Neither the formation of a Wigner crystal nor irregularities in the charge density along the well are consistent with all the aspects of our data [2]. In contrast, the
Fig. 2. The temperature dependence of the Knight shift (®lled symbols) and the linewidth (open symbols) in sample 10 W for ®lling factors: (A) 0.267 6 m 6 13 (adapted from [2]), and (B) m P 13. Lines are to guide the eye.
Knight shift and thus the total spin polarization drops monotonically below m 13, allowing the local spin polarization P ~ R0 ) to be spatially inhomogeneous (Fig. 3). This suggests that localization of spin-reversed regions is responsible for the behavior shown in Figs. 1 and 2. The time scale of this localization may be inferred from the simple model in which, after every jump time sJ , the local electron polarization at each nuclear site instantaneously assumes values of either 1 or ÿ0:15, with probability p or 1 ÿ p ), respectively [2]. Fig. 4 shows how the simulated spectra depend upon the value of sJ , for the case p 0.85, in qualitative agreement with the corresponding data in Fig. 1(A). When sJ is very fast, all nuclei see the same time-averaged local polarization, equal to the total polarization. At the other extreme (sJ ! 1), the motion is fro-
N.N. Kuzma et al. / Physica B 256±258 (1998) 121±124
Fig. 3. The evolution of the well linewidth and Knight shift as the temperature of sample 10 W is varied from 1.5 K (lower left corner) to 0.3 K (upper right corner), for ®lling factors below 13. Same data as in Fig. 2(A).
Fig. 4. Simulated OPNMR spectra using the model described in the text. KSint is set to 12 kHz for P 1. The barrier is suppressed (ab 0) for clarity. Adapted from [2].
zen out, and the single resonance splits into two lines. In the intermediate motion regime, the FWHM of the well peak goes through a maximum when sJ 40 ls.
123
Based upon this simple model, the peaks in the FWHM at Tloc 0:5 K (Fig. 2(A)) re¯ect the localization of reversed spins, such that they fail to cover the sample uniformly over 40 ls. The self-similar curves in Fig. 2(A) suggest that Tloc is not a strong function of the ®lling factor (or the density of reversed spins) for m < 13. Note also that below 0.5 K, the measured KS m < 13) increases toward KS m 13), as seen in the model. However, even down to T 0.3 K, the spectra do not appear to match the frozen limit of our simulation. Surprisingly, as m is varied below 13, the trends in the KS and FWHM data (Fig. 2(A) and 3) continue smoothly through m 27 without interruption. High-®eld magnetotransport measurements on samples taken from the same wafer as 10 W show much more structure, with well-developed minima in qxx at m 13 ; 25 ; 27, and 15 at T 0:3 K [8,9]. Additional measurements of the linewidth for m > 13 in sample 10 W are consistent with the above picture (Fig. 2(B)). The observed spectra contain more information than our simple simulation has revealed. A more sophisticated model might explain the non-trivial behavior shown in Fig. 3 and should probably include: (i) a detailed structure for the reversed spin regions present below m 13, (ii) the 2D dynamics of these reversed spins, and (iii) the eects of thermally excited spin ¯ips, since Tsat is not that much greater than Tloc . All of the above results were obtained using a weak rf pulse long after optical pumping to probe the equilibrium properties of the 2DES. The nonequilibrium properties of the 2DES can also be studied by varying these parameters, with a number of remarkable results at m 13. The rf tipping pulse for the NMR experiment is produced by a coil wrapped around the sample (Fig. 5(C), inset), which generates a linearly polar! ized (perpendicular to z0 ) magnetic ®eld of amplitude 2H1 at fo 155:93 MHz. The equilibrium value of KS (T) is independent of the rf pulse parameters for weak H1 (that is, H1 5 Gauss, spulse 20 ls). However, if stronger pulses are used at T < 0:5 K, the measured KS drops sharply below the equilibrium value, even though the lattice temperature is unaected by the pulses. The equilibrium measurements [1] of KS (T) (Fig. 5 (A)
124
N.N. Kuzma et al. / Physica B 256±258 (1998) 121±124
The non-equilibrium spectra remain motionally narrowed, and appear indistinguishable from the corresponding equilibrium spectra measured at a higher lattice temperature. Thus, the electron spin system achieves internal equilibrium prior to our measurement, justifying our use of Tspin . However, our measurement also shows that Tspin remains greater than T long after the rf pulse is turned o, which implies that the electron spin-lattice relaxation time s1s > 100 ls, for T < 0:5 K at m 13. The fact, that the measured Knight shifts [2] are essentially independent of sD after the ®rst 0.5 s, places an upper bound on s1s . Thus we ®nd 100 ls < s1s < 500 ms, for T < 0:5 K at m 13. While this value of s1s is at least a factor of 1000 longer than recent measurements of the transverse relaxation time s2 in bulk GaAs [10], it is consistent with a previous theoretical prediction [11] which assumed conditions rather similar to our experiment. Fig. 5. Top: T KS calibration curves based on the equilibrium KS (T) data for: (A) sample 40 W and (B) sample 10 W. Error bars for KS are shown. Bottom: The dependence of the eective spin temperature on the rf pulse length (H1 7 Gauss) for (C) sample 40 W and (D) sample 10 W. The intercept of the straight line ®t was constrained to be the lattice temperature: T 0:31 K (®lled circles 10 W and 40 W), T 0:42K (open circles 40 W), and T 0:44 K (open circles 10 W). The inset shows the top (along z0 ) and the front (along the rotation axis) views of the grooved sapphire platform holding a sample in a 5-turn rf coil. Adapted from [2].
and (B)), can be used to convert the measured Knight shift into an eective electron spin temperature Tspin ; Tspin rises linearly above the lattice temperature T as the duration of the rf pulse spulse increases, for H1 7G (Fig. 5(C) and (D)). The increase of Tspin drops o sharply with increasing lattice temperature and is not observable for T > 0:5 K. Furthermore, the apparent heating depends strongly on the rf ®eld strength and scales as H1g 2 < g < 5, which rules out nuclear spins and ohmic heating as the heat sources. Rather, these data provide evidence for a direct coupling between the rf pulse and the spins in the 2DES. The exact mechanism is not known, since the pulse frequency is well below the electron spin resonance at 74 GHz. Impurities in the bulk or edge states may be involved.
Acknowledgements We thank S.M. Girvin, D.A. Huse, A.H. MacDonald, N. Read, S. Sachdev, and S.L. Sondhi for helpful discussions. We also thank K.E. Gibble, R.L. Willett, and K.W. Zilm for experimental assistance. This work was supported by NSF CAREER Grant #DMR-9501925. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
P. Khandelwal et al., Phys. Rev. Lett. 81 (1998) 673. N.N. Kuzma et al., Science 281 (1998) 686. P. Khandelwal et al., Physica B, submitted. S.E. Barrett et al., Phys. Rev. Lett. 74 (1995) 5112. C.P. Slichter, Principles of Magnetic Resonance, 3rd ed., Springer, New York, 1990. D. Heiman, The Spectroscopy of Semiconductors, in: D.G. Seiler, C.L. Littler (Eds.), Ch. 1, Academic Press, San Diego, 1992. J. Winter, Magnetic Resonance in Metals, Oxford University Press, London, 1971. L.N. Pfeier et al., Appl. Phys. Lett. 61 (10) (1992) 1211. R.L. Willett, private communication. J. Kikkawa, D. Awschalom, Phys. Rev. Lett. 80 (1998) 4313. D.M. Frenkel, Phys. Rev. B 43 (1991) 14228.