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Electron spin resonance and nuclear spin pumping in 2DEG quantum Hall system
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Physica E 21 (2004) 928 – 932 www.elsevier.com/locate/physe
Electron spin resonance and nuclear spin pumping in 2DEG quantum Hall system S. Teraokaa;∗ , A. Numatab , S. Amahab , K. Onob , S. Taruchaa; b a ERATO
Mesoscopic Correlation Project, JST, Morinosato-Wakamiya, Atsugi, Kanagawa, Japan of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, Japan
b Department
Abstract We prepare a microwave electron spin resource (ESR) cavity for detecting a response from a 2DEG in an n-AlGaAs/GaAs. The response is obtained as a change in the longitudinal resistance (3xx ) in = 3 quantum Hall region, particularly as a peak in 3xx for resonance. We use the data of ESR to evaluate the g-factor and the lower bound for dephasing time. The resonance magnetic 5eld su6ers from nuclear spin e6ects via the hyper5ne coupling, resulting in the ESR peak shift. We 5nd the ESR peak shift or Overhauser shift decays with two time constants, suggesting the existence of two di6erent origins for the relaxation. ? 2003 Elsevier B.V. All rights reserved. PACS: 76.30.−v; 73.43.−f Keywords: Electron spin resonance; 2DEG; Overhauser shift
1. Introduction Electron spin in semiconductor quantum dots is one of the candidates for implementing quantum bits in quantum computation [1,2]. The fundamental quantum gate operation such as XOR operation can be achieved by temporarily coupling two spins in two dots and by rotating a single spin in a dot. Focused on the spin manipulation, electron spin resonance (ESR) is a crucial technique for accessing individual dots because ESR microwave frequencies are sensitive to the local environment. In addition, ESR provides ∗ Corresponding author. Tarucha-Lab, Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan. Tel.: +81-3-5841-4162; fax: +81-3-5841-4162. E-mail address: [email protected] (S. Teraoka).
information about the dephasing and g-factor for electron spins, both of which are important parameters for the spin manipulation. However, there are only few works on the ESR technique for con5ned electronic systems in semiconductor. Dobers et al. [3] has revealed that the longitudinal resistance xx is a6ected by ESR using an AlGaAs/GaAs/AlGaAs quantum well and an AlGaAs/GaAs heterostructure. An ESR peak is observed around the odd 5lling factor , i.e. when the Fermi level is located between Zeeman subbands formed by the splitting of each Landau level. In this work we prepare a microwave ESR cavity for detecting a response from a 2DEG in an n-AlGaAs/GaAs. We observe an ESR peak in the =3 quantum Hall regime and evaluate the g-factor and lower bound for T2 . A complex behavior of relaxation time dependence of Overhauser shift is observed.
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.11.152
S. Teraoka et al. / Physica E 21 (2004) 928 – 932
929
2. Experiment The ESR response is measured as a change in the longitudinal resistance, 3xx , in the = 3 quantum Hall regime between on and o6 the microwave radiation. A 100 nm × 400 nm Hall bar sample was made by mesa-etching in n-AlGaAs/GaAs substrate, which has a 60 nm deep 2DEG. The carrier density and mobility of the 2DEG are 8:63 × 105 cm2 =V s and 2:93 × 1011 cm−2 , respectively. A short circuited waveguide is used to make a rectangular cavity (TE10n -mode). The sample is located at one short-end of the waveguide so that the magnetic 5eld of standing microwave is applied to the sample for the ESR measurement. The cavity is installed in a 1:5 K cryostat. Magnetoresistivity of the sample is measured using a four-probe method. For detecting a small change in the longitudinal resistivity due to ESR we initially use a double lock-in detection technique with 967 Hz AC source–drain driving current and 7:5 Hz amplitude modulation to microwave source power. Later we mainly use single lock-in detection with dc source–drain driving current and with 670 Hz amplitude modulation to microwave source power. External magnetic 5eld is swept under 5xed microwave frequencies (f ).
Fig. 1. Longitudinal resistance (xx ) vs. magnetic 5eld without microwave radiation (lower curve) and its change (3xx ) with microwave radiation (upper curve).
