This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 186.65.96.106

This content was downloaded on 18/10/2013 at 08:18

Please note that terms and conditions apply.

Origin of the decoherence of the extended electron spin state in Ti-doped β-Ga2O3

View the table of contents for this issue, or go to the journal homepage for more

2013 J. Phys.: Condens. Matter 25 316002

(http://iopscience.iop.org/0953-8984/25/31/316002)

Home Search Collections Journals About Contact us My IOPscience

IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 25 (2013) 316002 (8pp) doi:10.1088/0953-8984/25/31/316002

Origin of the decoherence of the extendedelectron spin state in Ti-doped β-Ga2O3

F Mentink-Vigier1,2, L Binet1,2, D Gourier1,2 and H Vezin2,3

1 Ecole Nationale Superieure de Chimie de Paris (Chimie-ParisTech), Laboratoire de Chimie de laMatiere Condensee de Paris, UMR CNRS 7574, 11 rue Pierre et Marie Curie, F-75231 Paris Cedex 05,France2 TGE Reseau National de RPE Interdisciplinaire (RENARD), FR-CNRS 3443, France3 Universite des Sciences et Technologies de Lille 1, Laboratoire de Spectrochimie Infrarouge et Raman,UMR CNRS 8516, Batiment C5, F-59655 Villeneuve d’Ascq Cedex, France

Received 12 April 2013, in final form 6 June 2013Published 9 July 2013Online at stacks.iop.org/JPhysCM/25/316002

AbstractThe mechanism of decoherence of the electron spin of Ti3+ in β-Ga2O3 was investigated bypulsed electron paramagnetic resonance. At 4.2 K, both instantaneous and spectral diffusioncontribute to the decoherence. For electron spin concentrations ≈1025 m−3 in the studiedsamples, calculations indicate that electron–electron couplings and electron couplings with69Ga and 71Ga nuclei yield similar contributions to the spectral diffusion, but thatelectron–nuclei interactions could become the dominant cause of spectral diffusion for onlyslightly lower spin concentrations. Above 20 K, an additional contribution to the decoherenceas well as to the spin–lattice relaxation arises from a two-optical-phonon Raman process,which becomes the leading decoherence mechanism for T > 39 K. Rabi oscillations with adamping time of about 79 ns at 4.2 K could also be observed. The damping of the Rabioscillations, independent of the oscillation frequency, is suspected to arise fromelectron–nuclei interactions.

(Some figures may appear in colour only in the online journal)

1. Introduction

Paramagnetic centers in solids, such as phosphorous donorin silicon [1], NV centers in diamond [2], Si or GaAsquantum dots [3, 4], Ce3+ in CaF2 [5], Fe3+ in ZnO [6]or Ti3+ in β-Ga2O3 [7] are considered as potential systemsfor spin-based quantum information. For such systems,understanding the electron spin dynamics and specifically thedecoherence mechanisms is a key requirement to achieve aperfect control of the spin qubit states. A special case ariseswhen the electron spin interacts with non-zero nuclear spins,such as in GaAs quantum dots or in the spin-bus scheme [5].In the latter case, though the quantum algorithm is applied tothe nuclear spins, the nuclei are initialized and their state readout by the electron spin so that the electron spin dynamics isas crucial as the nuclear spin dynamics in controlling the timescale of the whole computation sequence and the fidelity ofthe readout. In previous works [7, 8], we investigated the Ti3+

center in β-Ga2O3 as a potential spin-bus system. The Ti3+

dopant substitutes in the octahedral sites of the structure asshown in figure 1. The cation sites are organized as doublechains of edge-sharing octahedra running along the binary baxis and surrounded by chains of tetrahedra (figure 1). Aninteresting consequence of this chain structure was observedon the spin dynamics of electrons from intrinsic donors, whichinteract with gallium nuclei along octahedral chains andgive an interesting phenomenon of bistability of the electronparamagnetic resonance (EPR) signal [9, 10]. In titaniumdoped β-Ga2O3, we showed by electron nuclear doubleresonance (ENDOR) spectroscopy that the unpaired electronof Ti3+ (configuration 3d1) has also a diffuse wavefunctionextending preferentially along the b axis [7]. The unpaired3d electron thus undergoes strong hyperfine interactions (upto 130 MHz) of Fermi contact type with at least eightneighboring Ga nuclei (69Ga, I = 3/2, natural abundance60.1% and 71Ga, I = 3/2, natural abundance 39.9%). Owingto the number of Ga nuclei monitored by this single unpairedelectron spin and the high nuclear spin multiplicity, the system

10953-8984/13/316002+08$33.00 c© 2013 IOP Publishing Ltd Printed in the UK & the USA

