- Email: [email protected]
Coherent electrical manipulation of nuclear spins in semiconductors
ARTICLE IN PRESS
Physica E 25 (2004) 142–149 www.elsevier.com/locate/physe
Coherent electrical manipulation of nuclear spins in semiconductors T. Machidaa,*, T. Yamazakib, K. Ikushimab, S. Komiyamab a
Nanostructure and Material Property, PRESTO, Japan Science and Technology Agency, Building 16-622, Komaba 3-8-1, Meguro-ku, Tokyo 153-8902, Japan b Department of Basic Sciences, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8902, Japan Available online 28 July 2004
Abstract We have utilized the electron–nuclear spin interaction in quantum-Hall edge channels for manipulating local nuclear spins in semiconductors. Nuclear spins in a limited region along spin-resolved quantum-Hall edge channels are strongly polarized through the hyperfine interaction. Quantum states of the nuclear spins evolve coherently during a pulsedmode application of RF magnetic field generated by a built-in micro-metal strip. The final nuclear-spin state reached after the pulsed-mode operation is read out by the edge-channel transport. The amplitude of nuclear magnetic resonance oscillates as a function of the pulse duration, i.e. the Rabi oscillation of nuclear spins in semiconductor devices. r 2004 Elsevier B.V. All rights reserved. PACS: 73.43.f; 76.60.k Keywords: Quantum-Hall effect; Nuclear magnetic resonance; Edge channels; Quantum bit
1. Introduction Electron–nuclear-spin interaction has recently been found to have striking effects on spindependent electron transport in quantum-Hall systems [1–7]. Electron spins are coupled to nuclear spins of the host material through the hyperfine interaction. This causes the simultaneous flip–flop reversals of a nuclear spin and an electron *Corresponding author. Present address: Institute of Industrial Science, University of Tokyo, Komaba 4-6-1, Tokyo 1538505, Japan. Fax: +81-3-5452-6743. E-mail address: [email protected] (T. Machida).
spin. When the electron-spin distribution is brought out of a thermal equilibrium, the flip– flop mechanism dynamically polarizes nuclear spins. The resultant dynamic nuclear polarization (DNP), in turn, affects the electron transport, thus leading to a novel physical phenomena, such as a spin-dependent enhancement/suppression of the inter-edge-channel scattering [1–3,7] and an unusually slow development of magnetoresistance [4–6]. Nuclear spin is an ideal system for storing quantum information because its decoherence time is extremely long. Nuclear-spin quantum states are coherently controlled in the liquid-state nuclear
1386-9477/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2004.06.011
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magnetic resonance (NMR) to realize a quantum algorithm [8]. In exploring its scalability, the schemes of NMR quantum computers have recently been proposed [9,10]. However, its experimental implementation is still beyond the present fabrication technology. Thus, the coherent control of nuclear spins in solids is a key ingredient for developing quantum information devices based on nuclear-spin systems. In semiconductors, the contact hyperfine interaction with conduction electrons provides a promising access to nuclear spins. Optical manipulation of local nuclear spins exploiting circularly polarized light [11] and resistive detection of NMR in quantum Hall systems [1–7] suggest a possibility of manipulating nuclear spins in semiconductors. However, coherent control of nuclear-spin quantum states has not been achieved so far. In this work, we demonstrate coherent control of local nuclear spins in semiconductor electronic devices [12,13]. First, nuclear spins along spinresolved quantum-Hall edge channels are strongly polarized via the hyperfine interaction by selectively populating the edge channels. Secondly, a pulsed radio-frequency (RF) magnetic field at a NMR frequency is generated by a micro-metal strip lithographically fabricated close to the edge channels, which causes the nuclear-spin state to evolve coherently. Finally, the nuclear-spin state reached during the pulse application is read out through the edge-channel conductance, which oscillates as a function of the pulse duration, i.e. all-electrical detection of the Rabi oscillation in solid-state nuclear-spin system. The technique of controlling nuclear-spin polarization via electron systems opens up new possibilities of developing nuclear-spin quantum information devices.