3. Results and discussion Fig. 1 shows Schbnikov de Haas oscillations in longitudinal resistance (xx ) of the Hall bar sample without microwave radiation and its change (3xx ) due to 19:137 GHz microwave radiation. A small sharp peak (arrow) is an ESR peak and the other large oscillations in the background are all due to non-resonative heating through the microwave radiation [3]. From DC xx measurement with and without the microwave radiation we determine the sign of 3xx and thus conclude that the magnetoresistivity does increase by ESR absorption. There can be two reasons for the positive 3xx , though we cannot distinguish them in the present experiment. The 5rst is spin Oip excitation of electrons in the = 3 edge channel, followed by scattering of the Oipped electrons to an edge channel going in the opposite direction or a bulk channel.
Fig. 2. ESR peaks observed in the 3xx curves for various microwave frequencies.
The second is spin-Oip excitation of electrons in the bulk channel, followed by back-scattering between bulk-edge gives rise to an increasing resistivity. Fig. 2 shows the 3xx curve including an ESR peak for various microwave frequencies. The ESR peaks are only observed in = 3 quantum Hall region (see Fig. 3). Note the ESR response su6ers from the nuclear spin e6ects (Overhauser e6ect) [4,5] because nuclear spins are dynamically polarized via the hyper5ne coupling (DNP). Care is taken to avoid unexpected
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S. Teraoka et al. / Physica E 21 (2004) 928 – 932
Fig. 3. Frequency vs. 5eld diagram for observed ESR peak.
inOuence of DNP. Each data is taken by sweeping up of a magnetic 5eld B with microwave radiation after sweeping down the magnetic 5eld to the initial value without microwave. The microwave source power is reduced to the lowest level but enough for detecting the ESR peak. Nevertheless the ESR peak shape is asymmetric, showing a sharp cuto6 at high B 5eld side due to DNP. The inOuence of DNP is later discussed in detail. Frequency dependencies of peak positions in Fig. 2 are plotted in Fig. 3. Here we show the SdH oscillations as a guide to the 5lling factor. The frequency vs. B 5eld for the ESR peak is well 5tted by an parabolic function (thin line in Fig. 3). The g-factor de5ned by hfres = −g B B is then magnetic 5eld dependent and well reproduced by g = −0:387 + 0:0175B. This value is similar to that given in Ref. [3]. We now evaluate the dephasing time for spin ensemble T2 from the ESR peak in Fig. 3. Bloch equation gives a Lorentz curve for an ESR line vs. frequency. FWHM 3f is equivalent to 2=T2 . 3f is given 3f=0:387 B 3B=h from the equivalent resonance full width 3B of the ESR peak by neglecting B dependent term of the g-factor. Then we obtain T2 ∼7 ns using a mean value of 3B = 0:0085 T. Measured peak can be broadened by spatial inhomogeneity over a large sample and the DNP e6ect. So we consider that the obtained value T2 is the lower bound. Note that the observed peak is asymmetric due to DNP. This e6ect is neglected in the evaluation of 3B.
Fig. 4. 3xx for upward sweep of B 5eld (solid line) and that for the downward sweep the (dotted line).
Next we will discuss nuclear spin relaxation. When ESR takes place, the neighboring nuclear spins are polarized via the hyper5ne coupling. Then the e6ective nuclear magnetic 5eld shifts the ESR condition to the low external magnetic 5eld. However, once the external B 5eld passes by the ESR peak, the e6ective B 5eld increases progressively to turn o6 the ESR condition. This gives rise to an asymmetric peak when the magnetic 5eld is swept up. On the other hand, when the magnetic 5eld is swept down across the ESR peak, the ESR condition can be sustained to keep the maximum if the magnetic 5eld is swept slowly in the presence of large microwave source power. If the external B 5eld is too low, the ESR condition can no longer be sustained due to the relaxation of nuclear spins. Fig. 4 shows such an example. We stop the magnetic 5eld at such a low magnetic 5eld and wait for some time with microwave turned o6. During this waiting at low magnetic 5eld, the nuclear spins relax to the equilibrium. We sweep up the magnetic 5eld again with microwave turned on and observe a shift of the ESR to the low B 5eld (Overhauser shift) (Fig. 5). For the downward sweep of B 5eld we choose a relatively large microwave source power of 10 dB m and a suf5cient slow sweep rate for magnetic 5eld so that the nuclear spin pumping rate is large enough for locking the ESR condition. The larger shift is observed for the shorter wait time, reOecting the smaller relaxation
S. Teraoka et al. / Physica E 21 (2004) 928 – 932
Fig. 5. An example of Overhauser shifts of the ESR peak at 1:5 K. After downward sweep (red line) nuclear spins are polarized. By waiting at Bwait (=4:1 T) with RF turned o6 for various times ESR peaks go back to its equilibrium position reOecting nuclear spin 1 depolarization. Note that the downward spectrum is shown in 10 scale.