J. Phys.: Condens. Matter 25 (2013) 316002 F Mentink-Vigier et al

Figure 1. Structure of β-Ga2O3 showing a Ti3+ dopant substitutinga Ga3+ in an octahedral site.

constitutes an exceptionally large nuclear qubit register [7, 8].The EPR spectrum of Ti3+ is made of a broad signaldue to the multiple hyperfine couplings (figure 2). Themain features of the partially resolved hyperfine structureof the EPR spectrum are accounted for by the hyperfineinteractions with the first and second Ga neighbors of Ti3+

along the b axis, which are the most strongly couplednuclei [7], the interactions with the other neighboring nucleibeing unresolved by EPR. Another outstanding consequenceof the strong hyperfine couplings is the existence of anindirect electron-mediated interaction between Ga nuclei,the so-called pseudo-dipolar interaction [8]. This interactionbetween the two nearest Ga neighbors flanking Ti3+ is about0.1 MHz when the isotopes are different (71Ga and 69Ga)and amplified up to about 1 MHz when the isotopes areidentical (two 71Ga or two 69Ga). Owing to the numberof nuclei coupled to the unpaired electron, the strengthof the hyperfine interactions and this strong inter-nucleipseudo-dipolar interaction, the Ti3+ center in β-Ga2O3 hasspecific features distinguishing it from other centers so farconsidered for quantum information. The purpose of thispaper is to investigate the electron spin dynamics of this centerin order to analyze the underlying mechanisms and determinethe different contributions (phonons, electron–nuclei ornucleus–nucleus interactions) to electron spin decoherence.

2. Experimental details

Single crystals of β-Ga2O3 doped with TiO2 were grownby the floating zone method [11]. The powder of β-Ga2O3(Alpha Aesar, 99.99%) and 1 wt% of TiO2 (Sigma Aldrich,99.9%) were mixed, pressed, and the resulting bar wassintered at 1000 ◦C. The crystal was grown from the sinteredbar and an undoped seed of β-Ga2O3 obtained from a previoussynthesis. The seed was aligned such as the crystal grew alongthe b axis. The resulting crystal was transparent and slightlyred. The ratio Ti/Ga ≈ 0.003, corresponding to a titaniumconcentration of 2.0×1026 m−3, was measured by inductivelycoupled plasma mass spectroscopy (ICP-MS) analysis atService Central d’Analyse du CNRS, Solaize, France. The

Figure 2. Top: continuous-wave EPR spectrum of Ti3+ for B0 ‖ cat X-Band and 30 K. Bottom: theoretical EPR line positions due tohyperfine coupling with nearest neighboring Ga nuclei for the threepossible isotopic configurations. Lines are labeled with values of thetotal nuclear spin quantum number M or with the spin quantumnumbers 69m or 71m of individual isotopes.

electron spin dynamics was studied with an X-band (9.4 GHz)Bruker Elexsys E580 FT spectrometer equipped with an MD5dielectric cavity and an ER 4188CF cryostat. Temperaturewas varied from 4.2 to 60 K. All the experiments weredone on oriented single crystals. The longitudinal relaxationtime T1 was measured with a picket-fence sequence madeof a series of 28 π -pulses separated by 100 ns followedby a π/2–τ–π -echo detection sequence with an inter-pulsedelay τ = 168 ns. The π and π/2 pulse durations were32 ns and 16 ns respectively. The echo intensity wascalculated by integration of the echo signal over a 100–120 nswindow. It was then monitored as a function of the timeseparation t between the 28-pulse train and the detectionsequence. The microwave attenuation was set at 6 dB for thisexperiment. The phase memory time Tm was measured by aπ/2–τ–π -echo Hahn sequence, with again π and π/2 pulsedurations of 32 ns and 16 ns respectively. The echo intensitywas monitored as a function of the inter-pulse delay τ . For thestudy of Tm as a function of the microwave field B1, it wasnecessary to determine the relation between the microwaveattenuation A and B1. In our case where only a small part of theEPR spectrum could be excited by the microwave pulses, wefollowed the procedure detailed in the appendix of [12]. Weobtained B1 (mT) = 0.54×10−A(dB)/20. The Rabi oscillationswere measured with an echo detection scheme. The sequenceconsisted of two pulses. A first pulse of varying length from12 ns to 1–2 µs was applied to achieve the Rabi oscillationsthen a second π pulse of 32 ns was applied after a waiting

2

J. Phys.: Condens. Matter 25 (2013) 316002 F Mentink-Vigier et al

Figure 3. Evolution of the longitudinal magnetization at 4.2 K forB0 ‖ c under the application of a picket-fence sequence (shown ininset). The theoretical full line was calculated with equation (1) andT1 = 500 µs.

time of 168 ns producing an echo, the intensity of which wasmonitored as a function of the first pulse length.

