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Physica E journal homepage: www.elsevier.com/locate/physe

Magnetic ﬁeld sensing using a driven double quantum dot G. Giavaras a,, J. Wabnig a,b, B.W. Lovett a, J.H. Jefferson c, G.A.D. Briggs a a b c

Department of Materials, University of Oxford, Oxford OX1 3PH, UK Cavendish Laboratory, Department of Physics, University of Cambridge, Cambridge CB3 0HE, UK QinetiQ, Technology Centre, Malvern WR14 3PS, UK

a r t i c l e in fo

abstract

Article history: Received 29 August 2009 Received in revised form 15 November 2009 Accepted 25 November 2009 Available online 5 December 2009

A microwave-driven double dot system is investigated as a detector for probing a magnetic ﬁeld gradient. The two dots are connected to metallic leads and a source–drain current ﬂows under microwave irradiation. The induced current as a function of the external magnetic ﬁeld exhibits resonance effects that depend directly on the local ﬁeld gradient and any difference in the g-factors between the dots. The properties of the dot detector with respect to temperature, interdot hopping and the magnitude of the ﬁeld gradient are examined. We demonstrate that the main factor limiting sensitivity is spin relaxation and discuss the issues in applying the method to single spin detection. & 2009 Elsevier B.V. All rights reserved.

Keywords: Driven double dot Detector Spin resonance

1. Introduction Double dot (DD) systems are promising nanostructures for applications in the ﬁelds of nanoelectronics and spintronics. Signiﬁcant experimental progress has been achieved in manufacturing and controlling DDs. In particular, coherent single spin rotations [1] and entanglement generation [2] have been demonstrated in electrostatically deﬁned GaAs quantum dots. Magnetic ﬁeld sensing on the nanoscale is important as well as challenging [3] and in this work we show that a microwave-driven DD can be used to detect a magnetic ﬁeld gradient. We consider two tunnel-coupled quantum dots under microwave irradiation which are connected to metallic leads. The source–drain current as a function of a static magnetic ﬁeld exhibits an electronspin-resonance effect [1] which is sensitive to an asymmetry in the induced Zeeman splitting of the dots. Such asymmetry could arise from a magnetic ﬁeld gradient and/or a small difference in the gfactors of the two dots. We identify these two cases and show that the magnitude of the asymmetry that can be probed depends crucially on the interdot hopping as well as the microwave intensity. Unlike in a single quantum dot resonant behaviour occurs even at temperatures much higher than the energy scale set by the Zeeman splitting since the charging energy is the relevant energy scale. We demonstrate that the main limiting factor for the sensitivity of the DD detector is spin relaxation. The system we suggest has also potential for single spin read-out since a nearby spin which interacts with the spins of the DD induces

Corresponding author.

E-mail address: [email protected] (G. Giavaras). 1386-9477/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2009.11.138

an effective Zeeman splitting that can be in principle different in the two dots. It has been shown for a single dot that when the interaction is relatively strong a narrow-band microwave radiation induces a resonant current that depends on the state of the nearby spin [4]. Here we extend that analysis to the case of a DD detector and show that it can be operated at higher temperatures than a single dot due to the use of the Pauli blocking effect.

2. Physical model The DD system is described by a Hamiltonian HDD ¼ H þHB , where H is an extended two-site Hubbard Hamiltonian H¼

2 X

ei ni t

2 X y X ðc1s c2s þh:c:Þ þ U nim nik þ Vn1 n2 :

s

i¼1

ð1Þ

i¼1

P P The number operator is deﬁned as ni ¼ s nis ¼ s ciys cis for dot i ¼ f1; 2g with spin s ¼ fm; kg. The operator ciys (cis ) creates (destroys) an electron on dot i with on-site energy ei . t is the tunnel coupling between the two dots, U is the charging energy and V the interdot Coulomb energy. The Hamiltonian part due to the applied magnetic ﬁelds is HB ¼

2 X Di i¼1

2

szi þ

2 X

‘ Oi cosðo0 tÞsxi ;