of the nuclear spins. The relaxation actually proceeds during the sweep time. So the wait time indicated in Fig. 5 include the sweep time needed for reaching the ESR peak. For the upward sweep the microwave source power is reduced to 0 dBm to minimize an additional Overhauser shift, and the sweep rate is chosen to be as high as possible to minimize the inOuence of 5eld dependency of relaxation [5]. Fig. 6 shows the ESR peak shift in ln(3B) as a function of wait time measured for f = 19:137 GHz. Interestingly, we 5nd that the relaxation is only 5tted by a bi-exponential function with two time constants T1A and T1B . This suggests the existence of two di6erent origins for the relaxation. One possible explanation is given by considering di6erent isotopes having di6erent relaxation times. Previously Berg et al. [5] obtain single exponential decay for a AlGaAs/GaAs/AlGaAs quantum well. The discrepancy from their result is not clear yet. We have measured Bwait dependence of T1A and T1B and plot the inversed relaxation time constants, 1=T1A and 1=T1B in Fig. 7. Both of 1=T1A and 1=T1B show the same tendency as given for the single exponential decay in Ref. [5], reOecting our relaxation time constants are related to nuclear spin–nuclear spin interaction via electron spin.
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Fig. 6. The logarithmic peak shifts are plotted as a function of waiting time. The results are only 5tted with bi-exponential function.
Fig. 7. Dependencies of di6erent time constants T1A and T1B on the waiting 5eld Bwait (lower panel). In the upper panel SdH oscillations of xx is also measured for the same sample shown for comparison.
Berg et al. [5] explained the magnetic 5eld dependence of the relaxation time based on the Slichter’s formula only using the hyper5ne interaction A for 75 As. The relaxation time ratio for both of 69 Ga and 71 Ga to 75 As is 4 to 5, reOecting di6erence in the hyper5ne interaction A. We 5nd in Fig. 7 that the ratio between T1A and T1B is in this range for any Bwait . Therefore, we can probably assign the short relaxation
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S. Teraoka et al. / Physica E 21 (2004) 928 – 932
time (T1A ) to 75 As and the long relaxation time (T1B ) to 69 Ga and 71 Ga. The relaxation times between 69 Ga and 71 Ga would be diQcult to separate from each other since the di6erence in A between them is small.
Acknowledgements Financial support from Grant-in-Aid for Scienti5c Research(A) and CREST-JST are gratefully acknowledged.
4. Summary We have prepared a microwave ESR cavity to study a response from a 2DEG in an n-AlGaAs/GaAs. The response has been detected as a peak in the change of longitudinal resistance in the = 3 quantum Hall regime. From the ESR peak we have evaluated the dephasing time of ∼7 ns as a lower bound. We have also observed an Overhauser shift of the ESR peak with two di6erent decay time constants.
References [1] G. Burkard, D. Loss, D.P. Divicenzo, Phys. Rev. B 59 (1999) 2070. [2] H.A. Engel, Daniel Loss, Phys. Rev. Lett. 86 (2001) 4648. [3] M. Dobers, K.V. Klitzing, G. Weimann, Phys. Rev. B 38 (1988) 5453. [4] C. Hillman, H.W. Jiang, Phys. Rev. B 64 (2001) 201308. [5] A. Berg, M. Dobers, R.R. Gerhardts, K.V. Klitzing, Phys. Rev. Lett. 64 (1990) 2563.