3. Electron spin–lattice relaxation

A typical recovery curve of the longitudinal magnetizationobtained with the picket-fence sequence is shown in figure 3.It can be simulated with a single exponential recoveryfunction

Mz(t) = Mz(0)(1− e−t/T1

), (1)

in the whole explored temperature range. The temperaturedependence of T−1

1 between 4 and 60 K is shown by theopen circles in figure 4. For temperatures below 18 K,T−1

1 weakly depends on temperature and remains in therange 500–600 µs. For T > 18 K, the spin–lattice relaxationrate T−1

1 rapidly increases upon increasing temperature. Thetemperature dependence of T−1

1 could be simulated by:

T−11 = AT + B exp

(−1

kT

)(2)

where the first term corresponds to a direct relaxationprocess and the second one represents either an Orbachprocess or a two-optical-phonon Raman process [13]. TheOrbach process operates when the paramagnetic center hasan excited state within the acoustic phonon spectrum at anenergy 1 above the ground state. The two-optical-phononRaman process involves modes of even parity and may beeffective due to the high density of states of optical modes,which compensates their low thermal occupancy factor. Theactivation energy obtained from the simulation by equation (2)of the spin–lattice relaxation rate is 1 = 164 ± 3 cm−1.This energy corresponds to a maximum of density of statesof optical phonon modes in β-Ga2O3 with even parity (Ag)at the center of Brillouin zone [14]. This suggests thata two-optical-phonon Raman process may be the leadingspin–lattice relaxation process above 18 K.

Figure 4. Inverse of the spin–lattice relaxation time T1 (opencircles) and of the phase memory time Tm (filled circles) as afunction of temperature for the Ti3+ center in β-Ga2O3. The fulllines represent simulations from equation (2) withA = (2.5± 0.9)× 10−4 MHz K−1 and 1 = 164± 3 cm−1 for T−1

1and A = (2.8± 0.16)× 10−2 MHz K−1 and 1 = 146± 16 cm−1

for T−1m .

4. Electron spin decoherence during free evolution

4.1. Temperature dependence and spectral diffusion

The phase memory time, corresponding to the characteristicdecay time of the echo in a Hahn echo sequence, is generallyused as a measure of the decoherence time of the system,i.e. how long a quantum superposition is sustained in theabsence of microwave irradiation. The case of the Ti3+

center in β-Ga2O3 is of particular interest since the electronspin is strongly coupled to a bath of nuclear spins. Anotherinteresting feature is that the EPR spectrum of the Ti3+ centerin β-Ga2O3 is highly inhomogeneous due to the multipleunresolved hyperfine interactions with neighboring galliumnuclei [7, 8] and extends over 40 mT. Thus, when performingelectron spin decoherence measurement with a Hahn echosequence, only a narrow part of the EPR spectrum withB1 < 1 mT is excited, while the non-excited spin packetsremain at thermal equilibrium. In that case, in addition tocoupling with phonons, two decoherence mechanisms shouldbe considered, namely spectral diffusion and instantaneousdiffusion. Spectral diffusion corresponds to fluctuations ofthe resonance frequencies of the excited spins owing tofluctuations of the local fields created by the spins at thermalequilibrium. It occurs through dipolar interactions betweenexcited and non-excited spins or, when the electron spins arecoupled to nuclei, through random nuclear spin flips, whichchange the resonance fields of the spin packets [15, 16].Instantaneous diffusion is a decoherence mechanism causedby dipolar interactions between excited spins driven by theexciting microwave field, inducing fluctuations of the localmagnetic fields [17, 18]. Instantaneous diffusion operates onlyduring a microwave pulse and involves only the excited spins.The decay of the maximum of the Hahn echo with inter-pulse

3

J. Phys.: Condens. Matter 25 (2013) 316002 F Mentink-Vigier et al

delay τ can then be expressed in the most general way as:

M (2τ) = M0 exp[−

2τT2−

(2τ

TSD

)n][1+ f (2τ)] . (3)

In this equation, TSD is the characteristic time for spectraldiffusion, while T2 includes other contributions such ascoupling with phonons and instantaneous diffusion. Theexponent n, generally in the range 0.5–4, depends on the spindynamics, the spin dilution regime and the interactions in thespin system (possibly including nuclei) [15, 16, 18–22]. Thefunction f (2τ) in equation (3) represents the electron spinecho envelope modulations (ESEEM) created by the hyperfinecouplings with 69Ga and 71Ga nuclei. These modulationssuperimpose to a background given by the exponential factorin equation (3). Equation (3) shows that the decay of thebackground of the echo signal may not be exponential in thegeneral case. The presence of nuclear modulations in the echosignal generally makes the determination of the exact shapeof the background signal difficult. Therefore, to smooth theoscillations of the echo signal due to these modulations, itis generally more convenient to plot ln [M(2τ)] instead ofM(2τ) [23]. In this representation, the background B(2τ)mustfollow a polynomial law:

B(2τ) = ln (M0)−2τT2−

(2τ

TSD

)n

. (4)

Figure 5 shows the decay of the echo signal on a log scaleat two characteristic temperatures, 4.2 and 39 K. The signal ismade of strong oscillations due to nuclear modulations, whichare superimposed to a decaying background. The backgrounddecay at 4.2 K is clearly not linear and can be simulatedby equation (4) with n ≈ 2–3. This non-linear decay of thebackground can be observed up to 39 K. This indicates thatspectral diffusion has a significant contribution to the lossof coherence for T < 39 K. However, the values of theparameters T2, TSD and n in equation (4) cannot be accuratelydetermined because several different sets give very similartheoretical curves, as exemplified by the dotted and full linesin figure 5(a), corresponding to n = 2 and 3 respectively.The background decay becomes linear at higher temperature(39 K in figure 5(b)) indicating that it is dominated by theterm −2τ/T2 in equation (4) and consequently that the lossof coherence is no longer driven by spectral diffusion in thishigh temperature range. Owing to the impossibility to obtainreliable values of T2 and TSD from the simulation of thebackground decays with equation (4), the backgrounds weresimulated in the whole explored temperature range (4–60 K)with a more robust simply decaying exponential function:

M(2τ) = M0e−2τ/Tm . (5)

The decay time Tm in equation (5) is then an effective phasememory time possibly including implicitly spectral diffusioneffects. The temperature dependence of T−1

m is shown by thefilled circles in figure 4. For T > 20 K, T−1

m increases uponincreasing temperature and its variation can be simulated bya law similar to equation (2) with 1 = 146 ± 16 cm−1.This is close to the activation energy of the spin–latticerelaxation, indicating that the coupling with optical phonons

Figure 5. Logarithm of the echo amplitude as a function of twicethe inter-pulse delay of a Hahn sequence at (a) 4.2 K and (b) 39 Kand simulations with a linear function (dashed line) or withequation (4) with n = 2 (dotted line) and n = 3 (full line).

via a two-phonon Raman process significantly contributesto the loss of coherence above 20 K. For T < 20 K, T−1

mslightly increases upon decreasing temperature. However, therelevance of this slight increase may be questioned since, asdiscussed above, the exponential decay (equation (5)) is notthe most appropriate function to simulate the variations of T−1

mowing to the significant contribution of spectral diffusion inthis temperature range.

4.2. Instantaneous diffusion

Another possible cause for the loss of coherence isinstantaneous diffusion. This mechanism is controlled bydipolar interactions between excited spins during theapplication of a microwave pulse, so that it depends on thespin concentration C and on the turning angle θ of the secondpulse of the Hahn sequence [18]. The effective phase memorytime can then be written as:

T−1m =

(T−1

m

)0+

π

9√

3

µ0g2β2

hC

⟨sin2

(θ

2

)⟩. (6)

4

J. Phys.: Condens. Matter 25 (2013) 316002 F Mentink-Vigier et al

The first term contains contributions from the couplingwith phonons and in our case implicit effects of spectraldiffusion and the second term represents the contribution ofinstantaneous diffusion. The brackets in equation (6) representthe average value of sin2(θ/2) over the EPR spectrum [18].When the EPR spectrum is so large that only a small partis excited by the microwave pulse, this average value iscalculated by [12, 18]:⟨

sin2(θ

2

)⟩=

∫+∞

−∞

B12

B12 + (B− B0)

2

× sin2[γ

tp2

√B1

2 + (B− B0)2]

f (B) dB

(7)

where B1 is the microwave field, B0 the setting static field,tp the second pulse duration, f (B) the spectral density at afield B and γ the electron gyromagnetic ratio. The spectraldensity is simply given by the EPR absorption spectrumnormalized to unity surface. Instantaneous diffusion is bestevidenced at low temperature, where the phonon contributionto decoherence and the spin–lattice relaxation of the excitedspins are inefficient [18]. The effect of the instantaneousdiffusion was then studied at 4.2 K. It was evidenced bymeasuring the phase memory time at different setting fields onthe EPR spectrum (for a 6 dB attenuation and tp = 16 ns), bychanging the microwave field (or equivalently the attenuationfrom 0 to 10 dB by steps of 1 dB, with tp = 16 ns, anda setting field at the center of the EPR spectrum) or byvarying the second pulse duration from 6 to 40 ns by stepsof 2 ns (for a 6 dB attenuation, and a setting field at thespectrum center). For each value of the above parameters,⟨sin2(θ/2)