2. Quantum-Hall devices Figs. 1(a) and (b) show the Hall-bar devices, A and B. Micrographs of their central regions are shown in Figs. 1(c) and (d). They are fabricated on an Al0.3Ga0.7As/GaAs single heterostrucure crystal with a two-dimensional electron gas (2DEG) located 100 nm beneath the crystal surface. The
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electron mobility and the sheet electron density are m=30 m2/V s and ns=3.6 1015 m2 at 4.2 K, respectively. Ohmic contacts serve as source and drain reservoirs (S and D) as well as voltage probes (V). Two Schottky cross gates (G1 and G2) are prepared by depositing a layer of 150-nm-thick Au and 10-nm-thick Ti on top of the crystal surface. In the device A (Fig. 1(c)), a metal strip (MS) with 2-mm width and 20-mm length is deposited along a boundary of the 2DEG region sandwiched by the two gates, and it is connected to coplanar strips of a characteristic impedance of 50 O. In the device B, metal strips (MS1 and MS2) with 2-mm width and 6-mm length are placed along an edge in series (Fig. 1(d)). The devices are mounted in a 3He–4He dilution refrigerator with a base temperature of 50 mK. The four-terminal differential Hall resistance, R0 H qVH/qI, is studied by superposing an ACcurrent IAC=1 nA (18 Hz) on a background DCcurrent IDC. The Landau-level filling factors in the bulk (ungated) and gated (G1 and G2) regions are adjusted to nB=2 and nG=1, respectively, by applying an external static magnetic field Bext and a negative gate-bias voltage to the gates. Only the outer spin-up edge channel of the lowest Landau level is transmitted through the potential barriers underneath the gates as schematically shown in Figs. 1(a) and (b). In this condition, the spinresolved edge channels along the upper boundary between the two gates are unequally populated up to different electrochemical potentials mS and mD by the source and drain reservoirs, respectively: When a positive current ISD is applied between the source and drain reservoirs (mS > mD), the outer spin-up edge channel is overpopulated as shown in Fig. 2(a), whereas the inner spin-down edge channel is overpopulated for a negative ISD (mS o mD) as shown in Fig. 2(b). In either conditions, the value of R0 H is a sensitive probe of the inter-edge-channel scattering rate, i.e. R0 H = h/e2 when the edge channels are completely decoupled, while R0 H = h/2e2 when they are fully coupled through strong scattering [14]. The wave functions of spin-up and spin-down states of edge-channel electrons overlap significantly in space, but the states are well decoupled due to a small spin–flip scattering rate.
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Fig. 1. (a) Schematic representation of the Hall-bar device A. The dashed lines represent quantum-Hall edge channels. (b) The Hallbar device B. (c) A micrograph of a central region of the device A. (d) A scanning electron micrograph of the device B.
conduction electrons in edge channels, A HHyperfine ¼ AI S ¼ I þ S þ I S þ þ AIz Sz ; 2 ð1Þ
Fig. 2. (a) Schematic representation of unequally-populated spin-resolved edge channels for ISD > 0. (b) ISD o 0. (c) Schematic representation of the dynamic nuclear polarization. (d) Schematic representation of RF magnetic fields on a cross section of the device.
3. Dynamic nuclear polarization To initialize nuclear-spin state, we exploit the contact hyperfine interaction between the nuclear spin I of GaAs nuclei and the electron spin S of
with A (> 0) as the hyperfine constant. The terms within the brackets correspond to the simultaneous flip–flop of the electron spin and the nuclear spin. The second term is the hyperfine splitting, which reduces or increases the effective Zeeman splitting of the electron spins. Earlier experiments [1–3] have shown that electron spin–flip scattering between unequally populated spin-resolved edge channels induces strong dynamical nuclear polarization (DNP) along the edge channels due to the simultaneous flip–flop process implied by the term ðI þ S þ I Sþ Þ in Eq. (1). When the electrochemical potential of electrons in the source mS is higher than that of the drain mD, up-to-down spin–flip scattering takes place along the edge channel as schematically shown in Fig. 2(a). The dominant contribution to the spin–flip scattering arises from the spin-orbit interaction. However, there is a finite contribution from the contact hyperfine interaction. The electron spin–flip process associated with the
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inter-edge-channel scattering induces a positive DNP (Iz >0) through the hyperfine interaction, which in turn reduces the effective Zeeman energy as implied by the term AIzSz in Eq. (1) [1–3, 7]. The inter-edge-channel scattering is thereby selfaccelerating, forcing R0 H to approach the limiting value of h/2e2. On the contrary, with the negative polarity of ISD (mS o mD), a negative DNP (Iz o 0) is induced: The inter-edge-channel scattering is thereby self-limiting, R0 H approaching the other limiting value of h/e2. This current-polarity-dependent enhancement and suppression of the interedge-channel scattering causes a hysteretic nonlinearity of R0 H in a sweep of IDC with an unusually long relaxation time [7]. The effect is observed in the present device as shown in the time traces of R0 H for IDC = 4 nA and IDC = 4 nA in Figs. 3(a) and 3(b), respectively, where the traces are obtained after IDC has been maintained at the opposite current polarities for a sufficiently long time. The slow evolutions of R0 H in Figs. 3(a) and (b) represent the development of negative and positive nuclear spin polarizations, respectively. The relevant nuclear spins are limited to those in a narrow region along the edge channels as schematically shown in Fig. 2(c), because only those nuclear spins that are in direct contact with edge-channel electrons are polarized through the hyperfine interaction. The width and thickness of the area are determined by the spatial extension of the electron wavefunction in edge channels, i.e. wE d E 10 nm. Thus, simply by selecting current polarity, we can initialize the nuclear-spin state in
a limited region to either a spin-up state or a spindown state.
Fig. 3. Time evolution of R0 H at (a) IDC = 4 nA and (b) IDC = 4 nA. The insets schematically show the dynamically induced nuclear polarization.