Physica E 21 (2004) 928 – 932 www.elsevier.com/locate/physe
Electron spin resonance and nuclear spin pumping in 2DEG quantum Hall system S. Teraokaa;∗ , A. Numatab , S. Amahab , K. Onob , S. Taruchaa; b a ERATO
Mesoscopic Correlation Project, JST, Morinosato-Wakamiya, Atsugi, Kanagawa, Japan of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, Japan
b Department
Abstract We prepare a microwave electron spin resource (ESR) cavity for detecting a response from a 2DEG in an n-AlGaAs/GaAs. The response is obtained as a change in the longitudinal resistance (3xx ) in = 3 quantum Hall region, particularly as a peak in 3xx for resonance. We use the data of ESR to evaluate the g-factor and the lower bound for dephasing time. The resonance magnetic 5eld su6ers from nuclear spin e6ects via the hyper5ne coupling, resulting in the ESR peak shift. We 5nd the ESR peak shift or Overhauser shift decays with two time constants, suggesting the existence of two di6erent origins for the relaxation. ? 2003 Elsevier B.V. All rights reserved. PACS: 76.30.−v; 73.43.−f Keywords: Electron spin resonance; 2DEG; Overhauser shift
1. Introduction Electron spin in semiconductor quantum dots is one of the candidates for implementing quantum bits in quantum computation [1,2]. The fundamental quantum gate operation such as XOR operation can be achieved by temporarily coupling two spins in two dots and by rotating a single spin in a dot. Focused on the spin manipulation, electron spin resonance (ESR) is a crucial technique for accessing individual dots because ESR microwave frequencies are sensitive to the local environment. In addition, ESR provides ∗ Corresponding author. Tarucha-Lab, Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan. Tel.: +81-3-5841-4162; fax: +81-3-5841-4162. E-mail address: [email protected] (S. Teraoka).
information about the dephasing and g-factor for electron spins, both of which are important parameters for the spin manipulation. However, there are only few works on the ESR technique for con5ned electronic systems in semiconductor. Dobers et al. [3] has revealed that the longitudinal resistance xx is a6ected by ESR using an AlGaAs/GaAs/AlGaAs quantum well and an AlGaAs/GaAs heterostructure. An ESR peak is observed around the odd 5lling factor , i.e. when the Fermi level is located between Zeeman subbands formed by the splitting of each Landau level. In this work we prepare a microwave ESR cavity for detecting a response from a 2DEG in an n-AlGaAs/GaAs. We observe an ESR peak in the =3 quantum Hall regime and evaluate the g-factor and lower bound for T2 . A complex behavior of relaxation time dependence of Overhauser shift is observed.
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.11.152
S. Teraoka et al. / Physica E 21 (2004) 928 – 932
929
2. Experiment The ESR response is measured as a change in the longitudinal resistance, 3xx , in the = 3 quantum Hall regime between on and o6 the microwave radiation. A 100 nm × 400 nm Hall bar sample was made by mesa-etching in n-AlGaAs/GaAs substrate, which has a 60 nm deep 2DEG. The carrier density and mobility of the 2DEG are 8:63 × 105 cm2 =V s and 2:93 × 1011 cm−2 , respectively. A short circuited waveguide is used to make a rectangular cavity (TE10n -mode). The sample is located at one short-end of the waveguide so that the magnetic 5eld of standing microwave is applied to the sample for the ESR measurement. The cavity is installed in a 1:5 K cryostat. Magnetoresistivity of the sample is measured using a four-probe method. For detecting a small change in the longitudinal resistivity due to ESR we initially use a double lock-in detection technique with 967 Hz AC source–drain driving current and 7:5 Hz amplitude modulation to microwave source power. Later we mainly use single lock-in detection with dc source–drain driving current and with 670 Hz amplitude modulation to microwave source power. External magnetic 5eld is swept under 5xed microwave frequencies (f ).
Fig. 1. Longitudinal resistance (xx ) vs. magnetic 5eld without microwave radiation (lower curve) and its change (3xx ) with microwave radiation (upper curve).