⟩was calculated from equation (7) and the variation

of T−1m as a function of

⟨sin2(θ/2)

⟩is shown on figure 6. The

experimental variation of T−1m is almost linear as expected

from equation (6). This shows that instantaneous diffusionsignificantly contributes to electron spin decoherence at 4.2 K.The ordinate at origin(

T−1m

)0= (5.0± 0.15)× 105 s−1 (8)

gives the decoherence rate at 4.2 K dominated by spectraldiffusion and excluding effects of instantaneous diffusion.The slope of the variation simulated with equation (6) givesa Ti3+ concentration C = (1.60± 0.07) × 1025 m−3 in oursample. This concentration is one order of magnitude lowerthan the total titanium concentration 2 × 1026 m−3 obtainedfrom chemical analysis. This indicates that the major part oftitanium in β-Ga2O3 is either in the form of EPR silent Ti4+ orsegregated in small inclusions of titanium-rich parasitic phase.The latter hypothesis appears likely owing to the very lowsolubility of Ti3+ into β-Ga2O3 [24].

4.3. Origin of spectral diffusion in β-Ga2O3:Ti3+

Spectral diffusion is caused by fluctuations of the local fieldexperienced by the excited spins (A spins). These fluctuationsarise from flip-flops among the non-excited spins (B spins)

Figure 6. Inverse of the phase memory time Tm at 4.2 K as afunction of the average value of sin2(θ/2) calculated fromequation (7) upon varying the microwave field B1 (filled squares),the second pulse length tp of a Hahn sequence (open circles), or thestatic magnetic field B0 (filled circles).

surrounding the A spins. In our case, there are two kinds of Bspins, which can contribute to spectral diffusion: the electronspins, which are not excited by the microwave pulses, and thenuclear spins of 69Ga and 71Ga.

4.3.1. Contribution of nuclear spins to spectral diffusion.To calculate the spectral diffusion rate due to 69Ga and71Ga nuclei, we used the model developed by de Sousa andDas Sarma [25]. This model takes into account the flip-floprate T−1

nm of each individual nuclear pair (n,m) around theelectron spin. The inputs of this model are the hyperfinecoupling constant An for each nucleus n and the atomicpositions rn for the calculation of the An’s and of the nucleardipolar couplings. The hyperfine coupling constant for anucleus at position rn from the electron is given by:

An = Aiso(rn)−µ0

4πγeγnh

1− 3cos2 (θn)

r3n

h (rn − a0) (9)

where Aiso is the isotropic Fermi contact-type interaction,rn = ‖rn‖, θn the angle between rn and the magnetic field,and a0 the average radius of the electron wavefunction. his the step function [h(x) = 0 for x < 0 and h(x) = 1 forx > 0]. This function is introduced in the second term ofequation (9), which represents the electron–nucleus dipolarinteraction, because the expression is only valid in thepoint-dipole approximation, i.e. at distance greater than thesize of the electron wavefunction. Besides, at short distanceand in the case of our system, the hyperfine interactionis largely dominated by the isotropic contribution [7].Equation (9) does not take into account contributions tothe hyperfine interaction mediated through bonds. However,these contributions concern only the close Ga neighbors andare two orders of magnitude smaller than the Fermi contactinteraction [7]. To obtain an expression for the isotropic

5

J. Phys.: Condens. Matter 25 (2013) 316002 F Mentink-Vigier et al

term in equation (9), we would, in principle, need the exactmapping of the electron wavefunction. Unfortunately thisinformation is not known. We then used an approximateelectron spin distribution calculated from the estimated radiiof the wavefunction a‖ ≈ 0.8 nm and a⊥ ≈ 0.36 nm alongb and perpendicular to b respectively, obtained in a previouswork from ENDOR measurements [7], and by assuming anexponentially decaying spin density. In that case, we canwrite:

Aiso (rn) = ρ0 exp(−

2r‖a‖−

2r⊥a⊥

)A0

iso (10)

with A0iso = 4.3 × 10−10 rad s−1, an average value for 69Ga

and 71Ga of the hyperfine coupling constant for an electron ina pure 4s orbital of Ga [26] and ρ0 ≈ 0.035. The echo intensityis calculated according to [25] by:

M(2τ) =∏n<m

R−2nm exp

(−

2τTnm

)[1

T2nm

cosh (2Rnmτ)

+Rnm

Tnmsinh (2Rnmτ)−1

2nm

](11)

with1nm = |An − Am| /2 and R2nm = T−2

nm −12nm. The nuclear

flip-flop rates T−1nm are calculated from equations (46) and

(50) in [25]. Figure 7 shows the calculated echo decay whenconsidering the nuclei within a cutoff radius 2.0 nm about thecentral Ti3+ (we checked that above this value, the spectraldiffusion rate no longer depended on the cutoff radius). Thedecay can be simulated by:

M(2τ) = exp[− (2τ/TSD)

n] (12)

with n ≈ 2.9 and a spectral diffusion rate T−1SD ≈ 105 s−1.

This value approaches the experimental decoherence rate5 × 105 s−1 (equation (8)) at 4.2 K and thus indicates thatnuclei may provide a nonnegligible contribution to spectraldiffusion.

4.3.2. Contribution of electron spins to spectral diffusion.Another possible contribution to spectral diffusion may arisefrom non-excited unpaired electron spins on Ti3+ dopants.Here, we have to distinguish between the cases wherethe fluctuations of these spins are caused by spin–latticerelaxation (T1-sample) or by flip-flops between spins (T2-sample). The case of T1-samples generally refers to hightemperatures or low spin concentrations, while T2-samplesrefer to low temperatures or high spin concentrations, withhowever the discriminations between high and low cases notclearly defined. For a magnetically diluted T2-sample, there isa distribution of flip-flop rates with a maximum rate Wmax ≈

(µ0/4π)2 γ 4e h2C2

[ln(1ω/1ωdip

)]21ωdip/1ω

2 where Cis the concentration of non-excited spins, 1ω is the

spectrum width, and1ωdip = (µ0/4π)(

4π2/9√

3)γ 2

e hC the

broadening due to electron-dipole interactions only [27]. Witha spectrum width at half-height 1ω ≈ 3 × 109 rad s−1

and C ≈ 1.6 × 1025 m−3 (owing to the small excitationbandwidth, most of the spins are not excited so that theirconcentration can be approximated by the total concentration

Figure 7. Calculated amplitude (filled circles) of a two-pulse spinecho decay versus twice the inter-pulse delay τ , including the effectof nuclear spectral diffusion and simulation (full line) withequation (12).

determined above), we get Wmax . 103 rad s−1. WhenWmax . 103 rad s−1, the echo decay is given by [27]:

M(2τ) = M0 exp[−1ωdipWmax

4(2τ)2

]. (13)

This gives a spectral diffusion rate T−1SD =

√1ωdipWmax/2 ≈

9.4 × 104 s−1. In the case of a T1-sample, Wmax inequation (13) should be replaced by 2/T1 [27], with in ourcase is T1 ≈ 5 × 10−4 s at 4.2 K. This would then giveT−1

SD ≈ 1.2 × 105 s−1. The values of the spectral diffusionrates in the cases of T1 and T2 samples are very close andthemselves close to the rate due to nuclei. They all approachthe correct order of magnitude of the experimental spectraldiffusion rate T−1

SD = 5.0 × 105 s−1. The dipolar interactionsbetween unpaired spins on Ti3+ thus also provide a significantcontribution to the spectral diffusion for an electron spinconcentration C of the order of 1025 m−3. However since T−1

SDscales as C2, this contribution could be strongly reduced andmade smaller than the nuclear contribution just by a moderatedecrease in Ti3+ concentration by a factor less than two.

5. Electron Rabi oscillations and decoherence undermicrowave irradiation

Another important parameter for a quantum informationdevice is the ability to implement logical gates. Single-qubitgates can be achieved with microwave pulses to controlthe quantum state of the electronic spin. For instance,the NOT gate corresponding to a mere spin inversion ismade by applying a π pulse. More complex gates may becomposed of several pulses of variable durations. A key pointhere is decoherence during the pulses, i.e. upon microwaveirradiation, which limits the total number and the duration ofapplicable pulses. The coherent behavior of the spin systemupon microwave irradiation on a time scale shorter than the

6

J. Phys.: Condens. Matter 25 (2013) 316002 F Mentink-Vigier et al

spin–lattice relaxation time can be probed by a sequencemade of a resonant high turning angle pulse, whose lengtht is increased, separated by a fixed interval τ from a secondπ pulse as shown in the inset of figure 8(a). During thehigh turning angle pulse, the magnetization rotates aboutthe fixed microwave field in the rotating frame inducingRabi oscillations of both the longitudinal and transversecomponents of the magnetization. After the π pulse an echois formed the intensity of which is linked to the transversecomponent of the magnetization at the end of the highturning angle pulse. Figure 8(a) represents the evolution ofthe transverse magnetization as a function of the high turningangle pulse length at 4.2 K for a microwave attenuation of1 dB. It must be mentioned that a drift of the baseline ofthe Rabi oscillations was observed and removed from theexperimental data. The signal consists of about five observableRabi oscillations, equivalent to coherent oscillations between|−1/2〉. and |+1/2〉 spin states. To simulate these oscillations,we have to consider the fact that the EPR spectrum is large andinhomogeneous so that with B1 < 1 mT, only a small part ofthe spectrum is excited by the high turning angle pulse. In thatcase, the transverse magnetization during the pulse is givenby [28, 29]:

My ∝ e

(−

tTR

)J0 (2πνRt + φ) (14)

where TR is the damping time of the Rabi oscillations, νRthe Rabi frequency and J0 the zeroth order Bessel functionof the first kind. We have added a phase shift φ for a bettermatch between experimental and simulated Rabi oscillations.As shown in figure 8(a), the agreement between equation (14)and the experimental oscillations is fairly good. At 4.2 K,the nutation frequency is 21 MHz for a 1 dB attenuation andvaries between 24 and 11 MHz for attenuations ranging from0 and 7 dB. The damping time at 4.2 K and 1 dB attenuationis TR = 79 ns, which is much smaller than the phase memorytime Tm ≈ 1 µs. Also interestingly, TR is independent ofthe microwave field (figure 8(b)), of the temperature inthe explored range, and of the spectral density (results notshown). It remains between 60 and 90 ns whatever the valuesof the above parameters, in contrast with the behaviors ofTm. In the ideal case, TR = 2Tm [28]. Cases were howeverobserved where TR was smaller than 2Tm [30–32]. In thesecases, the damping time decreased upon increasing nutationfrequency i.e. upon decreasing microwave attenuation. Thepossible reasons for this behavior can be interactions betweenspins [31, 33], random fluctuations of the microwave field [31]or microwave field inhomogeneity along the sample org-factor distribution [34]. In our case, the fact that TR does notdepend on the microwave attenuation, hence on the nutationfrequency, rules out the above causes for the very shortobserved damping time. Another explanation could then bea loss of electron spin coherence caused by spin flip-flopbetween the electron spin and the surrounding nuclearspins. This effect is favored when the nutation frequencymatches the splitting between nuclear spin states [35]. Asreported in a previous work [7], the ENDOR spectrumof β-Ga2O3:Ti3+ contains many lines in the whole range

Figure 8. (a) Rabi oscillations at 4.2 K for B0 ‖ c and a 1 dBmicrowave attenuation. The full line is a simulation withequation (14). (b) Damping time of the Rabi oscillation versusmicrowave field at 4.2 K.

10–25 MHz corresponding to the nutation frequencies. As aconsequence, the nutation frequency, whatever its value in thisrange, can be tuned to some nuclear spin energy splitting. Thiscould explain the fast damping of the Rabi oscillations andwhy TR does not depend on the nutation frequency.

6. Conclusion

The original feature of β-Ga2O3:Ti3+ lies in the large nuclearspin system coupled to the unpaired electron of Ti3+ [7, 8].The unpaired electron and the nuclear spins are bothpotential qubit carriers for quantum information processing.In either case, the electron spin dynamics plays a majorrole. We thus investigated the spin dynamics of the unpairedelectron of Ti3+ in β-Ga2O3 to analyze the respectivecontributions of the electron–phonon, electron–electron andelectron–nuclei interactions to the electron spin decoherence.The wavefunction of this electron extends over severalangstroms around the central titanium, thus encompassinga great number of nuclear spins of 69Ga and 71Ga. Inthis respect, the system shares some similarities with GaAsquantum dots. The decoherence below 18 K is determinedboth by instantaneous and spectral diffusions, while above thistemperature optical phonons become the major contribution todecoherence. This means that for quantum computation, thesystem should be used below 18 K where the decoherencetime and spin–lattice relaxation time are the longest. Besides,

7

J. Phys.: Condens. Matter 25 (2013) 316002 F Mentink-Vigier et al

the EPR signal of Ti3+ is no longer detectable above 250 K,which precludes any use at room temperature. The othercontrol parameter for the system is the Ti3+ concentration.With an electron spin concentration about 1025 m−3 inthe studied sample, dipolar interactions between electronspins and interactions between electron and nuclei contributeon the same footing to spectral diffusion at 4.2 K. Thisconcentration appears as an optimal one. On the one hand,increasing the Ti3+ concentration C would strongly reducethe electron spin decoherence time through instantaneousand electron–electron spectral diffusion processes, whichscale as C and C2, respectively. On the other hand,reducing the dopant concentration would not significantlyincrease the decoherence time since then nuclei wouldprovide the dominating contribution to spectral diffusion.The decoherence time due to electron–nuclei couplings wasestimated to≈10 ms, which is the same order of magnitude asthe decoherence time calculated for GaAs quantum dots [25].Interactions with nuclei are also suspected to be the cause for arapid loss of coherence during application of long microwavepulses. Therefore decoherence processes involving nuclei setan upper limit to the electron spin decoherence time at lowtemperature. The possibility to increase this decoherencetime using more sophisticated refocusing pulse sequencesthan the simple Hahn sequence, such as XYXY or KDDsequences [36–38], should then be investigated.

