Spin relaxation in charged quantum dots measured by coherent ...
Solid State Communications 140 (2006) 381–385 www.elsevier.com/locate/ssc
Spin relaxation in charged quantum dots measured by coherent optical phase modulation spectroscopy Jun Cheng a , Yanwen Wu a , Xiaodong Xu a , Dong Sun a , D.G. Steel a,∗ , A.S. Bracker b , D. Gammon b , Wang Yao c , L.J. Sham c a FOCUS, H. M. Randall Laboratory of Physics, The University of Michigan, Ann Arbor, MI 48109, USA b The Naval Research Laboratory, Washington DC 20375, USA c Department of Physics, The University of California, San Diego, La Jolla, CA 92093, USA
Received 23 August 2006; accepted 24 August 2006 by R. Merlin Available online 14 September 2006
Abstract Measurements of the electron spin relaxation rate (T1 ) in charged GaAs quantum dots are made using a new nonlinear optical phase-modulation spectroscopy technique. The measured results of T1 as a function of magnetic field and temperature are best explained in terms of the spin relaxation by the phonon-assisted Dresselhaus spin–orbit scattering. T1 approaches 34 µs at zero magnetic field at 4.5 K. c 2006 Elsevier Ltd. All rights reserved.
PACS: 39.30.+w; 42.65.-k; 72.25.Rb Keywords: A. Quantum dots; D. Spin dynamics; E. Phase modulation; E. Nonlinear spectroscopies
Interest in electron spin relaxation has intensified in recent years, in part because the new developments in nano-structures have led to the potential of reduced spin relaxation rates for application to spin-based devices and the observation of new spin relaxation physics [1–7]. Moreover, several all-optically manipulated spin schemes using stimulated Raman excitation have been proposed [8,9] where the trion state provides a strong optical enhancement through resonant coupling. These schemes take full advantage of femtosecond technology and flexible optical shaping for all-optical manipulations [10]. All-optical studies of electron spin relaxation in quantum dots (QDs) are important for developing an optically driven spinbased quantum computing scheme [11,12]. In addition, such measurements avoid the complications in spin physics which arise when the dots are too close to electric gates, e.g., Kondolike interaction of the dot spin with an electron Fermi sea in the electrode [13]. ∗ Corresponding address: The H. M. Randall Laboratory of Physics, 450 Church Street, Ann Arbor, MI 48109, USA. E-mail addresses: [email protected] (J. Cheng), [email protected] (D.G. Steel).
c 2006 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2006.08.035
Spin relaxation in QDs is greatly suppressed because the quantization of the electron energy levels results in the difficulty to simultaneously satisfy energy and momentum conservation. Among the various mechanisms leading to spin relaxation in QDs, spin–orbit interactions assisted by phonons have been predicted to be the dominant mechanism [1–3]. In recent experiments on electric gate confined QDs [14–16] and electric gate aided self-assembled QDs [17], spin–orbit coupling is shown to be the most significant contribution to spin relaxation under magnetic fields. In this article, we present all-optical measurements of spin relaxation in fluctuation QDs using a new kind of coherent nonlinear optical hole burning spectroscopy based on phase modulation. The measurements lay the groundwork for the alloptical addressing and the manipulation of the electron spin in a QD system. Measurements are performed as a function of magnetic field (typical in design of quantum logic operations [8, 9,18]) and temperature, leading to the conclusion that the Dresselhaus spin–orbit coupling with phonon scattering [19] is the dominant process in spin relaxation in QDs. The sample consists of an ensemble of QDs formed by ˚ wide GaAs QW layer. Electrons interface fluctuations in a 42 A
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Fig. 1. (a) The experiment setup for nonlinear optical phase-modulation spectroscopy. (b) Energy level diagram of a negatively charged QD in Voigt configuration, with ground states |x±i denoting electron spin projections ±1/2 along the x-axis. The trion states |t±i are labeled by the heavy-hole angular momentum projection ±3/2 along the growth axis (z). Solid (dashed) lines denote transitions excited by σ − (σ + ) light. If co-circularly polarized pump and probe fields (σ − ) are used, the relevant states are reduced to a three-level Λ-system enclosed in the dashed box.
Fig. 2. (a) The center sharp spike and two symmetric sidebands on the top of broad trion spectral hole burning in the nearly degenerate DT experiment at B = 3.3 T, where the pump is fixed at the lower monolayer trion ensemble center, as indicated by the arrow. The inset shows the zoomed in sharp spike at ∆ ∼ 0, where the spin relaxation dynamics can be examined and broadened by a spectral diffusion process. (b) tan φm at various circular modulation frequency (ωm ) at B = 3.3 T and temperature is 4.5 K. The solid line is a fitted function (Eq. (4)) by taking two different relaxation processes into account.
are incorporated into the dots by modulation of Si doping in the barrier [20]. In the Voigt geometry with the application of a magnetic field along the QW plane (x-axis), the energy level diagram of the negatively charged QD is schematically shown in Fig. 1(b). The transitions from the electron spin states |x±i to the charged exciton (trion) state |t−i (|t+i) are coupled by σ − (σ + ) polarized light according to the selection rules [21]. The relevant level scheme is reduced to the Λ-system (enclosed in the dashed box in Fig. 1(b)) since optical fields involved are all σ − polarized in the experiment. Spin relaxation rates are determined by analyzing the phase angle shift following amplitude modulation of the pump beam. Earlier use of ultra-high resolution coherent nonlinear spectroscopy techniques showed the important role of spectral diffusion of excitonic energy as well as spin flip induced hole burning in quantum well structures [22–24]. The earlier measurements were based on conventional four-wave mixing. In the present measurements, a standard continuous wave (CW) pump and probe setup is used based on the frequency stabilized laser, as shown in Fig. 1(a). Since both the pump and probe fields are mutually coherent, the conventional differential transmission setup actually measures the homodyne detected four-wave mixing response based on the third order nonlinear optical susceptibility.
Fig. 2(a) shows the large scale non-degenerate nonlinear optical response showing a complex structure: an overall broad structure due to spectral hole burning and spectral diffusion of the inhomogeneously broadened trion resonance, a central sharp resonance at the peak, and the Stokes and anti-Stokes sidebands due to resonant stimulated Raman excitation of the spin coherence. Detailed discussion of a systematic study and the corresponding theory of the lineshape profile will be given elsewhere [25]. Central to this discussion are the two sidebands showing that resonant trion excitation leads to optical coupling of the spin states as expected in the Voigt geometry and because of the large splitting (relative to the trion linewidth) there is no contribution from spontaneously generated spin coherence [21]. The central narrow feature characterizes a slow component associated with state relaxation (not decoherence) [22–24]. Careful study of the prominent feature in the inset of Fig. 2(a) suggests the width of the peak is determined by the rate of spin spectral diffusion [25]. However, the theory then predicts a narrower feature due to the slow process of spin flip relaxation should also be present. The width of this feature is expected to be in the millisecond to microsecond range [1–3], below the resolution limit determined by interlaser jitter. Typically, then, the method of correlated optical fields technique would be used to obtain T1 [26]. However, significant noise due to scattering
J. Cheng et al. / Solid State Communications 140 (2006) 381–385
from the sample surfaces leading to heterodyning noise on the detector complicates these measurements. To circumvent this problem, we extended the method of phase modulation spectroscopy [27] to coherent nonlinear optical spectroscopy. 2 The pump beam (E 1 ) is intensity-modulated [I1 = E 1,0 (1 + cos ωm t)2 ] through a traveling-wave acoustic-optical modulator (AOM), where E 1,0 is the complex electrical field in the absence of modulation and the modulation frequency ωm can range from DC to 2π × 105 rad/s. The electron spin dynamics are probed by an un-modulated probe field (E 2 ) with the same optical frequency as E 1 . The nonlinear signal field is homodyne detected with the transmitted co-propagating probe beam E 2 which gives an effectively background free signal. This signal is extracted by a phase-sensitive lock-in amplifier at the frequency of the pump modulation ωm . For optically thin samples, the homodyne detected signal (3) field has the form of E N L = i z 0 k0 PN L /2ε where the third (3) order induced optical polarization PN L = χ (3) : E∗1 E1 E2 . In the above expression, k0 , z 0 , ε, and χ (3) are the signal field wavevector, the sample thickness, dielectric constant, and the third order nonlinear susceptibility, respectively. Because of the intensity modulation of E 1 , the nonlinear signal field will also exhibit a similar modulation pattern, but with a phase shift φm with respect to that of the pump field. We shall establish below that this phase shift originates from the spin relaxation process and serves as the foundation of this frequency domain approach to the measurement of the spin T1 time. For a fixed modulation frequency ωm , the nonlinear signal is detected as the optical frequencies of the pump and probe fields are scanned together. When the laser energy is tuned across trion state resonances, the phase sensitive detection with a lockin amplifier gives both X channel (the in-phase component ∝ cos ωm t) and Y channel signals (the out-of-phase component ∝ sin ωm t). The phase angle shift φm is calculated with its tangent (tan φm ) being the ratio of the Y and X channel signals at the ensemble trion peak. The modulation frequency is then scanned from DC to 2π × 105 rad/s and φm is measured as a function of ωm as shown in Fig. 2(b) at 3.3 T and temperature of 4.5 K. A powerful feature of phase-modulation spectroscopy is that it is less sensitive (i.e., no phase shift will be induced) to relaxation processes much faster than the modulation rate (e.g., the fast spin transverse decoherence rate 1/T2∗ and the trion decoherence and relaxation rates γt and Γt ), and sensitive only to those relaxation rates on the same order or slower than the modulation rate. As shown in Fig. 2(b), the biphasic tan φm response is due to two different timescales. When ωm < 103 rad/s, tan φm is almost a constant close to zero. The decrease of tan φm from zero to negative values in the range from 103 to 2 × 104 rad/s is due to a slow relaxation rate. The value of tan φm reaches its singularity point at ωm ≈ 2 × 104 rad/s, where its minimum (−∞) and maximum (+∞) are reached. A relatively fast relaxation rate begins to affect the phase φm at ωm > 2×104 rad/s, and the value of tan φm slowly drops back down to zero at larger ωm . As we will see below, this
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faster rate can be unambiguously associated with the spin flip process. The physical picture for this phase-modulation spectroscopy technique is succinctly illustrated by the solution to the equation of motion, n(t) ˙ = −Γ n(t) + A(1 + cos ωm t) A n(t) = n 0 + 2 (Γ cos ωm t + ωm sin ωm t) ωm + Γ 2
(1) (2)
where the quantity n exhibits a response with a phase shift of φm = tan−1 (−ωm /Γ ) with respect to the amplitude modulated driving term. This phase shift effect is most pronounced when the modulation frequency ωm is comparable to the damping rate Γ . If ωm Γ , φm would vanish and the oscillatory part of n(t) follows instantaneously with that of the driving. On the other hand, both oscillatory components of n(t) diminish when ωm Γ . To understand the relationship between the spin relaxation rate and the phase angle, we consider the pump and probe dynamics of the three-level Λ-system shown in Fig. 1(b). Using perturbation theory, the equation of motion for the second (2) order optically created population ρσ,σ in spin state |σ i has a (2) similar form to Eq. (1), where n is replaced by ρσ,σ , Γ is replaced by the spin relaxation rate Γs , and the driving term is (2) (1) replaced by ρt−,t− and E i ρt−,σ (i = 1 or 2 depending on the optical perturbation path). Both driving terms have an oscillatory part that instantaneously follows the pump modulation (∝ cos ωm t) because of the fast trion relaxation and decoherence rates (Γt , γt ωm ) associated with them. Additional terms are present depending on the various optical perturbation (2) paths. The solution to ρσ,σ (t) is then of a similar form to Eq. (2) which exhibits a phase-shifted response to the pump modulation if ωm is comparable with the spin relaxation rate Γs . (3) The third order optical polarization ρt−,σ instantaneously fol(2) (2) lows its driving term ∝ E i (ρσ,σ − ρt−,t− ). The second order (2) trion population ρt−,t− is by a factor of Γt /Γs smaller than the (2) second order spin population ρσ,σ . Therefore, the third order nonlinear signal field reflects the second order population on the spin states, which exhibits the phase shifted response due to the spin relaxation process. The full solution to the third order induced optical polarization in the three-level Λ-system, after averaging over the quantum dot ensemble, is, √ E 4 |E 1 |2 E 2∗ i N0 π |µ| ωc2 (3) ∼ PN L = constant + 4γt2 + ωc2 4 h¯ 3 γt ∆inh 2 /2Γ 2ωm 6Γs + ωm s t) + t) (3) · cos(ω sin(ω m m 2 + 4Γ 2 2 + 4Γ 2 ωm ωm s s where we have omitted the component that oscillates with 2ωm as the lock-in amplifier is set at frequency ωm . The first constant term will not be detected by the lock-in amplifier as well. In Eq. (3), µ E is the electrical transition dipole moment for |x±i → |t−i, h¯ ωc is the spin Zeeman splitting, N0 is the total number of QDs in the ensemble, and ∆inh is the inhomogeneous broadening width of the trion states in the QDs ensemble.
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Fig. 3. (a), Spin relaxation rate versus magnetic field at T = 4.5 K, where the arrow indicates the data point shown in Fig. 1(b). (b), The spin relaxation rate (Γs ) as a function of temperature at 3.3 T () and 6.6 T (•), respectively. The solid (dashed) line is the linear fitting curve for B = 3.3 T (6.6 T). The error bars in both (a) and (b) are from the standard deviation with the fitting equation.
In the case of interest, as the pump and probe fields are (3) degenerate in frequency, the dominant contribution to PN L is from the process where both fields couple resonantly to the transition |σ i → |t−i. With the pump field depleting the spin population on state |σ i, this process corresponds to a reduced absorption of the probe field and hence the differential transmission (DT) signal is positive. This contribution to the DT signal from the three-level Λ-system is in agreement with the positive tan φm measured at ωm > 2 × 104 rad/s, as shown in Fig. 2(b). The existence of another slower process is implied from the negative tan φm at low modulation frequency. While the physical mechanism for this slower process is not the focus of the present work, it is believed to be caused by a pump induced red-shift effect arising from the photovoltaic effect [28] or thermomodulation effect [29]. The same phenomenon is observed and reported on modulation spectroscopy of neutral dots at low modulation frequency [30]. The value of tan φm depends on the total third order optical polarization arising from those two distinct contributions, m − k1 k2 ω22ω +4Γ 2
tan φm = −
m
k2 where k2 ≡ |E 1 |2 |E 2 |2
s
2 /2Γ 6Γs +ωm s 2 +4Γ 2 ωm s
ωm 2 +Γ 2 ωm 1
− k1
Γ1 2 +Γ 2 ωm 1
√ ωc2 E 4 |E 1 |2 |E 2 |2 N0 π|µ| . 4 h¯ 3 γt ∆inh 4γt2 +ωc2
(4)
The values of k1 ∝
and Γ1 are the magnitude and the corresponding damping rate of the red-shift process from a phenomenological theory, respectively. The domination of the red-shift process at the low ωm gives rise to the negative tan φm but is suppressed at higher ωm where the dominance is taken over by the dynamics of the three-level Λ-system in the charged dots (Eq. (3)), resulting in a positive tan φm . In the intermediate region where the two mechanisms are comparable in magnitude, the denominator in Eq. (4) vanishes, explaining the divergence of tan φm in Fig. 2(b) at ωm ≈ 2 × 104 rad/s. The fitting of the data in Fig. 2(b) with Eq. (4) gives the slow relaxation rate Γ1 = (4.3 ± 0.2) × 103 s−1 and the spin relaxation rate Γs = (4.6 ± 1.2) × 104 s−1 , as shown by the solid curve in Fig. 2(b). To study the dependence on the field magnitude, we extract Γs at various magnetic fields by keeping the temperature fixed at 4.5 K and the data is plotted in Fig. 3(a). The spin relaxation rate increases nonlinearly with the applied magnetic field, and can be fitted by a power law
Γs = Γs,0 + α B 4 , which gives α = 119 ± 10 s−1 T−4 and Γs,0 = (2.9±0.8)×104 s−1 corresponding to T1 = 34±7 µs at zero field. The spin relaxation rate is also studied as a function of temperature with keeping magnetic field fixed at 3.3 T and 6.6 T, as shown in Fig. 3(b). To understand the cause of the spin relaxation, we compare the experimental results with the available theories. At a temperature of a few degrees Kelvin or less and under a field of ∼1–10 T, spin flip by the Dresselhaus spin–orbit coupling mediated by the scattering of one acoustic piezophonon is shown in theory to be most relevant [2,3], (gµ B B) ∼ kB T (5) = λp h¯ (h¯ ω0 )4 where λ p is a dimensionless constant related to the piezoelectric phonon coupling strength and h¯ ω0 is the QD electronic level spacing. We have included the finite temperature factor N + 1(N ) for phonon emission (absorption), N = 1/(e gµ B B/k B T − 1) ≈ k B T /gµ B B, where the last step comes from the fact that k B T is much larger than the Zeeman splitting in our piezo experiment. This factor changes the field dependence of Γs 5 4 from B at T = 0 K to B at a few degrees Kelvin, and also leads to a linear dependence on temperature. piezo We obtain Γs ≈ 2 × 101 B 4 T from Eq. (5) by taking λ p = 4 × 10−2 as deduced from typical material properties of GaAs [2,3], h¯ ω0 ≈ 1 meV in a typical fluctuation dot, and g = 0.13 from a previous study [20,21]. At T = 4.5 K, we have piezo Γs ≈ 1 × 102 B 4 , which is in good agreement with the field dependence from the fitting in Fig. 3(a). Similar superlinear dependence of spin-flip relaxation rate on magnetic field has been reported in pulsed electrical gate confined GaAs [16] and Ga(In)As QDs [17]. The temperature dependence of spin relaxation rate can be obtained from Eq. (5) by plugging in the field piezo strength and Γs ≈ 2.4 × 103 T at 3.3 T which is in good agreement with the linear fitting Γs = (3.3 ± 1) × 104 + (2.9 ± 0.7) × 103 T (solid line) in piezo Fig. 3(b). At 6.6 Tesla, Eq. (5) becomes Γs ≈ 3.8 × 104 T , which is also qualitatively consistent with the linear fitting equation Γs = (1.0 ± 0.5) × 105 + (1.6 ± 0.4) × 104 T , shown as a dashed line in Fig. 3(b). The extrapolated Γs,0 at vanishing magnetic field could be affected by a field independent two-phonon scattering process piezo
Γs
4
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studied in [3] or due to the hyperfine interaction with the lattice nuclei [4]. Further experimental investigations are needed to distinguish between the two cases. In conclusion, the spin-flip relaxation rate between Zeeman sublevels in charged GaAs QDs is measured by nonlinear optical phase-modulation techniques for various magnetic field and temperature values. From the experimental studies, we find that Γs strongly depends on the magnetic field strength and temperature. Spin-flip relaxation resulting from the Dresselhaus spin–orbit coupling mechanism has been discussed, and compared with our experimental results. The long spin-flip relaxation time promises long spin decoherence time under low magnetic field and modest temperature, which is particularly attractive for quantum computing and quantum information processing. Acknowledgments This work was supported in part by the ARO, ONR, NSA/ARDA, DARPA, and FOCUS-NSF. References [1] A.V. Khaetskii, Y.V. Nazarov, Phys. Rev. B 61 (1997) 12639. [2] A.V. Khaetskii, Y.V. Nazarov, Phys. Rev. B 64 (2001) 125316. [3] L.M. Woods, T.L. Reinecke, Y. Lyanda-Geller, Phys. Rev. B 66 (2002) 161318. [4] A.V. Khaetskii, D. Loss, L. Glazman, Phys. Rev. Lett. 88 (2002) 186802. [5] I.A. Merkulov, A.L. Efros, M. Rosen, Phys. Rev. B 65 (2002) 205309. [6] S.I. Erlingsson, Y.V. Nazarov, V.I. Fal’ko, Phys. Rev. B 64 (2001) 195306. [7] V.A. Abalmassov, R. Marquardt, Phys. Rev. B 70 (2004) 075313.
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[8] A. Imamoglu, D.D. Awschalom, G. Burkard, D.P. DiVincenzo, D. Loss, M. Sherwin, A. Small, Phys. Rev. Lett. 83 (1999) 4204. [9] P. Chen, C. Piermarocchi, L.J. Sham, D. Gammon, D.G. Steel, Phys. Rev. B 69 (2004) 075320. [10] P. Chen, C. Piermarocchi, L.J. Sham, Phys. Rev. Lett. 87 (2001) 067401. [11] T. Calarco, A. Datta, P. Fedichev, E. Pazy, P. Zoller, Phys. Rev. A 68 (2003) 012310. [12] S.C. Benjamin, B.W. Lovett, J.H. Reina, Phys. Rev. A 70 (2004) 060305. [13] J.M. Smith, P.A. Dalgarno, R.J. Warburton, A.O. Govorov, K. Karrai, B.D. Gerardot, P.M. Petroff, Phys. Rev. Lett. 94 (2005) 197402. [14] T. Fujisawa, D.G. Austing, Y. Tokura, Y. Hirayama, S. Tarucha, Nature 419 (2002) 278. [15] R. Hanson, B. Witkamp, L.M.K. Vandersypen, L.H. Willems van Beveren, J.M. Elzerman, L.P. Kouwenhoven, Phys. Rev. Lett. 91 (2003) 196802. [16] J.M. Elzerman, R. Hanson, L.H. Willems van Beveren, B. Witkamp, L.M.K. Vandersypen, L.P. Kouwenhoven, Nature 430 (2004) 431. [17] M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, J.J. Finley, Nature 432 (2004) 81. [18] D. Loss, D.P. DiVincenzo, Phys. Rev. A 57 (1998) 120. [19] G. Dresselhaus, Phys. Rev. 100 (1955) 580. [20] J.G. Tischler, A.S. Bracker, D. Gammon, D. Park, Phys. Rev. B 66 (2002) 081310. [21] M.V.G. Dutt, J. Cheng, B. Li, X. Xu, X. Li, P.R. Berman, D.G. Steel, A.S. Bracker, D. Gammon, S.E. Economou, R. Liu, L.J. Sham, Phys. Rev. Lett. 94 (2005) 227403. [22] H. Wang, M. Jiang, D.G. Steel, Phys. Rev. Lett. 65 (1990) 1255. [23] H. Wang, D.G. Steel, Phys. Rev. A 43 (1991) 3823. [24] H. Wang, M. Jiang, R. Merlin, D.G. Steel, Phys. Rev. Lett. 69 (1992) 804. [25] J. Cheng, X. Xu, D.G. Steel, A.S. Bracker, D. Gammon, W. Yao, L.J. Sham, in preparation. [26] D.G. Steel, S.C. Rand, Phys. Rev. Lett. 55 (1985) 2285. [27] E.A. Bailey Jr., G.K. Rollefson, J. Chem. Phys. 21 (1953) 1315. [28] W. Heller, U. Bockelmann, G. Abstreiter, Phys. Rev. B 57 (1998) 6270. [29] R.R.L. Zucca, Y.R. Shen, Phys. Rev. B 1 (1970) 2668. [30] N.H. Bonadeo, Ph.D. Thesis, The University of Michigan, 1999.
Spin relaxation in charged quantum dots measured by coherent optical phase modulation spectroscopy Jun Cheng a , Yanwen Wu a , Xiaodong Xu a , Dong Sun a , D.G. Steel a,∗ , A.S. Bracker b , D. Gammon b , Wang Yao c , L.J. Sham c a FOCUS, H. M. Randall Laboratory of Physics, The University of Michigan, Ann Arbor, MI 48109, USA b The Naval Research Laboratory, Washington DC 20375, USA c Department of Physics, The University of California, San Diego, La Jolla, CA 92093, USA
Received 23 August 2006; accepted 24 August 2006 by R. Merlin Available online 14 September 2006
Abstract Measurements of the electron spin relaxation rate (T1 ) in charged GaAs quantum dots are made using a new nonlinear optical phase-modulation spectroscopy technique. The measured results of T1 as a function of magnetic field and temperature are best explained in terms of the spin relaxation by the phonon-assisted Dresselhaus spin–orbit scattering. T1 approaches 34 µs at zero magnetic field at 4.5 K. c 2006 Elsevier Ltd. All rights reserved.
PACS: 39.30.+w; 42.65.-k; 72.25.Rb Keywords: A. Quantum dots; D. Spin dynamics; E. Phase modulation; E. Nonlinear spectroscopies
Interest in electron spin relaxation has intensified in recent years, in part because the new developments in nano-structures have led to the potential of reduced spin relaxation rates for application to spin-based devices and the observation of new spin relaxation physics [1–7]. Moreover, several all-optically manipulated spin schemes using stimulated Raman excitation have been proposed [8,9] where the trion state provides a strong optical enhancement through resonant coupling. These schemes take full advantage of femtosecond technology and flexible optical shaping for all-optical manipulations [10]. All-optical studies of electron spin relaxation in quantum dots (QDs) are important for developing an optically driven spinbased quantum computing scheme [11,12]. In addition, such measurements avoid the complications in spin physics which arise when the dots are too close to electric gates, e.g., Kondolike interaction of the dot spin with an electron Fermi sea in the electrode [13]. ∗ Corresponding address: The H. M. Randall Laboratory of Physics, 450 Church Street, Ann Arbor, MI 48109, USA. E-mail addresses: [email protected] (J. Cheng), [email protected] (D.G. Steel).
c 2006 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2006.08.035
Spin relaxation in QDs is greatly suppressed because the quantization of the electron energy levels results in the difficulty to simultaneously satisfy energy and momentum conservation. Among the various mechanisms leading to spin relaxation in QDs, spin–orbit interactions assisted by phonons have been predicted to be the dominant mechanism [1–3]. In recent experiments on electric gate confined QDs [14–16] and electric gate aided self-assembled QDs [17], spin–orbit coupling is shown to be the most significant contribution to spin relaxation under magnetic fields. In this article, we present all-optical measurements of spin relaxation in fluctuation QDs using a new kind of coherent nonlinear optical hole burning spectroscopy based on phase modulation. The measurements lay the groundwork for the alloptical addressing and the manipulation of the electron spin in a QD system. Measurements are performed as a function of magnetic field (typical in design of quantum logic operations [8, 9,18]) and temperature, leading to the conclusion that the Dresselhaus spin–orbit coupling with phonon scattering [19] is the dominant process in spin relaxation in QDs. The sample consists of an ensemble of QDs formed by ˚ wide GaAs QW layer. Electrons interface fluctuations in a 42 A
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Fig. 1. (a) The experiment setup for nonlinear optical phase-modulation spectroscopy. (b) Energy level diagram of a negatively charged QD in Voigt configuration, with ground states |x±i denoting electron spin projections ±1/2 along the x-axis. The trion states |t±i are labeled by the heavy-hole angular momentum projection ±3/2 along the growth axis (z). Solid (dashed) lines denote transitions excited by σ − (σ + ) light. If co-circularly polarized pump and probe fields (σ − ) are used, the relevant states are reduced to a three-level Λ-system enclosed in the dashed box.
Fig. 2. (a) The center sharp spike and two symmetric sidebands on the top of broad trion spectral hole burning in the nearly degenerate DT experiment at B = 3.3 T, where the pump is fixed at the lower monolayer trion ensemble center, as indicated by the arrow. The inset shows the zoomed in sharp spike at ∆ ∼ 0, where the spin relaxation dynamics can be examined and broadened by a spectral diffusion process. (b) tan φm at various circular modulation frequency (ωm ) at B = 3.3 T and temperature is 4.5 K. The solid line is a fitted function (Eq. (4)) by taking two different relaxation processes into account.
are incorporated into the dots by modulation of Si doping in the barrier [20]. In the Voigt geometry with the application of a magnetic field along the QW plane (x-axis), the energy level diagram of the negatively charged QD is schematically shown in Fig. 1(b). The transitions from the electron spin states |x±i to the charged exciton (trion) state |t−i (|t+i) are coupled by σ − (σ + ) polarized light according to the selection rules [21]. The relevant level scheme is reduced to the Λ-system (enclosed in the dashed box in Fig. 1(b)) since optical fields involved are all σ − polarized in the experiment. Spin relaxation rates are determined by analyzing the phase angle shift following amplitude modulation of the pump beam. Earlier use of ultra-high resolution coherent nonlinear spectroscopy techniques showed the important role of spectral diffusion of excitonic energy as well as spin flip induced hole burning in quantum well structures [22–24]. The earlier measurements were based on conventional four-wave mixing. In the present measurements, a standard continuous wave (CW) pump and probe setup is used based on the frequency stabilized laser, as shown in Fig. 1(a). Since both the pump and probe fields are mutually coherent, the conventional differential transmission setup actually measures the homodyne detected four-wave mixing response based on the third order nonlinear optical susceptibility.
Fig. 2(a) shows the large scale non-degenerate nonlinear optical response showing a complex structure: an overall broad structure due to spectral hole burning and spectral diffusion of the inhomogeneously broadened trion resonance, a central sharp resonance at the peak, and the Stokes and anti-Stokes sidebands due to resonant stimulated Raman excitation of the spin coherence. Detailed discussion of a systematic study and the corresponding theory of the lineshape profile will be given elsewhere [25]. Central to this discussion are the two sidebands showing that resonant trion excitation leads to optical coupling of the spin states as expected in the Voigt geometry and because of the large splitting (relative to the trion linewidth) there is no contribution from spontaneously generated spin coherence [21]. The central narrow feature characterizes a slow component associated with state relaxation (not decoherence) [22–24]. Careful study of the prominent feature in the inset of Fig. 2(a) suggests the width of the peak is determined by the rate of spin spectral diffusion [25]. However, the theory then predicts a narrower feature due to the slow process of spin flip relaxation should also be present. The width of this feature is expected to be in the millisecond to microsecond range [1–3], below the resolution limit determined by interlaser jitter. Typically, then, the method of correlated optical fields technique would be used to obtain T1 [26]. However, significant noise due to scattering
J. Cheng et al. / Solid State Communications 140 (2006) 381–385
from the sample surfaces leading to heterodyning noise on the detector complicates these measurements. To circumvent this problem, we extended the method of phase modulation spectroscopy [27] to coherent nonlinear optical spectroscopy. 2 The pump beam (E 1 ) is intensity-modulated [I1 = E 1,0 (1 + cos ωm t)2 ] through a traveling-wave acoustic-optical modulator (AOM), where E 1,0 is the complex electrical field in the absence of modulation and the modulation frequency ωm can range from DC to 2π × 105 rad/s. The electron spin dynamics are probed by an un-modulated probe field (E 2 ) with the same optical frequency as E 1 . The nonlinear signal field is homodyne detected with the transmitted co-propagating probe beam E 2 which gives an effectively background free signal. This signal is extracted by a phase-sensitive lock-in amplifier at the frequency of the pump modulation ωm . For optically thin samples, the homodyne detected signal (3) field has the form of E N L = i z 0 k0 PN L /2ε where the third (3) order induced optical polarization PN L = χ (3) : E∗1 E1 E2 . In the above expression, k0 , z 0 , ε, and χ (3) are the signal field wavevector, the sample thickness, dielectric constant, and the third order nonlinear susceptibility, respectively. Because of the intensity modulation of E 1 , the nonlinear signal field will also exhibit a similar modulation pattern, but with a phase shift φm with respect to that of the pump field. We shall establish below that this phase shift originates from the spin relaxation process and serves as the foundation of this frequency domain approach to the measurement of the spin T1 time. For a fixed modulation frequency ωm , the nonlinear signal is detected as the optical frequencies of the pump and probe fields are scanned together. When the laser energy is tuned across trion state resonances, the phase sensitive detection with a lockin amplifier gives both X channel (the in-phase component ∝ cos ωm t) and Y channel signals (the out-of-phase component ∝ sin ωm t). The phase angle shift φm is calculated with its tangent (tan φm ) being the ratio of the Y and X channel signals at the ensemble trion peak. The modulation frequency is then scanned from DC to 2π × 105 rad/s and φm is measured as a function of ωm as shown in Fig. 2(b) at 3.3 T and temperature of 4.5 K. A powerful feature of phase-modulation spectroscopy is that it is less sensitive (i.e., no phase shift will be induced) to relaxation processes much faster than the modulation rate (e.g., the fast spin transverse decoherence rate 1/T2∗ and the trion decoherence and relaxation rates γt and Γt ), and sensitive only to those relaxation rates on the same order or slower than the modulation rate. As shown in Fig. 2(b), the biphasic tan φm response is due to two different timescales. When ωm < 103 rad/s, tan φm is almost a constant close to zero. The decrease of tan φm from zero to negative values in the range from 103 to 2 × 104 rad/s is due to a slow relaxation rate. The value of tan φm reaches its singularity point at ωm ≈ 2 × 104 rad/s, where its minimum (−∞) and maximum (+∞) are reached. A relatively fast relaxation rate begins to affect the phase φm at ωm > 2×104 rad/s, and the value of tan φm slowly drops back down to zero at larger ωm . As we will see below, this
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faster rate can be unambiguously associated with the spin flip process. The physical picture for this phase-modulation spectroscopy technique is succinctly illustrated by the solution to the equation of motion, n(t) ˙ = −Γ n(t) + A(1 + cos ωm t) A n(t) = n 0 + 2 (Γ cos ωm t + ωm sin ωm t) ωm + Γ 2
(1) (2)
where the quantity n exhibits a response with a phase shift of φm = tan−1 (−ωm /Γ ) with respect to the amplitude modulated driving term. This phase shift effect is most pronounced when the modulation frequency ωm is comparable to the damping rate Γ . If ωm Γ , φm would vanish and the oscillatory part of n(t) follows instantaneously with that of the driving. On the other hand, both oscillatory components of n(t) diminish when ωm Γ . To understand the relationship between the spin relaxation rate and the phase angle, we consider the pump and probe dynamics of the three-level Λ-system shown in Fig. 1(b). Using perturbation theory, the equation of motion for the second (2) order optically created population ρσ,σ in spin state |σ i has a (2) similar form to Eq. (1), where n is replaced by ρσ,σ , Γ is replaced by the spin relaxation rate Γs , and the driving term is (2) (1) replaced by ρt−,t− and E i ρt−,σ (i = 1 or 2 depending on the optical perturbation path). Both driving terms have an oscillatory part that instantaneously follows the pump modulation (∝ cos ωm t) because of the fast trion relaxation and decoherence rates (Γt , γt ωm ) associated with them. Additional terms are present depending on the various optical perturbation (2) paths. The solution to ρσ,σ (t) is then of a similar form to Eq. (2) which exhibits a phase-shifted response to the pump modulation if ωm is comparable with the spin relaxation rate Γs . (3) The third order optical polarization ρt−,σ instantaneously fol(2) (2) lows its driving term ∝ E i (ρσ,σ − ρt−,t− ). The second order (2) trion population ρt−,t− is by a factor of Γt /Γs smaller than the (2) second order spin population ρσ,σ . Therefore, the third order nonlinear signal field reflects the second order population on the spin states, which exhibits the phase shifted response due to the spin relaxation process. The full solution to the third order induced optical polarization in the three-level Λ-system, after averaging over the quantum dot ensemble, is, √ E 4 |E 1 |2 E 2∗ i N0 π |µ| ωc2 (3) ∼ PN L = constant + 4γt2 + ωc2 4 h¯ 3 γt ∆inh 2 /2Γ 2ωm 6Γs + ωm s t) + t) (3) · cos(ω sin(ω m m 2 + 4Γ 2 2 + 4Γ 2 ωm ωm s s where we have omitted the component that oscillates with 2ωm as the lock-in amplifier is set at frequency ωm . The first constant term will not be detected by the lock-in amplifier as well. In Eq. (3), µ E is the electrical transition dipole moment for |x±i → |t−i, h¯ ωc is the spin Zeeman splitting, N0 is the total number of QDs in the ensemble, and ∆inh is the inhomogeneous broadening width of the trion states in the QDs ensemble.
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J. Cheng et al. / Solid State Communications 140 (2006) 381–385
Fig. 3. (a), Spin relaxation rate versus magnetic field at T = 4.5 K, where the arrow indicates the data point shown in Fig. 1(b). (b), The spin relaxation rate (Γs ) as a function of temperature at 3.3 T () and 6.6 T (•), respectively. The solid (dashed) line is the linear fitting curve for B = 3.3 T (6.6 T). The error bars in both (a) and (b) are from the standard deviation with the fitting equation.
In the case of interest, as the pump and probe fields are (3) degenerate in frequency, the dominant contribution to PN L is from the process where both fields couple resonantly to the transition |σ i → |t−i. With the pump field depleting the spin population on state |σ i, this process corresponds to a reduced absorption of the probe field and hence the differential transmission (DT) signal is positive. This contribution to the DT signal from the three-level Λ-system is in agreement with the positive tan φm measured at ωm > 2 × 104 rad/s, as shown in Fig. 2(b). The existence of another slower process is implied from the negative tan φm at low modulation frequency. While the physical mechanism for this slower process is not the focus of the present work, it is believed to be caused by a pump induced red-shift effect arising from the photovoltaic effect [28] or thermomodulation effect [29]. The same phenomenon is observed and reported on modulation spectroscopy of neutral dots at low modulation frequency [30]. The value of tan φm depends on the total third order optical polarization arising from those two distinct contributions, m − k1 k2 ω22ω +4Γ 2
tan φm = −
m
k2 where k2 ≡ |E 1 |2 |E 2 |2
s
2 /2Γ 6Γs +ωm s 2 +4Γ 2 ωm s
ωm 2 +Γ 2 ωm 1
− k1
Γ1 2 +Γ 2 ωm 1
√ ωc2 E 4 |E 1 |2 |E 2 |2 N0 π|µ| . 4 h¯ 3 γt ∆inh 4γt2 +ωc2
(4)
The values of k1 ∝
and Γ1 are the magnitude and the corresponding damping rate of the red-shift process from a phenomenological theory, respectively. The domination of the red-shift process at the low ωm gives rise to the negative tan φm but is suppressed at higher ωm where the dominance is taken over by the dynamics of the three-level Λ-system in the charged dots (Eq. (3)), resulting in a positive tan φm . In the intermediate region where the two mechanisms are comparable in magnitude, the denominator in Eq. (4) vanishes, explaining the divergence of tan φm in Fig. 2(b) at ωm ≈ 2 × 104 rad/s. The fitting of the data in Fig. 2(b) with Eq. (4) gives the slow relaxation rate Γ1 = (4.3 ± 0.2) × 103 s−1 and the spin relaxation rate Γs = (4.6 ± 1.2) × 104 s−1 , as shown by the solid curve in Fig. 2(b). To study the dependence on the field magnitude, we extract Γs at various magnetic fields by keeping the temperature fixed at 4.5 K and the data is plotted in Fig. 3(a). The spin relaxation rate increases nonlinearly with the applied magnetic field, and can be fitted by a power law
Γs = Γs,0 + α B 4 , which gives α = 119 ± 10 s−1 T−4 and Γs,0 = (2.9±0.8)×104 s−1 corresponding to T1 = 34±7 µs at zero field. The spin relaxation rate is also studied as a function of temperature with keeping magnetic field fixed at 3.3 T and 6.6 T, as shown in Fig. 3(b). To understand the cause of the spin relaxation, we compare the experimental results with the available theories. At a temperature of a few degrees Kelvin or less and under a field of ∼1–10 T, spin flip by the Dresselhaus spin–orbit coupling mediated by the scattering of one acoustic piezophonon is shown in theory to be most relevant [2,3], (gµ B B) ∼ kB T (5) = λp h¯ (h¯ ω0 )4 where λ p is a dimensionless constant related to the piezoelectric phonon coupling strength and h¯ ω0 is the QD electronic level spacing. We have included the finite temperature factor N + 1(N ) for phonon emission (absorption), N = 1/(e gµ B B/k B T − 1) ≈ k B T /gµ B B, where the last step comes from the fact that k B T is much larger than the Zeeman splitting in our piezo experiment. This factor changes the field dependence of Γs 5 4 from B at T = 0 K to B at a few degrees Kelvin, and also leads to a linear dependence on temperature. piezo We obtain Γs ≈ 2 × 101 B 4 T from Eq. (5) by taking λ p = 4 × 10−2 as deduced from typical material properties of GaAs [2,3], h¯ ω0 ≈ 1 meV in a typical fluctuation dot, and g = 0.13 from a previous study [20,21]. At T = 4.5 K, we have piezo Γs ≈ 1 × 102 B 4 , which is in good agreement with the field dependence from the fitting in Fig. 3(a). Similar superlinear dependence of spin-flip relaxation rate on magnetic field has been reported in pulsed electrical gate confined GaAs [16] and Ga(In)As QDs [17]. The temperature dependence of spin relaxation rate can be obtained from Eq. (5) by plugging in the field piezo strength and Γs ≈ 2.4 × 103 T at 3.3 T which is in good agreement with the linear fitting Γs = (3.3 ± 1) × 104 + (2.9 ± 0.7) × 103 T (solid line) in piezo Fig. 3(b). At 6.6 Tesla, Eq. (5) becomes Γs ≈ 3.8 × 104 T , which is also qualitatively consistent with the linear fitting equation Γs = (1.0 ± 0.5) × 105 + (1.6 ± 0.4) × 104 T , shown as a dashed line in Fig. 3(b). The extrapolated Γs,0 at vanishing magnetic field could be affected by a field independent two-phonon scattering process piezo
Γs
4
J. Cheng et al. / Solid State Communications 140 (2006) 381–385
studied in [3] or due to the hyperfine interaction with the lattice nuclei [4]. Further experimental investigations are needed to distinguish between the two cases. In conclusion, the spin-flip relaxation rate between Zeeman sublevels in charged GaAs QDs is measured by nonlinear optical phase-modulation techniques for various magnetic field and temperature values. From the experimental studies, we find that Γs strongly depends on the magnetic field strength and temperature. Spin-flip relaxation resulting from the Dresselhaus spin–orbit coupling mechanism has been discussed, and compared with our experimental results. The long spin-flip relaxation time promises long spin decoherence time under low magnetic field and modest temperature, which is particularly attractive for quantum computing and quantum information processing. Acknowledgments This work was supported in part by the ARO, ONR, NSA/ARDA, DARPA, and FOCUS-NSF. References [1] A.V. Khaetskii, Y.V. Nazarov, Phys. Rev. B 61 (1997) 12639. [2] A.V. Khaetskii, Y.V. Nazarov, Phys. Rev. B 64 (2001) 125316. [3] L.M. Woods, T.L. Reinecke, Y. Lyanda-Geller, Phys. Rev. B 66 (2002) 161318. [4] A.V. Khaetskii, D. Loss, L. Glazman, Phys. Rev. Lett. 88 (2002) 186802. [5] I.A. Merkulov, A.L. Efros, M. Rosen, Phys. Rev. B 65 (2002) 205309. [6] S.I. Erlingsson, Y.V. Nazarov, V.I. Fal’ko, Phys. Rev. B 64 (2001) 195306. [7] V.A. Abalmassov, R. Marquardt, Phys. Rev. B 70 (2004) 075313.
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