ð2Þ

i¼1

where Di ¼ gi mB Bi is the Zeeman splitting on dot i due to a static magnetic ﬁeld Bi along z and a g-factor gi . We explicitly write the Zeeman splitting on each dot as D1 ¼ ðg þ dgÞmB ðB þ dBÞ and D2 ¼ g mB B, (g ¼ 2) where dg and dB is the difference in g-factors and magnetic ﬁelds, respectively, between the two dots which we

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assume to be independent of the applied ﬁeld. The Rabi frequency is Oi and the frequency of the oscillating magnetic ﬁeld along x is P o0 . The operators ri ¼ ss0 ciys rss0 cis0 , where r is the vector of the 2 2 Pauli matrices. The leads are modelled by noninteracting electrons and are P P described by the Hamiltonian Hleads ¼ l ¼ fL;Rg ks elk dylks dlks , where dylks (dlks ) creates (destroys) an electron in lead l ¼ fL; Rg with momentum k and spin s. The interaction between the dots and the leads is given by a standard tunnelling Hamiltonian of the form X ðtL c1y s dLks þ tR c2y s dRks Þ þ h:c:; ð3Þ HT ¼ ks

where tL ðtR Þ is the tunnel coupling between dot 1(2) and lead LðRÞ. Henceforth we consider tL ¼ tR . To take into account spin relaxation we have considered a generic bosonic bath (reservoir) of harmonic oscillators that is P P described by the Hamiltonian HE ¼ j ‘ oj ay1;j a1;j þ j ‘ oj0 ay2;j a2;j , where for simplicity we have assumed that each dot is coupled to a different bath. The operators ay1;j (a1;j ) create (destroy) a boson in mode j and similarly for ay2;j . The interaction Hamiltonian between the bath and the DD is X X L1;j ay1;j þ s L2;j ay2;j þh:c:; ð4Þ HES ¼ s 1 2 j

j y cik cim .

L1;j ðL2;j Þ is the coupling constant between dot where s i ¼ 1(2) and the j th mode of the corresponding bath and oj are the frequencies of the bath modes. HES allows incoherent spin-ﬂip processes for electrons in the two dots via energy exchange with the corresponding bath. To study spin-dependent electrical transport through the DD we employ a master equation approach [5] and perform a rotating wave approximation [5,6]. In the rotating frame and within the Born and Markov approximations we derive an equation of motion for the reduced density operator of the DD, r, that is extracted from the total density matrix, rT , by tracing over the leads and bath degrees of freedom, i.e., r ¼ TrfrT g where at the initial time rT ¼ r rleads rE , with rleads , rE being the density matrix for the leads and the bosonic bath, respectively. In the rotating frame and having performed a rotating wave approximation the free evolution part of the master equation involves the Hamiltonian 0 ¼ Hþ HDD

2 X di i¼1

2

szi þ

2 X ‘ Oi i¼1

2

sxi ;

ð5Þ

with the magnetic ﬁeld detuning di ¼ Di ‘ o0 . In the rotating frame we have time dependent operators cim0 ¼ cim expðio0 t=2Þ and cik0 ¼ cik expð þio0 t=2Þ. We take into account all the possible many body states of the DD up to four electrons giving in total 16 quantum states for single orbital dots and determine the electrical current in the steady state (when r_ ¼ 0) in the sequential tunnelling regime. The current operator for the lead l is by deﬁnition P Il ¼ eN_ l ¼ ie½HT ; Nl =‘ , where Nl ¼ ks dylks dlks is the number operator for electrons in the lead l. Tracing out the leads we derive an expression for the expectation value, I, with I ¼ /IR S=-/IL S.