3. Results and discussion Fig. 1 shows Schbnikov de Haas oscillations in longitudinal resistance (xx ) of the Hall bar sample without microwave radiation and its change (3xx ) due to 19:137 GHz microwave radiation. A small sharp peak (arrow) is an ESR peak and the other large oscillations in the background are all due to non-resonative heating through the microwave radiation [3]. From DC xx measurement with and without the microwave radiation we determine the sign of 3xx and thus conclude that the magnetoresistivity does increase by ESR absorption. There can be two reasons for the positive 3xx , though we cannot distinguish them in the present experiment. The 5rst is spin Oip excitation of electrons in the = 3 edge channel, followed by scattering of the Oipped electrons to an edge channel going in the opposite direction or a bulk channel.
Fig. 2. ESR peaks observed in the 3xx curves for various microwave frequencies.
The second is spin-Oip excitation of electrons in the bulk channel, followed by back-scattering between bulk-edge gives rise to an increasing resistivity. Fig. 2 shows the 3xx curve including an ESR peak for various microwave frequencies. The ESR peaks are only observed in = 3 quantum Hall region (see Fig. 3). Note the ESR response su6ers from the nuclear spin e6ects (Overhauser e6ect) [4,5] because nuclear spins are dynamically polarized via the hyper5ne coupling (DNP). Care is taken to avoid unexpected
930
S. Teraoka et al. / Physica E 21 (2004) 928 – 932
Fig. 3. Frequency vs. 5eld diagram for observed ESR peak.
inOuence of DNP. Each data is taken by sweeping up of a magnetic 5eld B with microwave radiation after sweeping down the magnetic 5eld to the initial value without microwave. The microwave source power is reduced to the lowest level but enough for detecting the ESR peak. Nevertheless the ESR peak shape is asymmetric, showing a sharp cuto6 at high B 5eld side due to DNP. The inOuence of DNP is later discussed in detail. Frequency dependencies of peak positions in Fig. 2 are plotted in Fig. 3. Here we show the SdH oscillations as a guide to the 5lling factor. The frequency vs. B 5eld for the ESR peak is well 5tted by an parabolic function (thin line in Fig. 3). The g-factor de5ned by hfres = −g B B is then magnetic 5eld dependent and well reproduced by g = −0:387 + 0:0175B. This value is similar to that given in Ref. [3]. We now evaluate the dephasing time for spin ensemble T2 from the ESR peak in Fig. 3. Bloch equation gives a Lorentz curve for an ESR line vs. frequency. FWHM 3f is equivalent to 2=T2 . 3f is given 3f=0:387 B 3B=h from the equivalent resonance full width 3B of the ESR peak by neglecting B dependent term of the g-factor. Then we obtain T2 ∼7 ns using a mean value of 3B = 0:0085 T. Measured peak can be broadened by spatial inhomogeneity over a large sample and the DNP e6ect. So we consider that the obtained value T2 is the lower bound. Note that the observed peak is asymmetric due to DNP. This e6ect is neglected in the evaluation of 3B.
Fig. 4. 3xx for upward sweep of B 5eld (solid line) and that for the downward sweep the (dotted line).
Next we will discuss nuclear spin relaxation. When ESR takes place, the neighboring nuclear spins are polarized via the hyper5ne coupling. Then the e6ective nuclear magnetic 5eld shifts the ESR condition to the low external magnetic 5eld. However, once the external B 5eld passes by the ESR peak, the e6ective B 5eld increases progressively to turn o6 the ESR condition. This gives rise to an asymmetric peak when the magnetic 5eld is swept up. On the other hand, when the magnetic 5eld is swept down across the ESR peak, the ESR condition can be sustained to keep the maximum if the magnetic 5eld is swept slowly in the presence of large microwave source power. If the external B 5eld is too low, the ESR condition can no longer be sustained due to the relaxation of nuclear spins. Fig. 4 shows such an example. We stop the magnetic 5eld at such a low magnetic 5eld and wait for some time with microwave turned o6. During this waiting at low magnetic 5eld, the nuclear spins relax to the equilibrium. We sweep up the magnetic 5eld again with microwave turned on and observe a shift of the ESR to the low B 5eld (Overhauser shift) (Fig. 5). For the downward sweep of B 5eld we choose a relatively large microwave source power of 10 dB m and a suf5cient slow sweep rate for magnetic 5eld so that the nuclear spin pumping rate is large enough for locking the ESR condition. The larger shift is observed for the shorter wait time, reOecting the smaller relaxation
S. Teraoka et al. / Physica E 21 (2004) 928 – 932
Fig. 5. An example of Overhauser shifts of the ESR peak at 1:5 K. After downward sweep (red line) nuclear spins are polarized. By waiting at Bwait (=4:1 T) with RF turned o6 for various times ESR peaks go back to its equilibrium position reOecting nuclear spin 1 depolarization. Note that the downward spectrum is shown in 10 scale.
