Spin temperature concept verified by optical magnetometry of nuclear spins
M. Vladimirova, S. Cronenberger, D. Scalbert, I. I. Ryzhov, V. S. Zapasskii, G. G. Kozlov, A. Lemaître, K. V. Kavokin
SSpin temperature concept verified by optical magnetometry of nuclear spins
M. Vladimirova, S. Cronenberger, D. Scalbert, I. I. Ryzhov, V. S. Zapasskii, G. G. Kozlov, A. Lemaˆıtre, and K. V. Kavokin
2, 4 Laboratoire Charles Coulomb, UMR 5221 CNRS-Universit´e de Montpellier, F-34095, Montpellier, France Spin Optics Laboratory, St. Petersburg State University,1 Ul’anovskaya, Peterhof, St. Petersburg 198504, Russia Centre de Nanosciences et de nanotechnologies - CNRS - Universit´eParis-Saclay - Universit´e Paris-Sud, Route de Nozay, 91460 Marcoussis, France Ioffe Physico-Technical Institute of the RAS, 194021 St.Petersburg, Russia
We develop a method of non-perturbative optical control over adiabatic remagnetisation of thenuclear spin system and apply it to verify the spin temperature concept in GaAs microcavities.The nuclear spin system is shown to exactly follow the predictions of the spin-temperature theory,despite the quadrupole interaction that was earlier reported to disrupt nuclear spin thermalisation.These findings open a way to deep cooling of nuclear spins in semiconductor structures, with aprospect of realisation of nuclear spin-ordered states for high fidelity spin-photon interfaces.
The concept of nuclear spin temperature is one of thecornerstones of the nuclear magnetism in solids . It hasmade possible realisation of the cryogenic cooling intothe microKelvin range and observation of nuclear spinordering in metals and insulators . Such degree of con-trol of the nuclear spin system (NSS) in semiconductorheterostructures would allow enhancing the efficiency ofspin-based information storage and processing . How-ever, proving the validity of the spin temperature con-cept for semiconductor nano- and microstructures is chal-lenging due to the lack of techniques capable of pre-cise sensing of weak nuclear magnetisation in a smallvolume. In addition, recent experiments showed thatin quantum dots, where strong quadrupole-induced lo-cal fields have been reported, nuclear spin temperaturefailed to establish . In this context, NSS thermalisa-tion sensing in semiconductor heterostructures is one thecentral issues for both fundamental questions related tothe realisation of nuclear spin-ordered states, and forpotential applications, such as high fidelity spin-photoninterfaces .The basic postulates of the spin temperature theoryare illustrated in Fig. 1(a). It is assumed that duringthe characteristic time T determined by spin-spin inter-actions the NSS reaches the internal equilibrium. Thismeans that properties of the NSS are governed by a singleparameter, the spin temperature Θ N . When this tem-perature is made different from the lattice temperatureΘ L (e.g. by the optical pumping), the thermalisation ofthe NSS with the crystal lattice usually requires a muchlonger characteristic time T . Fig.1(b) illustrates one ofthe main predictions of the spin temperature theory: ifthe NSS is subjected to a slowly varying magnetic field,such that dB/dt < B L /T , then Θ N and the nuclear spinpolarisation P N change obeying universal expressions:Θ N / (cid:113) B + B L = Θ Ni /B i ; P N = B k B Θ N ¯ h (cid:104) γ N ( I +1) (cid:105) . (1)Here γ N is the gyromagnetic ratio of the nuclear spin I , angular brackets denote the averaging over all nuclear species, k B is the Boltzman constant, and Θ Ni is the spintemperature at strong magnetic field B i >> B L , where B L is the local field induced by the fluctuating nuclearspins. These generic relations are based on the princi-ple of entropy conservation in a thermodynamic systemduring adiabatic process. They constitute the basis forthe nuclear spin cooling by adiabatic demagnetisation, awidely used cryogenic technique . The nuclear spintemperature may take either positive or negative values,in the latter case the magnetisation being anti-parallel tothe applied field.Various optical and magnetic techniques have been em-ployed to measure nuclear spin temperature, mostly bythe magnetisation measurement at a fixed value of theexternal magnetic field . On the other hand, adirect measurement of the nuclear magnetisation as afunction of slowly varying magnetic field is extremelychallenging and has never been realised to the best ofour knowledge. Such an experiment is required to checkrigorously the validity of the concept of spin temperatureas applied to a specific system.In this Letter we report on realisation of such a proof-of-concept experiment in microcavities, semiconductormicrostructures with enhanced light-matter coupling .The principle of our experiment is sketched in Fig. 1(c).Prior to the measurement, the NSS of the n-GaAs layerembedded in a microcavity is polarised by optical pump-ing in the presence of the longitudinal magnetic field.Nuclear spin polarisation is probed by linearly polar-ized cavity mode photons with the photon energy in thetransparency band of GaAs. Polarisation of the lightbeam transmitted through the cavity is sensitive to the Overhauser field , an effective magnetic field created byNSS and acting on electron spins . Two methods ofdetection of nuclear spin polarisation are used: (i) theFaraday effect induced by the Overhauser field and(ii) the spin noise spectroscopy of resident electrons sub-ject to the Overhauser field . The main features ofthe behaviour of the optically cooled NSS under varyingexternal magnetic fields are demonstrated in the experi- a r X i v : . [ c ond - m a t . o t h e r] J un FIG. 1. (a) Sketch of the two heat reservoirs, the atomic lattice at temperature Θ L , and the nuclear spin system (NSS)at temperature Θ N . The equilibrium within the NSS is established during the spin-spin relaxation time T << T , thespin-lattice relaxation time. (b) Evolution of the nuclear spin temperature (dashed lines) and polarisation (solid lines) in theadiabatic de(re)-magnetisation process starting from either positive (red lines) or negative initial spin temperature Θ Ni undermagnetic field B i , as described by equation (1). The lowest nuclear spin temperature Θ N that can be reached in the adiabaticdemagnetisation procedure is determined by the initial temperature of the nuclei Θ Ni in the strong magnetic field B i andthe local field B L . (c) Schematic view of the sample and the detection stage of Faraday rotation and spin noise experiments.NSS in the cavity probed using two optical technics, that allow us to trace the evolution of the initially prepared nuclearspin polarization P N and temperature Θ N along the demagnetisation process. Spin noise spectrum is obtained as Fouriertransformation of the stochastic Faraday rotation. The spectral peak frequency is directly related to the Overhauser fieldacting on electrons in the presence of the in-plane magnetic field. ment where the Faraday rotation angle is measured whileramping the longitudinal magnetic field across zero (Fig.2). The experiment is conducted in two steps: prepa-ration and measurement (Fig. 2 (a)). The measuredsignal (Fig. 2(b)) contains two contributions: Faradayrotation directly induced by the external field (shown bysolid lines, it remains unchanged for all the scans), andthe Faraday rotation induced by the Overhauser field φ N ( shown separately in Fig. 2(e) for the first scan), whichis proportional to the nuclear spin polarisation. In eachconsecutive scan, φ N diminishes due to the nuclear spin-lattice relaxation, but the behaviour of nuclear polari-sation is described by Eqs. (1): the polarisation is anodd function of the applied field, there is no remanentmagnetisation at B = 0, and B L = 8 ± n d : an insulating sample with n d = 2 · cm − (Sample A) and a sample characterisedby a metallic conductivity ( n d = 2 · cm − , SampleB), for NSS prepared either at positive, or at negativespin temperature. The value of B L obtained for bothsamples is the same within our experimental accuracy.We complemented these results by spin-noise measure-ments of nuclear remagnetisation under magnetic fieldperpendicular to the light and the structure axis (Fig.3). Color maps in Figs. 3b,c show the evolution of theelectron spin noise spectra under varying magnetic fields.The narrow peak in the spectra appears at the frequency ν of the electron Larmor precession in the total effectivemagnetic field acting upon the electron spins. This fieldis given by the sum of the external and the Overhauserfield, which allows us to extract the nuclear spin polar- !FIG. 2. Nuclear spin magnetometry by Faraday rotation. (a) Timeline of the experiment. The preparation (blue area)consists in pumping under longitudinal magnetic field B pz , waiting for eventual nuclear relaxation in the vicinity of the localisedelectrons during t w and fast demagnetisation down to B iz . Faraday rotation of the probe beam is measured during successivescans of the magnetic field across zero (pink area, only first scan is shown). (b) Raw measurements of the Faraday rotationin Sample B (circles). NSS is prepared at Θ N <
0. During nine successive scans of the magnetic field ( t m = 5 s, directionshown by arrows) conventional Faraday rotation remains constant, this contribution is shown by solid lines. The remainingcontribution to the signal is due to the nuclear spin polarisation. It is shown separately in (e) for the first scan. (c-d) Faradayrotation induced in Sample A by nuclear spin prepared either at negative (c) or at positive (d) temperature (circles). Lines in(c-e) are calculated from Eq. (1), assuming different values of the local field, see Supplemental Material (SM). isation. The asymmetry of the recorded sets of spectrawith respect to zero magnetic field is due to nuclear spin-lattice relaxation. We have taken it into account whenfitting equation (1) to the data (black dashed lines inFig. 3(b-e)). For both samples and both signs of thenuclear spin temperature, the value of the local field wasfound to be B L = 12 ± B L ≈
10 G. Indeed, the spin-spin interactions in GaAsare dominated by magnetic dipole-dipole coupling, whichyields a much weaker local field B dd = 1 . .To elucidate the origin of this striking discrepancy,we performed spin noise measurements with the bulkGaAs layer without a microcavity, Sample C (Fig. 3(f-g)). Although the signal is much weaker, the best fitusing Eqs. (1) and taking into account spin-lattice re-laxation during the measurement yields B L = 2 G andΘ N = ± µ K . This comparison shows unambiguouslythe enhanced value of local field in the microcavities,compared to that in the bulk GaAs. Within the ther-modynamic description of the NSS, the local field whichenters Eqs. (1) is defined as : B L = T r ( H S ) /T r ( M B ) , (2) where H S is the Hamiltonian of all nuclear spin interac-tions, excluding Zeeman part (typically it includes themagnetic dipole-dipole interactions, and the indirect ex-change), and M B is the parallel to the magnetic fieldcomponent of the nuclear magnetic moment. In n-GaAs,magnetic dipole-dipole interaction is well-studied, and B L = 2 G measured in bulk GaAs agrees well with theprevious estimations for B dd .The only plausible explanation for the unexpect-edly strong local field detected in microcavities is thequadrupole splitting hν Q of the nuclear spin states in-duced by an uniaxial strain. In Eq.(2) it can be accountedfor by introducing H S = H dd + H Q , where H Q is theHamiltonian of the quadrupole interaction H Q = (cid:88) i =1 hν iQ I z − I ( I + 1)3 ) . (3)Here the index i stands for the summation over the threeisotopes ( Ga , Ga , As ), and ˆ I z is the projection onthe nuclear spin operator on the growth (strain) axis.Using equation (2) and the parameters of strain-inducedquadrupole splittings in GaAs , one can estimate thatthe strain as weak as 0 .
01% induces the local field B L =10 G in GaAs . FIG. 3. Nuclear spin magnetometry by spin noise spectroscopy. (a) Timeline of the experiment. The preparation (blue area)consists in pumping under oblique magnetic field, waiting during the time t w required for nuclear relaxation in the vicinity oflocalised electrons and fast demagnetisation down to B i ⊥ . Spin noise spectra of the probe beam are measured while scanning B ⊥ across zero (pink area). (b-g) Color maps of the spin noise spectra during adiabatic demagnetisation procedure at positive(c, e and g) and negative (b, d and f) spin temperature (measured in the signal to shot noise ratio units) for two microcavitysamples A ( b-c), B (d-e) and a bulk sample C (f-g). Black lines in (b-e) and red line in (f-g) are fits to Eqs. 1, that determinethe values of B L and Θ N indicated on the figure (see also SM). Red lines in (b-e) illustrate how the the value B L = 2 G failsto describe the experiment. Because B L >> B dd , it is the quadrupole interactionthat determines the capacity of the NSS to store the en-ergy in the internal degrees of freedom. But in contrastwith dipole-dipole interaction, the quadrupole interac-tion does not provide any coupling between the spins,and can not establish the thermodynamic equilibriumwithin the NSS. Indeed, in quantum dots, where strongquadrupole-induced local fields have been reported, nu-clear spin temperature failed to establish . From ourdata we can estimate the lower limit of 50 G for the mix-ing field B m , at which Zeeman and internal energy reser-voirs come to equilibrium between each other, so that theNSS can be described by the unique spin temperature (see Supplemental material).The question remains, how can the thermodynamicequilibrium be established under magnetic field B m >
50 G, much larger than the characteristic field of thedipole-dipole interaction B dd = 1 . ν N are illustrated in Fig. 4as functions of the magnetic field in the absence (Fig. 4a)and in the presence of the quadrupole splitting of the nu-clear spin states along z -axis (Fig. 4 (c, e)). The spin flip-flop transitions involving different isotopes ensure the en-ergy transfer between the Zeeman and quadrupole energyreservoirs, with total energy conservation of the NSS. These transitions are broadened by dipole-dipole inter-actions. It is usually assumed that the efficient equili-bration of energy reservoirs is ensured at detuning fromthe resonance less than δν N = 5 B dd / (2 π (cid:104) γ N (cid:105) = 8 kHz.One can see in Fig. 3d,f, that for both orientations of themagnetic field, the transitions involving such a small de-tuning are available at B <
50 G, and the mixing remainsas efficient as in the absence of the quadrupole splitting(Fig. 3(b)).Our results show that the strain-induced nuclearquadruple splittings in semiconductor microcavity do nothinder the establishment of the thermodynamic equilib-rium within the nuclear spin system. The quadrupoleeffects result in the increase of the local field, indicat-ing that the heat capacity of the NSS is dominated bythe quadrupole energy reservoir. The energy transferbetween the Zeeman and quadrupole reservoirs duringadiabatic demagnetisation is made possible by dipole-dipole interaction via spin flip-flop transitions involv-ing different isotopes. Thus, deep cooling of the NSSdown to microKelvin temperature range via adiabaticdemagnetisation is possible in photonic microstructures.This paves the way towards realisation of nuclear mag-netically ordered states and their applications, includingspin-photon interfaces with reduced thermal noise. B z (G) " N ( k H z ) Ga - Ga As - Ga As - Ga B (G) " N ( k H z ) Ga - Ga As - Ga As - Ga B ? (G) " N ( k H z ) Ga - Ga As - Ga As - Ga B z (G) N ( k H z ) B z (G) N ( k H z ) Ga Ga As B ? (G) N ( k H z ) (a)(b) (c)(d) (e)(f) FIG. 4. (a) Nuclear spin flip transition frequencies ν N for three GaAs isotopes, and (b) the differences ∆ ν N between themas functions of magnetic field in the absence of the quadrupole splitting. (c, d) Same as (a) and (b), respectively, but in thepresence of the quadrupole spitting in z -direction. (d, f), Same as (c) and (d), respectively, but the magnetic field is appliedin the plane of the structure. The blue area ν N < ACKNOWLEDGMENTS
This work was supported by the joint grant of theRussian Foundation for Basic Research (RFBR, GrantNo. 16-52-150008) and National Center for Scientific Re-search (CNRS, PRC SPINCOOL No. 148362), as well as French National Research Agency (Grant OBELIX, No.ANR-15-CE30-0020-02). IIR, VSZ and GGK acknowl-edge Russian Foundation for Basic Research (grant No.17-12-01124) for the financial support of their experimen-tal work.
Supplemental Material
I. SAMPLES
The studied microcavity structures consist of Si-doped GaAs 3 λ/ n e = 2 × cm − (Sample A) and n e = 4 × cm − (Sample B). The front (back) mirrors are distributed Bragg reflectorscomposed of 25 (30) pairs of AlAs/Al . Ga . As layers, grown on a 400 µ m thick GaAs substrate. Due to multipleround trips in the cavity, the Faraday rotation (FR) is amplified by a factor of N ∼ L = 0 . Q ∼ µ m-thick GaAs layer grown by liquid-phase epitaxy, with Si donor concentration of n d = 4 × cm .All these samples have been studied previously . II. EXPERIMENTAL TECHNIQUES
Both the spin-noise (SN) and FR techniques have been used previously for studies of the NSS and aredescribed in in detail in Ref. 29. They have an advantage of being virtually non-perturbative for the NSS, becausepumping and measurement stages are separated in time, and cooled NSS is optically probed via the polarisationrotation of the light beam with photon energy tuned below (here 20 meV) the band gap of the studied GaAs layer. Ina typical measurement, the sample is placed in a cold finger cryostat at T = 5 K, B z = 180 G. At the first stage, it isoptically pumped during 3-15 minutes by the circularly polarised laser diode with photon energy 1 .
57 eV and power P = 10 mW, focused on 1 mm spot on the sample surface. In the case of SN experiments, the transverse field B ⊥ is also applied during pumping. After the pumping stage, we wait for t w = 1 minute before lowering down B z (Fig.2(a), 3(a)), to be sure that nuclear spins situated under the orbits of the donors and characterised by the relativelyshort T do not contribute to the signal . At the last preparation step, B z is lowering down to the value fromwhich the measurement stage starts ( B z = B iz = 50 G for FR and B z = 0 for SN). FR and SN experiments mainlydiffer by the measurement stage. In FR experiment, the rotation of the linearly polarised probe beam is detected inthe presence of the slowly varying longitudinal magnetic field B z . In the SN experiment, the spin noise of the residentelectrons is measured in the presence of the slowly varying transverse magnetic field B ⊥ via the fluctuation spectrumof the Faraday rotation angle. The probe beam has the photon energy 20 meV below GaAs band gap, power of0 . µ m spot on the sample surface. II.1. Faraday rotation
To extract nuclear spin temperature and the local field from the Faraday rotation angle measured as a functionof the slowly varying magnetic field B z (the duration of each scan is ∼
10 s so that dB/dt = 10 G/s), we proceedas follows. First, we subtract the external field contribution from the total signal. This contribution to the signalremains unchanged for all consecutive scans and depends linearly on the magnetic field. The remaining part of theFR is induced by the Overhauser field B N , which is proportional to the nuclear spin polarisation P N . φ N = B N V N L = b N P N V N L, (S1)where b N = 5 . , V N is the nuclear Verdetconstant, L is the effective optical length of the sample accounting for by multiple round trips of light in the cavity .Therefore, from Eqs. (1) we get: φ N = φ Ni B/ (cid:113) B + B L (S2)where φ Ni is the Faraday rotation angle at the saturation field B iz . Fitting φ N to equation (S2) we determine B L = 8 ± V N = 0 . L = 0 . we alsoextract Θ N = − µ K in sample B (Fig. 2 (e) in the main text) from Eqs. (S1), (S2) and (1). The values of B L andΘ N extracted from FR measurement are averaged over the crystal volume, since the signal is given by the electronband spin splitting . II.2. Spin noise spectroscopy
The electron spin noise spectrum exhibits a pronounced peak at the electron Larmor frequency ν corresponding tothe total ( B ⊥ and B N ) field , so that: ν = γ e ( B ⊥ + B N ) = γ e ( B ⊥ + b N P N ) , (S3)where γ e = 0 .
64 MHz/G is the gyromagnetic ratio of the electrons in the conduction band of GaAs . Thus, bymeasuring ν as a function of B ⊥ and fitting equations (1) and (S3) to the data we obtain the values of Θ N and B L . Because each field scan takes 100 s (10 times longer than in the case of the Faraday rotation measurements),the spin-lattice relaxation of the NSS is not negligible on this time-scale. It manifests itself in the asymmetry ofthe recorded sets of spectra with respect to zero magnetic field. For the quantitative comparison with the theorypredictions given by Eqs. (1) and (S3) we measured the magnetic field-dependent relaxation times in an independentset of experiments . Note that in metallic samples the SN signal is mediated by the electron gas, and is thereforecontributed by all the nuclei. In the insulating samples, only the nuclei situated under the donor orbits can bedetected. However, the polarisation of the nuclear spins situated in the core of the donor orbit decays rapidly, andvanishes during t w . Thus the SN signal comes from the nuclei situated in the periphery of the donor orbits, so thatthe extracted Θ N and B L are close to those of the bulk nuclei. III. ESTIMATION OF THE MIXING FIELD
The mixing field B m , is the field at which Zeeman (for each of three isotopes) and internal energy reservoirs cometo equilibrium between each other . Only the Zeeman reservoirs can be cooled down via dynamic nuclear polarisationat strong field. By measuring nuclear polarisation, we get access to the average energy of the Zeeman reservoirs (cid:104) E Z (cid:105) = BP N (cid:104) ¯ hγ N (cid:105) . During adiabatic demagnetisation, energy transfer and the thermalisation between Zeeman andinternal energy reservoirs is achieved at B = B m . At this field, a part of Zeeman energy is transferred to the internalenergy reservoir, which results in the modification of the nuclear polarisation. The nonadiabaticity of this processwould lead to a deviation from Eqs. (1), quantified by the nonadiabaticity factor f na = B m / (cid:112) B + B m . Comparingthe magnetisation measured at B = 50 G before and after the passage through zero field, we have not observed anydifference within the experimental precision of 2%. This yields f na > .
98, and therefore B m >
50 G. M. Goldman,
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