LETTER pubs.acs.org/NanoLett
Resistively Detected Nuclear Magnetic Resonance in n- and p-Type GaAs Quantum Point Contacts Z. K. Keane,*,† M. C. Godfrey,† J. C. H. Chen,† S. Fricke,† O. Klochan,† A. M. Burke,† A. P. Micolich,† H. E. Beere,‡ D. A. Ritchie,‡ K. V. Trunov,§ D. Reuter,§ A. D. Wieck,§ and A. R. Hamilton† †
School of Physics, University of New South Wales, Sydney, NSW 2052, Australia Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE, U.K. § Lehrstuhl f€ur Angewandte Festk€orperphysik, Ruhr-Universit€at Bochum, 44801 Bochum, Germany ‡
ABSTRACT: We present resistively detected NMR measurements in induced and modulation-doped electron quantum point contacts, as well as induced hole quantum point contacts. While the magnitude of the resistance change and associated NMR peaks in n-type devices is in line with other recent measurements using this technique, the effect in p-type devices is too small to measure. This suggests that the hyperfine coupling between holes and nuclei in this type of device is much smaller than the electron hyperfine coupling, which could have implications in quantum information processing. KEYWORDS: Quantum Hall effect, nuclear magnetic resonance, spin-dependent transport, quantum point contacts
T
he use of spins in solid-state devices to convey information has drawn significant attention in recent years, particularly for purposes of quantum information processing. In principle, it is possible to store, manipulate, and extract quantum information in the form of a carrier spin localized to a quantum dot, but the technical challenges involved in so doing are considerable. For example, in n-type GaAs-based quantum dots, perhaps the most well-developed low-dimensional semiconductor system, the hyperfine interaction keeps the electron spin lifetime very short, making it difficult to accomplish any useful tasks before the spin information is lost. One way to skirt this limitation is to base spintronic devices on 28Si or another spinless lattice, such as 12C or certain isotopes of germanium; however, single-electron transistor technology is significantly more advanced in GaAs.1 Another possibility is to look to holes in GaAs devices. In general, the spinorbit interaction severely limits hole spin lifetimes in GaAs, but if the orbital motion is quantized (as would be the case in a quantum dot), spin lifetimes could potentially be much longer.2,3 Furthermore, the p-wave symmetry of holes eliminates the contact hyperfine interaction, which is predicted to account for approximately 90% of the electronnucleus hyperfine coupling.4 Optical measurements in self-assembled InGaAs5 and InP/GaInP6 quantum dots indicate that the hole hyperfine coupling in these systems is on the order of 10% of the electron hyperfine coupling. However, as these are highly strained systems in materials related to, but different from GaAs, the relevance of these results to low-dimensional GaAs devices is unclear. It would be useful, then, to directly quantify the relative strengths of the hyperfine interactions for electrons and holes in GaAs. One potential avenue for exploring the hyperfine interaction in low-dimensional semiconductor devices is through the recently developed technique of dynamic nuclear polarization and r 2011 American Chemical Society
resistively detected NMR (Figure 1). In this technique, an external magnetic field is applied to bring a two-dimensional carrier gas into the quantum Hall regime. A quantum point contact can then be tuned to transmit only a single spin population, reflecting the opposite spin. At high current densities in the contact, the quantum Hall effect breaks down, leading some carriers to move between Landau levels. 712 Since the inter-Landau level tunneling consistently entails a transition from spin-up to spin-down, the tunneling electrons exert a net torque on the nuclear spins. This tends to polarize the nearby nuclei.13,14 This polarization can be detected by a change in the four-terminal resistance of the QPC. Finally, if an radio frequency (rf) magnetic field is at the Larmor frequency of one of the substrate’s constituent nuclei, the nuclear spins rotate, and eventually randomize, with an accompanying resistance change. Typically, the overall resistance change is approximately 1% of the overall four-terminal resistance in electron devices;15,16 the relative magnitude of this effect in holes could give a measure of the strength of their hyperfine coupling relative to that of electrons. The detailed mechanism for the generation and resistive detection of dynamic nuclear polarization in low-dimensional devices12 is beyond the scope of this article, but a brief overview of the physics is helpful. Both of these effects rely on the hyperfine coupling between nuclei and charge carriers. Conduction electrons, as s-waves, have no orbital angular momentum; hence, their hyperfine interaction is entirely due to contact between the electron and nuclear wave functions, and is Received: April 12, 2011 Revised: June 28, 2011 Published: June 29, 2011 3147
dx.doi.org/10.1021/nl201211d | Nano Lett. 2011, 11, 3147–3150
Nano Letters
LETTER
otherwise similar to that of electrons4 H LH ¼ ¼
Mp ½2Ix Sx 2Iy Sy þ Iz Sz 3 Mp ½ðI þ S þ I Sþ Þ þ Iz Sz 3
ð2Þ
Again we see the flip-flop term which should generate a nuclear polarization, and the Overhauser term which enables its detection. Heavy holes are another matter: since they have spin 3/2, the exchange of spin angular momentum between a heavy hole and a nucleus would entail a three-photon transition. Accordingly, the heavy-hole hyperfine Hamiltonian omits the flip-flop term but retains the Overhauser term3,4 H HH ¼ Mp Sz Iz
Figure 1. Schematic representation of the dynamic nuclear polarization process. Red and blue lines represent spin-up and spin-down currents, respectively; green circles represent host nuclei. (a) At bulk filling factor 2 in the quantum Hall regime, the carriers are organized into dissipationless edge states, split into spin-polarized subbands by the Zeeman energy gμBB. The side gates are tuned to transmit only a single spin population. (b) At high current densities, the quantum Hall effect breaks down, leading to carrier spin “flips” and nuclear spin “flops”, and building up a nuclear spin polarization. This polarization is detected via the fourterminal resistance. (c) After some time, the system reaches a steady polarization state; the nuclei may now be randomized by an rf magnetic field applied at the Larmor frequency with an associated resistance change.
written17 1 H electrons ¼ Ms I 3 S ¼ Ms ðIx Sx þ Iy Sy þ Iz Sz Þ 2 1 ¼ Ms ðIþ S þ I Sþ Þ þ As Iz Sz 2
ð1Þ
Here, the first term is a “flip-flop” term which describes an electron and nucleus exchanging one quantum of spin angular momentum. It is this term which is responsible for the buildup of nuclear polarization in n-type devices. The second term is an effective Zeeman term, which describes the effective magnetic field due to a net nuclear polarization; this term is responsible for the Overhauser shift, which in turn generates the resistance changes associated with dynamic nuclear polarization. The situation becomes more complicated when holes enter the picture. As p-waves, holes have no contact hyperfine interaction; instead, their hyperfine Hamiltonian is due to dipole dipole interactions between the holes and the nuclei, and is therefore about 10 times weaker than the electrons’ contact hyperfine interaction. For light holes, the Hamiltonian is
ð3Þ
In essence, while only light hole states would be available to generate a nuclear polarization, both light and heavy holes should contribute to resistive detection. In this Letter, we describe measurements of dynamic nuclear polarization and associated nuclear magnetic resonances in modulation-doped and induced electron QPCs in the quantum Hall regime. We find conductance changes consistent with previously reported results on dynamic nuclear polarization, as well as resonant peaks at frequencies appropriate to the GaAs substrate. This effect is robust to small changes in the applied magnetic field and gate voltages and is present at a wide range of carrier densities. We report similar measurements in induced hole QPCs, where we find no evidence for dynamic nuclear polarization or NMR, despite noise levels more than 50 times lower than the signal in the electron measurement. This suggests that the hyperfine coupling for holes in this system may be weaker than the expected 10% of the electron hyperfine, which in turn suggests that holes in GaAs/AlGaAs heterostructures may be a favorable system for quantum information processing. Materials and Methods. All devices were fabricated on highquality GaAs/AlGaAs single heterojunctions by photolithography to define a Hall bar, followed by electron-beam lithography to define quantum point contacts approximately 500 nm wide by 500 nm long. Modulation-doped electron devices were fabricated on wafer A2899, with the 2DEG 90 nm below the surface, with a carrier density