Fig. 4. Spatial distribution of RF magnetic field parallel to 2DEG plane (B//). IRF = 44 mA (100 MHz). (a) Gray-scale plot. (b) Distribution of B// along the thick solid line in (a).
4. Resistive detection of NMR We prepare an initial condition for NMR in the device A by maintaining IDC = 4 nA until R0 H decreases to a steady value, i.e. the positive DNP fully develops. We then transmit RF current through the metal strip MS of the device A. The RF current generates RF magnetic field BRF exclusively at the nuclear spins underneath MS in the direction parallel to the 2DEG (Fig. 2(d)). Spatial distribution of BRF is calculated by the finite element method. Fig. 4(a) shows a grayscale plot of the parallel component of BRF at the 2DEG layer, 100-nm depth from the surface. Fig. 4(b) shows its distribution along the thick solid line in Fig. 4(a). The distribution of BRF is local, and rapidly decays outside the region beneath MS. Since the MS is located only 100 nm above the edge channels, the small RF
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current suffices to generate the RF magnetic fields of necessary amplitude for NMR. The effective electron temperature is accordingly kept unchanged after the application of BRF, which is essential for the electrical detection of NMR that follows. When the frequency f of BRF is scanned, a distinct resonance peak is found in R0 H at the NMR frequency for 75As, 69Ga, and 71Ga as shown in Figs. 5(a)–(c). The resonance structure in the curve of R0 H versus f arises because the nuclear spins underneath MS, positively polarized in the initial condition, are depolarized by the NMR.
Fig. 5. NMR spectra (cw mode) of the device A: (a) 69 Ga, (c) 71Ga.
75
As, (b)
Additional experiments carried out with IDC o 0 provide similar NMR signals with the opposite polarity for all the nuclei of GaAs. To prove the locality of the present NMR, we make similar experiments using metal strips MS1 and MS2 in the device B (Fig. 1(d)). We transmit RF current through either MS1 or MS2. Similar NMR signals for 75As are obtained for both cases, but the amplitude of those signals are different. As shown in Fig. 6(a), the NMR signal obtained by MS1 is slightly larger than that obtained by MS2 when the external static magnetic field Bext points toward +z-axis as shown in Fig. 1(b). This is because MS2 is located downstream of the electron propagation along the edge channels, and the degree of unequal population between the edge channels is therefore smaller in the region underneath MS2. To confirm that the difference of the NMR amplitudes is not a consequence of extrinsic origins, we reversed the polarity of Bext so as to reverse the propagation direction of electrons in the edge channels. The results are displayed in Fig. 6(b), where R0 H is studied by exchanging the current contacts (S and D) and the two voltage probes (V). The NMR signals exchange their relative amplitudes between MS1 and MS2. We thus conclude to have achieved local excitation and detection of NMR in a quantum-Hall device [12].
Fig. 6. NMR spectra (cw mode) of the device B for different polarities of the static magnetic field: (a) Bext // +z, (b) Bext // z.
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5. Rabi oscillations By taking advantage of the high efficiency of the initial polarization as well as the local controllability of nuclear spins via the micro-metal strips, nuclear spins can be coherently controlled and electrically detected. We prepare an initial nuclearspin state by maintaining IDC = +4 nA in the device A. The ensemble of nuclear spins in a narrow region along the edge channels are initialized to the up-spin state |0S. Superposition of the up-spin and down-spin states, |0Sand |S is generated by transmitting a pulse of RF electrical current IRF through the metal strip MS. The nuclear-spin state evolves with time during the pulse application. The final state reached at the end of a pulse, jtS ¼ aðtÞj0S þ bðtÞj1S; is read out via the change in R0 H, which gives a measure of jbðtÞj2 : Fig. 7 displays time evolutions of R0 H when a pulse of BRF with pulse duration tpulse and a peakto-peak RF-current amplitude of IRF = 62 mA at a frequency of f = 94.8 MHz is applied at t=0. The increases in R0 H arise from a reduction of |a|2, or an increase of |b|2. The change in R0 H gives a measure of |b(t)|2 for t=tpulse. The NMR
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response, DR0 H, shows an oscillatory behavior in scanning tpulse. We have carried out a number of similar measurements by applying RF pulses with different pulse widths (0 p tpulse p 200 ms) at IRF = 62 mA, and find that the amplitude of DR0 H oscillates as a function of tpulse as shown by the top curve in Fig. 8. Reducing IRF increases the period of oscillation as exemplified by the lower curves in Fig. 8, where the period is inversely
Fig. 8. Coherent evolution of nuclear-spin quantum state in a time domain, i.e. Rabi oscillations. Values of DR0 are plotted as a function of the pulse duration of BRF. The lines are offset for clarity.
Fig. 7. Time evolution of R0 H in pulsed-mode NMR of the device A. A pulse of RF magnetic field at the NMR frequency of 71Ga is applied to the nuclear spins at t=0. The lines are offset for clarity.