of the nuclear spins. The relaxation actually proceeds during the sweep time. So the wait time indicated in Fig. 5 include the sweep time needed for reaching the ESR peak. For the upward sweep the microwave source power is reduced to 0 dBm to minimize an additional Overhauser shift, and the sweep rate is chosen to be as high as possible to minimize the inOuence of 5eld dependency of relaxation [5]. Fig. 6 shows the ESR peak shift in ln(3B) as a function of wait time measured for f = 19:137 GHz. Interestingly, we 5nd that the relaxation is only 5tted by a bi-exponential function with two time constants T1A and T1B . This suggests the existence of two di6erent origins for the relaxation. One possible explanation is given by considering di6erent isotopes having di6erent relaxation times. Previously Berg et al. [5] obtain single exponential decay for a AlGaAs/GaAs/AlGaAs quantum well. The discrepancy from their result is not clear yet. We have measured Bwait dependence of T1A and T1B and plot the inversed relaxation time constants, 1=T1A and 1=T1B in Fig. 7. Both of 1=T1A and 1=T1B show the same tendency as given for the single exponential decay in Ref. [5], reOecting our relaxation time constants are related to nuclear spin–nuclear spin interaction via electron spin.
931
Fig. 6. The logarithmic peak shifts are plotted as a function of waiting time. The results are only 5tted with bi-exponential function.
Fig. 7. Dependencies of di6erent time constants T1A and T1B on the waiting 5eld Bwait (lower panel). In the upper panel SdH oscillations of xx is also measured for the same sample shown for comparison.
Berg et al. [5] explained the magnetic 5eld dependence of the relaxation time based on the Slichter’s formula only using the hyper5ne interaction A for 75 As. The relaxation time ratio for both of 69 Ga and 71 Ga to 75 As is 4 to 5, reOecting di6erence in the hyper5ne interaction A. We 5nd in Fig. 7 that the ratio between T1A and T1B is in this range for any Bwait . Therefore, we can probably assign the short relaxation
932
S. Teraoka et al. / Physica E 21 (2004) 928 – 932
time (T1A ) to 75 As and the long relaxation time (T1B ) to 69 Ga and 71 Ga. The relaxation times between 69 Ga and 71 Ga would be diQcult to separate from each other since the di6erence in A between them is small.
Acknowledgements Financial support from Grant-in-Aid for Scienti5c Research(A) and CREST-JST are gratefully acknowledged.
4. Summary We have prepared a microwave ESR cavity to study a response from a 2DEG in an n-AlGaAs/GaAs. The response has been detected as a peak in the change of longitudinal resistance in the = 3 quantum Hall regime. From the ESR peak we have evaluated the dephasing time of ∼7 ns as a lower bound. We have also observed an Overhauser shift of the ESR peak with two di6erent decay time constants.
References [1] G. Burkard, D. Loss, D.P. Divicenzo, Phys. Rev. B 59 (1999) 2070. [2] H.A. Engel, Daniel Loss, Phys. Rev. Lett. 86 (2001) 4648. [3] M. Dobers, K.V. Klitzing, G. Weimann, Phys. Rev. B 38 (1988) 5453. [4] C. Hillman, H.W. Jiang, Phys. Rev. B 64 (2001) 201308. [5] A. Berg, M. Dobers, R.R. Gerhardts, K.V. Klitzing, Phys. Rev. Lett. 64 (1990) 2563.