n = 1.39 1011/cm2 and mobility μ = 4.5 105 cm2/(V s). Induced hole devices were fabricated18,19 on wafer B13180 with the 2DHG 190 nm below the surface, with a peak mobility μ = 3 105 cm2/(V s) at density n = 1.5 1011/cm2. Induced electron devices and additional induced hole devices were fabricated on wafer B13520, with the 2D layer 317 nm below the surface. This wafer reached an electron mobility of 4 106 cm2/(V s) at density n = 1.0 1011/cm2 and a hole mobility of 3.5 105 cm2/(V s) at the same density. The fabrication process for these devices will be described in detail elsewhere.20 All measurements were carried out in a 3 He/4He dilution refrigerator with a base temperature of 30 mK; dc magnetic fields were applied perpendicular to the plane of the device to bring the bulk filling factor to approximately 2, and rf magnetic fields were applied via a two-turn coil wound around the device. All resistance measurements were carried out via standard lock-in techniques. A diagonal four-terminal resistance measurement was used to eliminate the contact and Hall resistances from the measurement. Results and Discussion. All measurements were carried out at bulk filling factors near 2, so as to have one spin-up channel and 3148
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Figure 2. (a) Typical pinchoff trace for an electron device. Note the presence of hysteresis at conductances near e2/h (highlighted in green); this is consistent with a nuclear polarization process. (b) Typical pinchoff trace for a hole device. Hysteresis is apparent at all conductance values. Data for (b) are taken at a side gate voltage of 0.5 V, indicated by a vertical arrow.
Figure 4. NMR data for n- and p-type devices. Red traces are taken with increasing frequency; blue traces are taken with decreasing frequency. (a, c) Conductance as a function of rf frequency for an electron QPC at the 75 As and 69Ga resonances, respectively. (b, d) Equivalent data in a hole QPC. The NMR frequency in (b) is expected to be 12.68 MHz; in (d) the expected frequency is 17.77 MHz. Figure 3. Conductance as a function of time after the application of a 100 μV ac excitation. Scale bars indicate a 1% change in conductance: (a) n-type conductance traces showing characteristic decay on the time scale of a few minutes; (b) p-type conductance trace showing no change as a function of time.
one spin-down channel available for transport. Two- and fourterminal resistances were measured via standard lock-in techniques, typically with a 100 μV ac excitation. As part of the initial characterization, the side gate voltage was swept to determine an operating point at which a single spin state was transmitted; we would expect dynamic nuclear polarization, if present, to occur under these conditions. Figure 2 shows typical results of this sidegate sweep in both n- and p-type devices. Both types of device show broad conductance plateaus at νQPC = 1 (conductance values near e2/h). In addition, each device shows resonant structure at conductance below e2/h due to imperfections in the channel, which is not relevant to the DNP process we study. Of primary interest is the hysteresis evident near νQPC = 1. Previous reports have noted that hysteresis in this region may be associated with a dynamic nuclear polarization process. Notably, however, the hysteresis in the n-type device occurs only in the quantum Hall regime and at the edges of the ν = 1 plateau, while the p-type device shows a nearly constant hysteresis in terms of side gate voltage, irrespective of the dc magnetic field. This suggests that the hysteresis in the p-type device may be related to the behavior of the gates themselves rather than dynamic effects in the channel.21 A second signature of DNP in this type of device is timedependent conductance changes after the abrupt application of a current. This offers the possibility of testing the origin of the hysteresis in each device. Accordingly, once the initial characterization had been completed, the ac excitation was turned off for several minutes, the side gates were set to several operating points in the hysteretic region, and the ac excitation was reapplied. The conductance was monitored for time dependence. Figure 3 shows typical time dependences for electrons and holes in the first minutes after the ac excitation was reapplied. The
Figure 5. Resonant frequencies for each nucleus as a function of dc magnetic field. Measured slopes are 7.26 MHz/T for 75As, 10.13 MHz/T for 69Ga, and 12.89 MHz/T for 71Ga.
time-dependent conductance changes in the electron devices are of a magnitude and time scale consistent with previous reports; furthermore, the slow time evolution is consistent with a nuclear process. This time evolution of conductance consistently appears in n-type devices and is robust to changes of 10% in the dc magnetic field near bulk filling factor 2, as well as changes of a factor of 2 in the carrier density in induced devices. Additionally, this time dependence was present in n-type devices for ac excitations ranging from 50 to 500 μV. By contrast, the p-type QPC shows little to no change in conductance after the application of the ac excitation. Again, this turned out to be typical of all four p-type devices measured; time dependence was absent at ac excitations ranging from 5 to 500 μV. After the conductance reached steady state, an rf magnetic field was applied to the device. The frequency was swept through the Larmor frequency of each nucleus present in turn to randomize the nuclear spin populations. In the interest of noise reduction, these sweeps were repeated up to 100 times and the results averaged. Typically this procedure reduced the noise in the averaged conductance traces to below 1 part in 20000, suggesting that nuclear-related conductance changes as small as 0.01% might be detectable. Figure 4 shows typical averaged NMR data for n- and p-type devices. 3149
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Nano Letters As expected, the n-type devices show significant conductance changes as the rf magnetic field is swept through the resonant frequencies of the nuclei. As with the time dependence, these effects are robust to variations in the carrier density, dc magnetic field, and ac excitation voltage. The frequency at which these resonances appear scales linearly with the dc magnetic field, as expected for a nuclear process, and the ratios of the nuclear resonant frequencies match the expected values to within the line width of the resonances. Figure 5 shows the location of each NMR peak as a function of the magnetic field. The frequency scaling in electron devices allowed us to calibrate out any error in the dc magnetic field and precisely locate the expected resonant frequencies for use with holes. Significantly, we still found no evidence for nuclear magnetic resonance in any p-type device, despite noise levels that would allow detection of a signal approximately 2 orders of magnitude smaller than that seen in electrons. While it was expected that the DNP effect would be smaller in holes than electrons, it is somewhat surprising that no signal at all was evident under any conditions for p-type QPCs. We examined several p-type devices fabricated on multiple wafers, always with the same result. In order to eliminate any possibility that the materials were at issue, we fabricated a device in which the carriers in the channel could be changed from electrons to holes and measured it with both polarities of carrier on exactly the same lithographic structure20—again the electrons showed the time dependence and resonant behavior characteristic of DNP, while the holes showed nothing. There are a few possible reasons for this behavior. Heavy hole states have spin 3/2, which would entail a three-photon transition for the “flip-flop” process that generates the nuclear polarization. This would tend to suppress DNP by more than the expected factor of 10. However, light-hole states are expected to be present under the conditions in this study,22 which implies that the nuclear polarization process should occur as expected. While the exact behavior of the dynamic nuclear polarization mechanism is unclear, the “polarization-dragging” behavior exploited in optical measurements5 suggests that the polarization process is limited by the Overhauser shift. In such circumstances, we would expect the polarization at steady state, even with a relatively small fraction of light holes, to equal that achieved with electrons. Since both light and heavy hole states should contribute to readout, it is unclear in this scenario why we would not see a signal in p-type devices. It is possible that the weaker Overhauser term for holes would tend to yield a lower steady-state polarization than in electrons; in this case the signal would be suppressed in holes by a factor of about 100 relative to electrons. A signal of that size would be difficult to detect unambiguously in this scheme, but we would have expected to see at least some time dependence. An alternative explanation is that the hyperfine coupling between holes and nuclei in GaAs may be significantly weaker than currently predicted. In any case, it is clear that holenuclear coupling is not a large contributor to spin flips in GaAs/AlGaAs single heterojunctions. Conclusions. In conclusion, we have measured dynamic nuclear polarization and nuclear magnetic resonance in a variety of n-type quantum point contacts. Similar measurements undertaken in p-type QPCs show that the nuclear polarization process, if present, is much smaller than would be expected given the predicted strength of the hole hyperfine coupling. This may indicate that the hole hyperfine coupling is weaker than previously predicted in this system, which in turn would suggest that holes in GaAs/AlGaAs-based devices may be viable for quantum information processing.
LETTER
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT This work was funded by the Australian Research Council under the Discovery Projects scheme. A.R.H. acknowledges an Australian Professorial Fellowship, A.P.M. a Future Fellowship (FT0990285). H.E.B. and D.A.R. acknowledge financial support from the EPSRC. K.T., D.R., and A.D.W. thank the BMBFnanoQuit, QuaHLRep, and QUIMP as well as the DFG SPP1285 for financial support. We thank J. Cochrane for technical support, and D. McCamey, U. Z€ulicke, Y. Hirayama, F. Sfigakis, T. Machida, and M. Kawamura for helpful discussions and advice. ’ REFERENCES (1) Zutic, I.; Fabian, J.; Sarma, S. D. Rev. Mod. Phys. 2004, 76, 323–410. (2) Bulaev, D. V.; Loss, D. Phys. Rev. Lett. 2005, 95, 076805. (3) Fischer, J.; Coish, W. A.; Bulaev, D. V.; Loss, D. Phys. Rev. B 2008, 78, 155329. (4) Testelin, C.; Bernardot, F.; Eble, B.; Chamarro, M. Phys. Rev. B 2009, 79, 195440. €slmaz, S.; Imamoglu, A. Phys. Rev. Lett. 2010, (5) Fallahi, P.; YA 105, 257402. (6) Chekhovich, E.; Krysa, A.; Skolnick, M.; Tartakovskii, A. Phys. Rev. Lett. 2011, 106, 027402. (7) Wald, K. R.; Kouwenhoven, L. P.; McEuen, P. L.; van der Vaart, N. C.; Foxon, C. T. Phys. Rev. Lett. 1994, 73, 1011–1014. (8) Dixon, D. C.; Wald, K. R.; McEuen, P. L.; Melloch, M. R. Phys. Rev. B 1997, 56, 4743–4750. (9) Kawamura, M.; Takahashi, H.; Sugihara, K.; Masubuchi, S.; Hamaya, K.; Machida, T. Appl. Phys. Lett. 2007, 90, 022102. (10) Yang, C. L.; Zhang, J.; Du, R. R.; Simmons, J. A.; Reno, J. L. Phys. Rev. Lett. 2002, 89, 076801. (11) Komiyama, S.; Astafiev, O.; Machida, T. Phys. E (Amsterdam, Neth.) 2003, 20, 43–56. (12) Hirayama, Y.; Yusa, G.; Hashimoto, K.; Kumada, N.; Ota, T.; Muraki, K. Semicond. Sci. Technol. 2009, 24, 023001. (13) Kane, B. E.; Pfeiffer, L. N.; West, K. W. Phys. Rev. B 1992, 46, 7264–7267. (14) Kane, B. E.; Tsui, D. C.; Weimann, G. Phys. Rev. Lett. 1988, 61, 1123–1126. (15) Corcoles, A.; Ford, C. J. B.; Pepper, M.; Jones, G. A. C.; Beere, H. E.; Ritchie, D. A. Phys. Rev. B 2009, 80, 115326. (16) Yusa, G.; Muraki, K.; Takashina, K.; Hashimoto, K.; Hirayama, Y. Nature 2005, 434, 1001. (17) Paget, D.; Lampel, G.; Sapoval, B.; Safarov, V. I. Phys. Rev. B 1977, 15, 5780–5796. (18) Klochan, O.; Clarke, W. R.; Danneau, R.; Micolich, A. P.; Ho, L. H.; Hamilton, A. R.; Muraki, K.; Hirayama, Y. Appl. Phys. Lett. 2006, 89, 092105. (19) Chen, J. C. H.; Klochan, O.; Micolich, A. P.; Hamilton, A. R.; Martin, T. P.; Ho, L. H.; Z€ulicke, U.; Reuter, D.; Wieck, A. D. New J. Phys. 2010, 12, 033043. (20) Chen, J. C. H. Wang, Q.; Micolich, A. P.; Hamilton, A. R.; das Gupta, K.; Sfigakis, F.; Ritchie, D. A.; Reuter, D.; Wieck, A. D. Unpublished work. (21) Csontos, M.; Komijani, Y.; Shorubalko, I.; Ensslin, K.; Reuter, D.; Wieck, A. D. Appl. Phys. Lett. 2010, 97, 2110. (22) Winkler, R. Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems; Springer: Berlin and New York, 2003.
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