3. Results The internal parameters of the DD are adjusted to zero energy detuning, i.e., we choose Eð1; 1ÞEð0; 2Þ ¼ 0, with Eðn; mÞ the energy of the charge state ðn; mÞ with nðmÞ electrons on dot 1(2). For practical realizations this would be achieved via gate electrodes. For our numerical evaluations we choose e1 ¼ 5 meV,

e2 ¼ 10 meV, U ¼ 10 meV, V ¼ 5 meV. The interdot hopping is t 5 U and the bias voltage Vsd ¼ 5 mV is applied symmetrically,

mL ¼ þeV sd =2 and mR ¼ eV sd =2, with mL ðmR Þ being the chemical potential of the left(right) lead. For this set of parameters the DD system operates in the spin-blockade regime (for kB T tU=80) in which the triplet states are almost fully populated without spin relaxation and microwaves and the current as a function of source–drain bias is suppressed [7,8]. For a ﬁxed bias voltage the microwave-induced current can ﬂow only when n ¼ D1 D2 a 0, since otherwise the spins in the two dots rotate at the same rate in the triplet subspace [1,9,10]. This is the basic principle of the DD detector. However, even when n a0 current might not ﬂow or be too small for detection and below we identify the optimum conditions for the detector. First we examine the DD system when there is no spin relaxation and therefore set L1;j ¼ L2;j ¼ 0 in Eq. (4). Fig. 1(a) shows the current as a function of Zeeman splitting D2 for two values of o0 when t ¼ 0:3 meV, O1;2 ¼ 90 MHz and T ¼ 1:2 K. In a simpliﬁed approach the left- and right-hand-side resonances occur when either the spins on dot 1 or 2 rotate due to the oscillating magnetic ﬁeld though because of the interdot hopping the positions of the resonances are somewhat shifted from the expected position (D1;2 ¼ ‘ o0 ) when the spins are independent. Intradot spin rotations mix two-electron states and this enables interdot tunnelling and hence current ﬂows. The antiresonance at ðD1 þ D2 Þ=2 ¼ ‘ o0 occurs since at this value the detuning in Eq. (5) is jd1 j ¼ jd2 j resulting in a zero occupation of the (0,2) state and the current is suppressed. For a ﬁxed temperature the width of the resonances is proportional to the Rabi frequency and the size of the Zeeman asymmetry n. The height of the resonances depends on O1 ðO2 Þ, t and n; the current peaks when ‘ O1 ot which allows interdot hopping while spin rotations take place. In the opposite limit spin rotations dominate yielding a negligible current. Fig. 1(b) shows the current when dB ¼ 0 and dg ¼ 0:02. In this case the height of the resonances changes with o0 via its dependence on n ¼ dg mB B which increases linearly with B. In Fig. 1(b), in contrast to Fig. 1(a), the heights of the peaks are independent of B, reﬂecting constant n ¼ g mB dB. This suggests a way of probing a difference in the gfactors between the two dots, though detecting a small difference in g-factors requires a relatively high magnetic ﬁeld. The quantity of interest is the relative current that we deﬁne as Ir ¼ Ip I0 , where Ip is the current on resonance (maximum current) of the right-hand-side peak and I0 is the background current, i.e., the current off resonance. The resonances can be resolved when the relative current is large. Fig. 1(c) shows Ir a function of o0 and as explained above this changes only when the two dots have different g-factors. An important parameter for the DD detector is the tunnel coupling t since interdot hopping takes place simultaneously with intradot spin rotations. In Fig. 2 we plot the relative current as a function of the Zeeman asymmetry when dg ¼ 0. In the regime t 5 n interdot tunnelling is suppressed, while in the opposite regime the mixing of the two-electron states is too weak. A maximum current ﬂows for t n and there is thus a competition between detectable current and minimum ﬁeld gradient for optimization of detector efﬁciency. The small values of t used in this work can be achieved experimentally [9] and in the regime of interest t 5U, yield the same pattern for the current as seen in Fig. 2. Since it might be difﬁcult to tune t to arbitrarily small values the sensitivity could alternatively be increased by exploiting a large g-factor. Further, the background current increases with temperature. This effect is caused by the tail of the Fermi-Dirac distribution describing the lead electrons, which lengthens at high temperatures leading to the opening of additional transport channels. All

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897

I (fA)

200 100 0 9

10

11 Δ2 (µeV)

12

13

9

10

11 Δ2 (µeV)

12

13

I (fA)

300 200 100 0

Ir (fA)

1000

500

0 20

0

40

60 h ω0 (µeV)

100

80

120

Fig. 1. (a) