Jul 16, 2013 - Spin Noise Spectroscopy (SNS) is an experimental approach to obtain correlators of ... where Ï is the vector of Pauli matrices, H is t...

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Yuriy V. Pershin Department of Physics and Astronomy, University of South Carolina, Columbia, SC, 29208 USA

arXiv:1307.4426v1 [cond-mat.mes-hall] 16 Jul 2013

Valeriy A. Slipko Department of Physics and Astronomy, University of South Carolina, Columbia, SC, 29208 and Department of Physics and Technology, V. N. Karazin Kharkov National University, Kharkov 61077, Ukraine

N. A. Sinitsyn∗ Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545 USA (Dated: November 9, 2018) Spin Noise Spectroscopy (SNS) is an experimental approach to obtain correlators of mesoscopic spin fluctuations in time by purely optical means. We explore the information that this technique can provide when it is applied to a weakly non-equilibrium regime when an electric current is driven through a sample by an electric field. We find that the noise power spectrum of conducting electrons experiences a shift, which is proportional to the strength of the spin-orbit coupling for electrons moving along the electric field direction. We propose applications of this effect to measurements of spin orbit coupling anisotropy and separation of spin noise of conducting and localized electrons.

Introduction. Optical spin noise spectroscopy (SNS) [1] has been recently introduced as a promising approach for probing local fluctuations of spins in semiconducting materials [2–8], atomic gases [9–14] and quantum dots [15–17]. In particular, if Sz (t) is the time-dependent polarization of spins in a mesoscopic region of a semiconductor, SNS can be used to obtain the spin noise power spectrum [18–23]: Z ∞ P (ω) = 2 dt cos (ωt)hSz (t)Sz (0)i. (1) 0

The advantage of the optical SNS over the other measurement approaches (e.g., optical pump-probe [24, 25] or STM measurements of a single spin [26]) is usually associated with the minimal energy dissipated, i.e., it can probe spin dynamics at thermodynamic equilibrium. In addition, the SNS allows accumulation of a large statistics that smoothes out the statistical noise in the data, so one can study subtle details at the tails of spin-spin correlators [16]. Sensitivity of this approach is continuously improving, e.g. the recently introduced Ultrahigh Bandwidth SNS can resolve spin correlations with picosecond resolution [8]. At the thermodynamic equilibrium, the fluctuationdissipation theorem predicts that the knowledge of the spin correlator hSz (t)Sz (0)i, which is obtained by SNS, is formally equivalent to the information that can be obtained from a linear response pump-probe measurements. So, for systems at the thermodynamic equilibrium, SNS probes characteristics that, at least in prin-

ciple, can be obtained from more traditional approaches based on characterizing system’s linear response. In this letter we explore the possibility to apply SNS to semiconductors in a non-equilibrium steady state. The behavior of the spin-spin correlator in a non-equilibrium regime may no longer be the subject of the fluctuationdissipation theorem. Hence, even if a perturbation from the equilibrium is weak (in our case it will be a weak electric field that induces an electric current), identifying non-equilibrium contribution to the spin correlator may provide the information about the system that cannot be obtained from the linear response characteristics. For a demonstration, we consider the effect of an electric field on spin fluctuations of conducting elections in 2D electron gas with Rashba and Dresselhaus spin orbit couplings and spin-independent scatterings. We develop an approach that allows us not only to derive the equations for mean spin polarization [27] but also to relate parameters of spin fluctuations to the shot noise at microscopic scattering events. Stochastic dynamics of spin fluctuations. Consider spin fluctuations from the mean steady state of an electron system. Let ρˆk be the spin density matrix in the momentum space, which is a 2×2 matrix in spin indexes. We assume that the observation region is much larger than the spin diffusion length, so that we can consider dynamics only in the momentum space [28]. The evolution of the spin density matrix in a momentum space volume k is described by the quantum Boltzmann equation: ˆ 0 , ρˆk ] = Iˆcoll , ρˆ˙ k − eE · ∇k ρˆk + i[H

∗

[email protected]

(2)

in which Iˆcoll is the collision term due to elastic scattering ˆ 0 is the scattering-free part of the on impurities, and H

2 ωk,k0 = |Tkk0 |2 /t, we then find

Hamiltonian: 2 ˆ 0 = k − 1 Ωk · σ − 1 H · σ, H 2m 2 2

(3)

hJk0 →k i = ωk,k0 (sk0 − sk ) , (7) µ µ 0 ν 0 ν hJk→k0 (t )Jk1 →k0 (t)i − hJk→k0 (t )ihJk1 →k0 (t)i = 1

1

where σ is the vector of Pauli matrices, H is the inplane magnetic field (with absorbed Bohr magneton and g-factor). The spin orbit coupling field can be written as a sum of two parts Ωk = kx Ω1 + ky Ω2 , with:

ωkk0 fk (1 − fk0 )δ(t − t0 )δµν (δkk1 δk0 k01 − δkk01 δk0 k1 ).(8) Let’s introduce variables describing coarse-grained spin characteristics: X X S0 = sk , S1,2 = sk kx,y , (9) k

Ω1 = 2(β x ˆ − αˆ y ), Ω2 = 2(αˆ x − β yˆ),

(4)

where α and β are strengths of, respectively, Rashba and Dresselhaus couplings. It is convenient to introduce the spin density sµk = Tr[σµ ρˆk ]/2, where µ = x, y, z, so that eq.(2) can be rewritten as s˙ k − eE · ∇k sk = sk × H + sk × Ωk +

X

Jk0 →k , (5)

k0

where Jk0 →k is the stochastic spin current, in the momentum space, due to scattering between states k0 and k. For spin conserving scatterings, Jk0 →k = −Jk→k0 . Consider a scattering channel that connects states at k0 and k. Let ak,↑ , ak,↓ be the annihilation operators of an electron with momentum k for spin up and down, respectively. For such scatterings, without spin-flipping, the evolution of spin-up and spin-down electrons during a small time interval t is described by a scattering matrix [29]: ak,↑/↓ (t) ak,↑/↓ (0) Rkk Tkk0 = , (6) Tk0 k Rk0 k0 ak0 ,↑/↓ (t) ak0 ,↑/↓ (0) where Tkk0 and Rkk are, respectively, time-dependent transmission and reflection amplitudes. The spin operator is defined as ˆsk = 12 Ψ†k σΨk , with Ψk = (ak,↑ , ak,↓ )T . If we assume that scattering is weak, then |Tkk0 |2 = |Tk0 k |2 1, |Rkk |2 = 1 − |Tkk0 |2 . The spin current R t 0 operator0 due to such scatterings is defined by: dt Jˆk0 →k (t ) ≡ ˆsk (t) − ˆsk (0). 0 To determine first two cumulants of the spin current [29], one can take the trace of the first and second powers of ˆsk (t) − ˆsk (0) with the density matrix ρˆst ˆk , k + ρ where ρˆst is the density matrix at the steady state and k ρˆk is due to the currently present spin fluctuation [30]. The steady state density matrix is approximated by a ˆ spin-diagonal matrix ρˆst k ≈ fk 1k , where fk = f (k ) is the Dirac-Fermi distribution over energy k and ˆ1k is a unit matrix in the spin space of fermions with momentum k. This is equivalent to disregarding terms O(E 2 ) and higher order corrections in Ω/F , where F is the Fermi energy, in the final expression for the spin correlator. We should also assume that there are no initial correlations between different phase space volumes and between spin currents at different time moments. Scattering probability, |Tkk0 |2 , is linearly growing with time when energies of k0 and k are the same. Introducing the scattering rate

k

and the transport life time τtr : X 1 = ωk,k0 [1 − cos(ϕ − ϕ0 )], τtr 0

(10)

k

with ϕ and ϕ0 are angles of, respectively, k and k0 taken at the Fermi surface. By summing over k in Eq. (5) we then obtain S˙ 0 = S0 × H + S1 × Ω1 + S2 × Ω2 . (11) Due to kB TP F , where T is temperature, we can ap2 proximate k kx,y sk ≈ (kF2 /2)S0 , where kF is the Fermi momentum. Multiplying Eq. (5) by kx and ky and summing over k we then find k2 S˙ 1 = −eEx S0 + F S0 × Ω1 − 2 k2 S˙ 2 = −eEy S0 + F S0 × Ω2 − 2

S1 + η1 , τtr S2 + η2 , τtr

(12) (13)

where we neglect the terms proportional to H because H 1/τtr . The relaxation terms in (12) and (13) originate from the mean value of the stochastic spin current, and the spin noise terms are defined as: X η1,2 = kx,y (Jk0 →k − hJk0 →k i), (14) k,k0

with their averages zero and correlations: hηiµ ηjν i = kF2 δµν δij DkB T δ(t − t0 )/(2τtr ), i, j = 1, 2, (15) where D is the density of states per spin in the full observation region. Linear dependence on temperature T appears in (15) after integration of fk (1 − fk ) over energy, i.e. it can be traced to the Dirac-Fermi statistics of electrons. Due to short correlation time of spin currents and due to fast relaxation, first harmonics S1µ and S2µ will change at fast time-scales, at which variables S0µ can be considered constant. This allows us to express first harmonics, e.g. S1 , as functions of S0 : S1 = −eEx τtr S0 +

kF2 τtr S0 × Ω1 + κ1 (t), 2

(16)

where κ1 is the solution of the equation, κ˙ 1 = − τκtr1 + η1 (t), which describes a noise with correlators hκµ1 (t)i = 0,

hκµ1 (t)κν1 (t0 )i =

kF2 DkB T −|t−t0 |/τtr e δµ,ν . 4 (17)

3 We also note that the correlator (17) has very short decay time τtr (less than 10ps in Rashba 2D electron gas), so at time scales of spin relaxation it can be safely approximated by a white noise hκµ1 (t)κν1 (t0 )i ≈

kF2 DkB T τtr δ(t − t0 )δµ,ν . 2

(18)

By similarly working out equations for S2 and substituting results into (11) we obtain equations for slowly changing spin density. In order to simplify notation, we introduce new parameters HSO = 2λeEτtr with λ = q p 2 2 2 2 α + β and E = Ex + Ey , and τ1s = (2λkF )2 τtr as well as the noise variables ξ: ξ = κ1 × Ω1 + κ2 × Ω2 .

(19)

Note that hξ x ξ y i = 6 0. Equations for dynamics of S0µ are: Sz S˙ 0z = (S0 × Heff )z − 0 + ξz , τs S y sin(2φ) Sx + ξx , S˙ 0x = (S0 × Heff )x − 0 − 0 2τs 2τs

(20)

S x sin(2φ) Sy S˙ 0y = (S0 × Heff )y − 0 − 0 + ξy , 2τs 2τs with Heff = H + HSO , in which HSO = HSO (− sin(θ + φ), cos(θ − φ), 0),

(21)

where θ denotes the angle that the in-plane electric field makes with x-axis, and tan φ = β/α. Eqs. (20) and (21) show that τs is a characteristic relaxation time for the out-of-plane spin component of the fluctuation, and HSO is the effective magnetic field which is induced by the electric current. By taking Fourier transform of (20), e.g. S0z (ω) = R Tm /2 dteiωt S0z (t) with Tm being the measurement time, −Tm /2 which is much larger than τs , we obtain the correlator in the frequency domain. When |Heff | 1/τs and a magnetic field is in-plane of the sample, we find that the noise power is given by P (ω) ≡

DkB T /τe hS0z (−ω)S0z (ω)i = , Tm (ω − ωL )2 + 1/τe2

(22)

where ωL = |Heff | and the effective spin relaxation time is τe−1 = τs−1 (3 − sin 2ϕ sin 2φ)/4 in which ϕ is the angle between the direction of Heff and the x-axis. Eq. (22) shows that the shape of the noise power spectrum is Lorentzian but the Larmor frequency is influenced by the electric field and the peak width is renormalized by a factor, which depends on the direction of the magnetic field and the spin orbit coupling anisotropy. Applications. A straightforward experimentally testable prediction of Eqs. (21) and (22) is that the electric field shifts the position of the maximum of the Lorentzian peak by the amount δωL ∼ 2eEλτtr , as we

FIG. 1. (a) The spin noise power spectrum without and with an in-plane electric field (black and red curves, respectively) in strained bulk GaAs. Maximum of the peak is normalized to 1. The peak is shifted by an amount of δωL ≡ ωL − H. Here, H = 100MHz, HSO = 15MHz, τs−1 = 10MHz. (b) The polar plot of the peak shift |δωL | in a 2D electron gas as a function of θ, i.e. the angle of an in-plane electric field with x-axis. Here E = 12V/cm, and the magnetic field is always perpendicular to the electric field. Red, black and green curves correspond to α = λ (β = 0), α = β and α = 0 (β = λ), respectively.

show in Fig. 1(a). The sign of the shift is proportional to e, i.e. depends on the sign of the carriers. Taking the values from [31] for Rashba coupling in 2D electron gas at GaAs/AlGaAs interface, λR = 1.5 · 10−13 eV · m, relaxation time 1/τtr = 10−3 eV, and assuming the electric field E = 12V/cm, we find HSO ∼ 800MHz. which is comparable to the spin relaxation rate in such systems [32]. Alternatively, a linear Rashba-type spin orbit coupling is induced in bulk 3D GaAs samples at imposed strains, i.e. α ∼ ε ≡ (εxx − εyy ), where εαβ are components of the strain tensor. The effective spin orbit field, HSO , induced by an electric field E = 9V/cm in a 3D GaAs sample with a strain ε = 0.015% was previously determined to be about ∼ 1Gauss by a local Hanle measurement approach [25]. Considering that strains can be increased by an order of magnitude, the field E ∼ 25V/cm should produce the shift of the conducting peak by 15MHz, which would be larger than its width (∼ 10MHz) and hence clearly observable (Fig. 1(a)). The magnitude of the peak-shift effect is sensitive to the anisotropy of the spin orbit coupling, and hence, in a 2D electron gas, depends on the direction of the electric field, as illustrated in Fig. 1(b). In fact, measuring the shift of the Larmor frequency at two transverse directions of the external electric field, one can determine strengths of the Rashba and Dresselhaus couplings separately. Another application of the peak-shift effect can be in studies of localized states at the presence of conducting electrons. At low doping, below the conductinginsulating phase transition [33, 34], there can be donor impurities that are well separated from each other. If the distance between impurities exceeds some critical value R ∼ 200nm, electron hopping between them will be strongly suppressed and localized electron states near such impurities become akin to localized states in quantum dots [6]. Here we predict that spin noise from con-

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FIG. 2. Estimate of the noise power spectrum (a) at zero electric field and (b) at an in-plane electric field E = 30V/cm for a GaAs sample with comparable numbers of conducting and strongly localized electrons (n ∼ 3 · 1014 cm−3 ). Spin relaxation times for conducting electrons is τs = 100ns and for localized electrons τloc = 500ns. Electric field E = 30V/cm at strain ε = 0.15% splits the peak at zero frequency into two peaks at ±HSO ∼ 15MHz due to conducting electrons and a peak that remains at zero frequency, which is produced by localized electrons.

ducting and localized electrons can be distinguished due to their different behavior in an applied electric field. Namely, the noise power spectrum for mobile electrons will experience a displacement in the electric field, as explained above, but the power spectrum of localized electron spins will stay intact, as is shown in Fig. 2. In order to estimate this effect, consider a 3D strained GaAs sample with an arbitrary donor concentration n. Let V = 4πR3 /3 be the volume that is needed for a localized electron of a donor impurity to be well separated from other impurities. The probability that a localized state is not overlapped with any other electron state is given by e−nV , so the concentration of impurities whose electron wave functions remain well separated from other donors is given by nloc (n) = ne−nV .

(23)

localized and thermally activated conducting electrons, and hence their contribution to the spin noise power, can be made comparable. The physics of spin relaxation of isolated localized electrons is expected to be dominated by the hyperfine coupling in essentially the same way as in the spin of electron-doped InGaAs quantum dots, which was discussed in [22]. The theory [22] predicts that if the magnetic field is set to zero the localized states of a single donor impurities produce a sharp noise power peak. At low temperatures (below 7-10K), this peak has a nonLorentzian power-law shape at frequencies below 1MHz with a broader shoulder, whose width is determined by the typical strength of the quadrupolar coupling of nuclear spins (Fig. 5b in [22]). For GaAs, the latter is in the order of several megahertz. At moderately large temperatures (7-30 K), phonon mediated mechanisms of localized spin relaxation make this peak shape Lorentzian [16]. Fig. 2 shows that a reasonably strong electric field is sufficient to shift the peak of conducting electrons and distinguish it from the peak of localized states. Conclusion. We predict that measuring the spin noise power spectrum at steady non-equilibrium conditions is a promising research direction with applications to parameter estimation and uncovering new phenomena. We showed that an electric field leads to a measurable shift of the noise power peak of conducting electrons, which can be used for characterizing the anisotropy of the spin orbit coupling and separating the spin noise of localized states from the spin noise of conducting electrons. Future research directions on the non-equilibrium SNS may include effects of an AC electric field, spin noise measured from optically polarized electrons, studies of high order fluctuation-dissipation relations [35], and spin noise in the non-Ohmic regime at strong electric fields [36].

ACKNOWLEDGMENTS

One can find that for R = 200nm and a doping above the metal-insulator transition (n ∼ 1016 cm−3 ), the number of such well localized states is negligibly small. However, at lower doping (n ∼ 1014 cm−3 ), the number of well

Authors thank S. Crooker, D. Smith, A. Saxena, and Yan Li for useful discussions. This work was funded by DOE under Contract No. DE-AC52-06NA25396.

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