References

[1] Kane B E 1998 Nature 393 133[2] Wrachtrup J and Jelezko F 2006 J. Phys.: Condens. Matter

18 S807[3] Loss D and Di Vincenzo D P 1998 Phys. Rev. A 57 9701055[4] Burkard G, Loss D and Di Vincenzo D P 1999 Phys. Rev. B

59 9808026[5] Mehring M and Mende J 2006 Phys. Rev. A 73 052303[6] Tribollet J, Behrends J and Lips K 2008 Europhys. Lett.

84 20009[7] Mentink-Vigier F, Binet L, Vignoles G, Gourier D and

Vezin H 2010 Phys. Rev. B 82 184414[8] Mentink-Vigier F, Binet L and Gourier D 2011 Phys. Rev. B

83 214409

[9] Aubay E and Gourier D 1992 J. Phys. Chem. 96 5513[10] Binet L and Gourier D 1996 J. Phys. Chem. 100 17630[11] Saurat M, Revcolevschi A and Collongues R 1970 C. R. Acad.

Sci. Paris Ser. C 271 58[12] Marrale M, Brai M, Barbon A and Brustolon M 2009 Radiat.

Res. 171 349[13] Huang C Y 1967 Phys. Rev. 154 215[14] Liu B, Gu M and Liu X 2007 Appl. Phys. Lett. 91 172102[15] Herzog B and Hahn E 1956 Phys. Rev. 103 148[16] Klauder J and Anderson P 1962 Phys. Rev. 125 912[17] Raitsimring A M, Salikhov K M, Umanskii B A and

Tsvekov Yu D 1974 Sov. Phys.—Solid Sate 16 492[18] Salikhov K M, Dzuba S A and Raitsimring A M 1981 J.

Magn. Reson. 42 255[19] Mims W 1968 Phys. Rev. 168 370[20] Hu P and Hartmann S R 1974 Phys. Rev. B 9 1[21] Witzel W and Das Sarma S 2006 Phys. Rev. B 74 035322[22] Liu R, Yao W and Sham L 2007 New J. Phys. 9 226[23] Hoffmann S K, Goslar J, Hilczer W,

Augustyniak-Jablokow M A and Kiczka S 2001 J. Magn.Reson. 153 56

[24] Tomm Y, Ko J M, Yoshikawa A and Fukuda T 2001 Sol.Energy Mater. Sol. cells 66 369

[25] de Sousa R and Das Sarma S 2003 Phys. Rev. B 68 115322[26] Rauber A and Schneider J 1966 Phys. Status Solidi b 18 125[27] Salikhov K M and Tsvetkov Yu D 1979 Time Domain Electron

Spin Resonance ed L Kevan and R N Schwartz (New York:Wiley) chapter 7

[28] Torrey H C 1949 Phys. Rev. 76 1059[29] Fedoruk G 2002 J. Appl. Spectrosc. 69 161[30] Boscaino R, Gelardi F and Korb J 1993 Phys. Rev. B 48 7077[31] Shakhmuratov R, Gelardi R and Cannas M 1997 Phys. Rev.

Lett. 79 2963[32] Bertaina S, Gambarelli S, Tkachuk A, Kurkin I, Malkin B,

Stepanov A and Barbara B 2007 Nature Nanotechnol. 2 39[33] Baibekov E I 2011 JETP Lett. 93 292[34] De Raedt H, Barbara B, Miyashita S, Michielsen K,

Bertaina S and Gambarelli S 2012 Phys. Rev. B 85 014408[35] van den Heuvel D J, Henstra A, Lin T S, Schmidt J and

Wenckebach W T 1992 Chem. Phys. Lett. 188 194[36] Gullion T, Baker D B and Conradi M S 1990 J. Magn. Reson.

89 479[37] Tyryshkin A M, Wang Z H, Zhang W, Haller E E, Ager J W,

Dobrovitski V V and Lyon S A 2010 arXiv:1011.1903v2[38] Souza A M, Alvarez G A and Suter D 2011 Phys. Rev. Lett.

106 240501

8