Fig. 9. The inverse of the oscillation period as a function of the peak-to-peak amplitude of the RF current.
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Fig. 10. Nuclear-spin echo signal obtained by applying p/2t1pt2p/2 pulse sequences of BRF.
proportional to IRF as shown in Fig. 9. The series of curves in Fig. 8 is thus a demonstration of the coherent nuclear-spin evolution in a time domain, i.e. the Rabi oscillation. Therefore we can achieve any desired superposition of the nuclear-spin states, a|0S + b|1S, by adjusting the pulse duration [13], thus achieving the coherent electrical manipulation of nuclear spins in the semiconductor device. To evaluate the decoherence time T2, we apply p/2t1pt2p/2 pulse sequences of BRF. The spin–echo signal has been detected resistively as shown in Fig. 10. From the decay of the echo signal, we have evaluated T2 B 80 ms. Details of the spin–echo experiments will be described elsewhere [15]. The number of relevant nuclei is estimated to be N B109, by noting the length of edge channels (B20 mm) and the spatial extension of the edgestate wave functions in parallel (B10 nm) and normal (B10 nm) to the heterostructure interface. This value is much smaller than that of the optical manipulation of nuclear spins [11], where the spot size of the circularly polarized light (diameter B50 mm) defines the number of relevant nuclear spins, Nb109 :MORE importantly, coherent evolution of nuclear-spin state is not probed by the optical method [11]. Standard pulsed-NMR experiments on bulk quantum-Hall systems have been reported [16],
where an external coil detects nuclear magnetic moments from the entire regions of 2DEG layers in multiple quantum wells. The number of relevant nuclear spins is thus larger than that in the present scheme by a factor of more than five orders in magnitude. In addition, the present work is distinguished from those standard NMR measurements by: (i) its efficient initialization of nuclearspin state through hyperfine interaction, (ii) the local controllability of nuclear spins in a welldefined narrow region along the edge channels, and (iii) the high sensitivity of detecting nuclear polarization via conductance through the edge channels. The GaAs nuclei forms a four-level system, because the nuclear-spin moment is I ¼ 32: The single-peaked NMR spectra in this device (Figs. 5(a)–(c)) indicate the nearly equidistant level spacing. However, different-level transitions are resolved in appropriate conditions [7]. It follows that the two-qubit operation is potentially possible in the present scheme [17]. Finally, the general importance of the present scheme lies in its flexible device-design capability and the resultant variety of potential aspects for further developments. The location of polarized/ detected nuclear spins is primarily defined by the incompressible strip between spin-resolved edge channels. The spatial pattern of edge channels can be tailored by a simple metal-gating technique:
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The location of the incompressible strip can be controlled on nano-meter-scale accuracy. Indirect interaction mediated via edge channels might serve as a controllable scalar coupling between different ensembles of nuclear spins, yielding more-thanone qubit systems.
6. Summary In summary, we have demonstrated coherent control of local nuclear spins based on pulsed NMR in a quantum-Hall device. An initial nuclear-spin state is prepared by strongly polarizing nuclei via the hyperfine interaction with electrons in unequally populated spin-resolved edge channels. A pulsed radio-frequency magnetic field is generated using a built-in micro-metal strip, causing the nuclear-spin state to evolve coherently. The final state of the nuclear spins is read out through the edge-channel conductance, which shows the Rabi oscillation. This technique will open up a new possibility of developing nuclearspin qubits in solid-state systems.
Acknowledgements This work is partly supported by Grant-in-Aid for specially promoted research from the Japanese Ministry of Education, Culture, Sports, Science and Technology, and by Solution Oriented Research for Science and Technology (SORST), Japan Science and Technology Corporation (JST).
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References [1] B.E. Kane, L.N. Pfeiffer, K.W. West, Phys. Rev. B 46 (1992) 7264. [2] K.R. Wald, L.P. Kouwenhoven, P.L. McEuen, N.C. van der Vaart, C.T. Foxon, Phys. Rev. Lett. 73 (1994) 1011. [3] D.C. Dixon, K.R. Wald, P.L. McEuen, M.R. Melloch, Phys. Rev. B 56 (1997) 4743. [4] S. Kronmuller, . W. Dietsche, J. Weis, K. von Klitzing, W. Wegscheider, M. Bichler, Phys. Rev. Lett. 81 (1998) 2526; Phys. Rev. Lett. 82 (1999) 4070. [5] J.H. Smet, R.A. Deutschmann, F. Ertl, W. Wegscheider, G. Abstreiter, K. von Klitzing, Nature 415 (2002) 281. [6] K. Hashimoto, K. Muraki, T. Saku, Y. Hirayama, Phys. Rev. Lett. 88 (2002) 176601. [7] T. Machida, S. Ishizuka, T. Yamazaki, S. Komiyama, K. Muraki, Y. Hirayama, Phys. Rev. B 65 (2002) 233304. [8] L.M.K. Vandersypen, M. Steffen, G. Breyta, C.S. Yannoni, M.H. Sherwood, I.L. Chuang, Nature (London) 414 (2001) 883. [9] B.E. Kane, Nature (London) 393 (1998) 133. [10] T.D. Ladd, J.R. Goldman, F. Yamaguchi, Y. Yamamoto, E. Abe, K.M. Itoh, Phys. Rev. Lett. 89 (2002) 017901. [11] G. Salis, D.T. Fuchs, J.M. Kikkawa, D.D. Awschalom, Y. Ohno, H. Ohno, Phys. Rev. Lett. 86 (2001) 2677. [12] T. Machida, T. Yamazaki, S. Komiyama, Appl. Phys. Lett. 80 (2002) 4178. [13] T. Machida, T. Yamazaki, K. Ikushima, S. Komiyama, Appl. Phys. Lett. 82 (2003) 409. [14] See, for instance, T. Machida, H. Hirai, S. Komiyama, T. Osada, Y. Shiraki, Phys. Rev. B 54 (1996) 14261. [15] T. Machida, T. Yamazaki, K. Ikushima, S. Komiyama, unpublished. [16] N. Freytag, Y. Tokunaga, M. Horvatic, C. Berthier, M. Shayegan, L.P. L!evy, Phys. Rev. Lett. 87 (2001) 136801. [17] M.N. Leuenberger, D. Loss, M. Poggio, D.D. Awschalom, Phys. Rev. Lett. 89 (2002) 207601.
Physica E 25 (2004) 142–149 www.elsevier.com/locate/physe
Coherent electrical manipulation of nuclear spins in semiconductors T. Machidaa,*, T. Yamazakib, K. Ikushimab, S. Komiyamab a
Nanostructure and Material Property, PRESTO, Japan Science and Technology Agency, Building 16-622, Komaba 3-8-1, Meguro-ku, Tokyo 153-8902, Japan b Department of Basic Sciences, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8902, Japan Available online 28 July 2004
Abstract We have utilized the electron–nuclear spin interaction in quantum-Hall edge channels for manipulating local nuclear spins in semiconductors. Nuclear spins in a limited region along spin-resolved quantum-Hall edge channels are strongly polarized through the hyperfine interaction. Quantum states of the nuclear spins evolve coherently during a pulsedmode application of RF magnetic field generated by a built-in micro-metal strip. The final nuclear-spin state reached after the pulsed-mode operation is read out by the edge-channel transport. The amplitude of nuclear magnetic resonance oscillates as a function of the pulse duration, i.e. the Rabi oscillation of nuclear spins in semiconductor devices. r 2004 Elsevier B.V. All rights reserved. PACS: 73.43.f; 76.60.k Keywords: Quantum-Hall effect; Nuclear magnetic resonance; Edge channels; Quantum bit
1. Introduction Electron–nuclear-spin interaction has recently been found to have striking effects on spindependent electron transport in quantum-Hall systems [1–7]. Electron spins are coupled to nuclear spins of the host material through the hyperfine interaction. This causes the simultaneous flip–flop reversals of a nuclear spin and an electron *Corresponding author. Present address: Institute of Industrial Science, University of Tokyo, Komaba 4-6-1, Tokyo 1538505, Japan. Fax: +81-3-5452-6743. E-mail address: [email protected] (T. Machida).
spin. When the electron-spin distribution is brought out of a thermal equilibrium, the flip– flop mechanism dynamically polarizes nuclear spins. The resultant dynamic nuclear polarization (DNP), in turn, affects the electron transport, thus leading to a novel physical phenomena, such as a spin-dependent enhancement/suppression of the inter-edge-channel scattering [1–3,7] and an unusually slow development of magnetoresistance [4–6]. Nuclear spin is an ideal system for storing quantum information because its decoherence time is extremely long. Nuclear-spin quantum states are coherently controlled in the liquid-state nuclear
1386-9477/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2004.06.011
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magnetic resonance (NMR) to realize a quantum algorithm [8]. In exploring its scalability, the schemes of NMR quantum computers have recently been proposed [9,10]. However, its experimental implementation is still beyond the present fabrication technology. Thus, the coherent control of nuclear spins in solids is a key ingredient for developing quantum information devices based on nuclear-spin systems. In semiconductors, the contact hyperfine interaction with conduction electrons provides a promising access to nuclear spins. Optical manipulation of local nuclear spins exploiting circularly polarized light [11] and resistive detection of NMR in quantum Hall systems [1–7] suggest a possibility of manipulating nuclear spins in semiconductors. However, coherent control of nuclear-spin quantum states has not been achieved so far. In this work, we demonstrate coherent control of local nuclear spins in semiconductor electronic devices [12,13]. First, nuclear spins along spinresolved quantum-Hall edge channels are strongly polarized via the hyperfine interaction by selectively populating the edge channels. Secondly, a pulsed radio-frequency (RF) magnetic field at a NMR frequency is generated by a micro-metal strip lithographically fabricated close to the edge channels, which causes the nuclear-spin state to evolve coherently. Finally, the nuclear-spin state reached during the pulse application is read out through the edge-channel conductance, which oscillates as a function of the pulse duration, i.e. all-electrical detection of the Rabi oscillation in solid-state nuclear-spin system. The technique of controlling nuclear-spin polarization via electron systems opens up new possibilities of developing nuclear-spin quantum information devices.
2. Quantum-Hall devices Figs. 1(a) and (b) show the Hall-bar devices, A and B. Micrographs of their central regions are shown in Figs. 1(c) and (d). They are fabricated on an Al0.3Ga0.7As/GaAs single heterostrucure crystal with a two-dimensional electron gas (2DEG) located 100 nm beneath the crystal surface. The
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electron mobility and the sheet electron density are m=30 m2/V s and ns=3.6 1015 m2 at 4.2 K, respectively. Ohmic contacts serve as source and drain reservoirs (S and D) as well as voltage probes (V). Two Schottky cross gates (G1 and G2) are prepared by depositing a layer of 150-nm-thick Au and 10-nm-thick Ti on top of the crystal surface. In the device A (Fig. 1(c)), a metal strip (MS) with 2-mm width and 20-mm length is deposited along a boundary of the 2DEG region sandwiched by the two gates, and it is connected to coplanar strips of a characteristic impedance of 50 O. In the device B, metal strips (MS1 and MS2) with 2-mm width and 6-mm length are placed along an edge in series (Fig. 1(d)). The devices are mounted in a 3He–4He dilution refrigerator with a base temperature of 50 mK. The four-terminal differential Hall resistance, R0 H qVH/qI, is studied by superposing an ACcurrent IAC=1 nA (18 Hz) on a background DCcurrent IDC. The Landau-level filling factors in the bulk (ungated) and gated (G1 and G2) regions are adjusted to nB=2 and nG=1, respectively, by applying an external static magnetic field Bext and a negative gate-bias voltage to the gates. Only the outer spin-up edge channel of the lowest Landau level is transmitted through the potential barriers underneath the gates as schematically shown in Figs. 1(a) and (b). In this condition, the spinresolved edge channels along the upper boundary between the two gates are unequally populated up to different electrochemical potentials mS and mD by the source and drain reservoirs, respectively: When a positive current ISD is applied between the source and drain reservoirs (mS > mD), the outer spin-up edge channel is overpopulated as shown in Fig. 2(a), whereas the inner spin-down edge channel is overpopulated for a negative ISD (mS o mD) as shown in Fig. 2(b). In either conditions, the value of R0 H is a sensitive probe of the inter-edge-channel scattering rate, i.e. R0 H = h/e2 when the edge channels are completely decoupled, while R0 H = h/2e2 when they are fully coupled through strong scattering [14]. The wave functions of spin-up and spin-down states of edge-channel electrons overlap significantly in space, but the states are well decoupled due to a small spin–flip scattering rate.
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Fig. 1. (a) Schematic representation of the Hall-bar device A. The dashed lines represent quantum-Hall edge channels. (b) The Hallbar device B. (c) A micrograph of a central region of the device A. (d) A scanning electron micrograph of the device B.
conduction electrons in edge channels, A HHyperfine ¼ AI S ¼ I þ S þ I S þ þ AIz Sz ; 2 ð1Þ
Fig. 2. (a) Schematic representation of unequally-populated spin-resolved edge channels for ISD > 0. (b) ISD o 0. (c) Schematic representation of the dynamic nuclear polarization. (d) Schematic representation of RF magnetic fields on a cross section of the device.
3. Dynamic nuclear polarization To initialize nuclear-spin state, we exploit the contact hyperfine interaction between the nuclear spin I of GaAs nuclei and the electron spin S of
with A (> 0) as the hyperfine constant. The terms within the brackets correspond to the simultaneous flip–flop of the electron spin and the nuclear spin. The second term is the hyperfine splitting, which reduces or increases the effective Zeeman splitting of the electron spins. Earlier experiments [1–3] have shown that electron spin–flip scattering between unequally populated spin-resolved edge channels induces strong dynamical nuclear polarization (DNP) along the edge channels due to the simultaneous flip–flop process implied by the term ðI þ S þ I Sþ Þ in Eq. (1). When the electrochemical potential of electrons in the source mS is higher than that of the drain mD, up-to-down spin–flip scattering takes place along the edge channel as schematically shown in Fig. 2(a). The dominant contribution to the spin–flip scattering arises from the spin-orbit interaction. However, there is a finite contribution from the contact hyperfine interaction. The electron spin–flip process associated with the
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inter-edge-channel scattering induces a positive DNP (Iz >0) through the hyperfine interaction, which in turn reduces the effective Zeeman energy as implied by the term AIzSz in Eq. (1) [1–3, 7]. The inter-edge-channel scattering is thereby selfaccelerating, forcing R0 H to approach the limiting value of h/2e2. On the contrary, with the negative polarity of ISD (mS o mD), a negative DNP (Iz o 0) is induced: The inter-edge-channel scattering is thereby self-limiting, R0 H approaching the other limiting value of h/e2. This current-polarity-dependent enhancement and suppression of the interedge-channel scattering causes a hysteretic nonlinearity of R0 H in a sweep of IDC with an unusually long relaxation time [7]. The effect is observed in the present device as shown in the time traces of R0 H for IDC = 4 nA and IDC = 4 nA in Figs. 3(a) and 3(b), respectively, where the traces are obtained after IDC has been maintained at the opposite current polarities for a sufficiently long time. The slow evolutions of R0 H in Figs. 3(a) and (b) represent the development of negative and positive nuclear spin polarizations, respectively. The relevant nuclear spins are limited to those in a narrow region along the edge channels as schematically shown in Fig. 2(c), because only those nuclear spins that are in direct contact with edge-channel electrons are polarized through the hyperfine interaction. The width and thickness of the area are determined by the spatial extension of the electron wavefunction in edge channels, i.e. wE d E 10 nm. Thus, simply by selecting current polarity, we can initialize the nuclear-spin state in
a limited region to either a spin-up state or a spindown state.
Fig. 3. Time evolution of R0 H at (a) IDC = 4 nA and (b) IDC = 4 nA. The insets schematically show the dynamically induced nuclear polarization.
Fig. 4. Spatial distribution of RF magnetic field parallel to 2DEG plane (B//). IRF = 44 mA (100 MHz). (a) Gray-scale plot. (b) Distribution of B// along the thick solid line in (a).
4. Resistive detection of NMR We prepare an initial condition for NMR in the device A by maintaining IDC = 4 nA until R0 H decreases to a steady value, i.e. the positive DNP fully develops. We then transmit RF current through the metal strip MS of the device A. The RF current generates RF magnetic field BRF exclusively at the nuclear spins underneath MS in the direction parallel to the 2DEG (Fig. 2(d)). Spatial distribution of BRF is calculated by the finite element method. Fig. 4(a) shows a grayscale plot of the parallel component of BRF at the 2DEG layer, 100-nm depth from the surface. Fig. 4(b) shows its distribution along the thick solid line in Fig. 4(a). The distribution of BRF is local, and rapidly decays outside the region beneath MS. Since the MS is located only 100 nm above the edge channels, the small RF
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current suffices to generate the RF magnetic fields of necessary amplitude for NMR. The effective electron temperature is accordingly kept unchanged after the application of BRF, which is essential for the electrical detection of NMR that follows. When the frequency f of BRF is scanned, a distinct resonance peak is found in R0 H at the NMR frequency for 75As, 69Ga, and 71Ga as shown in Figs. 5(a)–(c). The resonance structure in the curve of R0 H versus f arises because the nuclear spins underneath MS, positively polarized in the initial condition, are depolarized by the NMR.
Fig. 5. NMR spectra (cw mode) of the device A: (a) 69 Ga, (c) 71Ga.
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As, (b)
Additional experiments carried out with IDC o 0 provide similar NMR signals with the opposite polarity for all the nuclei of GaAs. To prove the locality of the present NMR, we make similar experiments using metal strips MS1 and MS2 in the device B (Fig. 1(d)). We transmit RF current through either MS1 or MS2. Similar NMR signals for 75As are obtained for both cases, but the amplitude of those signals are different. As shown in Fig. 6(a), the NMR signal obtained by MS1 is slightly larger than that obtained by MS2 when the external static magnetic field Bext points toward +z-axis as shown in Fig. 1(b). This is because MS2 is located downstream of the electron propagation along the edge channels, and the degree of unequal population between the edge channels is therefore smaller in the region underneath MS2. To confirm that the difference of the NMR amplitudes is not a consequence of extrinsic origins, we reversed the polarity of Bext so as to reverse the propagation direction of electrons in the edge channels. The results are displayed in Fig. 6(b), where R0 H is studied by exchanging the current contacts (S and D) and the two voltage probes (V). The NMR signals exchange their relative amplitudes between MS1 and MS2. We thus conclude to have achieved local excitation and detection of NMR in a quantum-Hall device [12].
Fig. 6. NMR spectra (cw mode) of the device B for different polarities of the static magnetic field: (a) Bext // +z, (b) Bext // z.
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5. Rabi oscillations By taking advantage of the high efficiency of the initial polarization as well as the local controllability of nuclear spins via the micro-metal strips, nuclear spins can be coherently controlled and electrically detected. We prepare an initial nuclearspin state by maintaining IDC = +4 nA in the device A. The ensemble of nuclear spins in a narrow region along the edge channels are initialized to the up-spin state |0S. Superposition of the up-spin and down-spin states, |0Sand |S is generated by transmitting a pulse of RF electrical current IRF through the metal strip MS. The nuclear-spin state evolves with time during the pulse application. The final state reached at the end of a pulse, jtS ¼ aðtÞj0S þ bðtÞj1S; is read out via the change in R0 H, which gives a measure of jbðtÞj2 : Fig. 7 displays time evolutions of R0 H when a pulse of BRF with pulse duration tpulse and a peakto-peak RF-current amplitude of IRF = 62 mA at a frequency of f = 94.8 MHz is applied at t=0. The increases in R0 H arise from a reduction of |a|2, or an increase of |b|2. The change in R0 H gives a measure of |b(t)|2 for t=tpulse. The NMR
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response, DR0 H, shows an oscillatory behavior in scanning tpulse. We have carried out a number of similar measurements by applying RF pulses with different pulse widths (0 p tpulse p 200 ms) at IRF = 62 mA, and find that the amplitude of DR0 H oscillates as a function of tpulse as shown by the top curve in Fig. 8. Reducing IRF increases the period of oscillation as exemplified by the lower curves in Fig. 8, where the period is inversely
Fig. 8. Coherent evolution of nuclear-spin quantum state in a time domain, i.e. Rabi oscillations. Values of DR0 are plotted as a function of the pulse duration of BRF. The lines are offset for clarity.
Fig. 7. Time evolution of R0 H in pulsed-mode NMR of the device A. A pulse of RF magnetic field at the NMR frequency of 71Ga is applied to the nuclear spins at t=0. The lines are offset for clarity.
Fig. 9. The inverse of the oscillation period as a function of the peak-to-peak amplitude of the RF current.
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Fig. 10. Nuclear-spin echo signal obtained by applying p/2t1pt2p/2 pulse sequences of BRF.
proportional to IRF as shown in Fig. 9. The series of curves in Fig. 8 is thus a demonstration of the coherent nuclear-spin evolution in a time domain, i.e. the Rabi oscillation. Therefore we can achieve any desired superposition of the nuclear-spin states, a|0S + b|1S, by adjusting the pulse duration [13], thus achieving the coherent electrical manipulation of nuclear spins in the semiconductor device. To evaluate the decoherence time T2, we apply p/2t1pt2p/2 pulse sequences of BRF. The spin–echo signal has been detected resistively as shown in Fig. 10. From the decay of the echo signal, we have evaluated T2 B 80 ms. Details of the spin–echo experiments will be described elsewhere [15]. The number of relevant nuclei is estimated to be N B109, by noting the length of edge channels (B20 mm) and the spatial extension of the edgestate wave functions in parallel (B10 nm) and normal (B10 nm) to the heterostructure interface. This value is much smaller than that of the optical manipulation of nuclear spins [11], where the spot size of the circularly polarized light (diameter B50 mm) defines the number of relevant nuclear spins, Nb109 :MORE importantly, coherent evolution of nuclear-spin state is not probed by the optical method [11]. Standard pulsed-NMR experiments on bulk quantum-Hall systems have been reported [16],
where an external coil detects nuclear magnetic moments from the entire regions of 2DEG layers in multiple quantum wells. The number of relevant nuclear spins is thus larger than that in the present scheme by a factor of more than five orders in magnitude. In addition, the present work is distinguished from those standard NMR measurements by: (i) its efficient initialization of nuclearspin state through hyperfine interaction, (ii) the local controllability of nuclear spins in a welldefined narrow region along the edge channels, and (iii) the high sensitivity of detecting nuclear polarization via conductance through the edge channels. The GaAs nuclei forms a four-level system, because the nuclear-spin moment is I ¼ 32: The single-peaked NMR spectra in this device (Figs. 5(a)–(c)) indicate the nearly equidistant level spacing. However, different-level transitions are resolved in appropriate conditions [7]. It follows that the two-qubit operation is potentially possible in the present scheme [17]. Finally, the general importance of the present scheme lies in its flexible device-design capability and the resultant variety of potential aspects for further developments. The location of polarized/ detected nuclear spins is primarily defined by the incompressible strip between spin-resolved edge channels. The spatial pattern of edge channels can be tailored by a simple metal-gating technique:
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The location of the incompressible strip can be controlled on nano-meter-scale accuracy. Indirect interaction mediated via edge channels might serve as a controllable scalar coupling between different ensembles of nuclear spins, yielding more-thanone qubit systems.
6. Summary In summary, we have demonstrated coherent control of local nuclear spins based on pulsed NMR in a quantum-Hall device. An initial nuclear-spin state is prepared by strongly polarizing nuclei via the hyperfine interaction with electrons in unequally populated spin-resolved edge channels. A pulsed radio-frequency magnetic field is generated using a built-in micro-metal strip, causing the nuclear-spin state to evolve coherently. The final state of the nuclear spins is read out through the edge-channel conductance, which shows the Rabi oscillation. This technique will open up a new possibility of developing nuclearspin qubits in solid-state systems.
Acknowledgements This work is partly supported by Grant-in-Aid for specially promoted research from the Japanese Ministry of Education, Culture, Sports, Science and Technology, and by Solution Oriented Research for Science and Technology (SORST), Japan Science and Technology Corporation (JST).
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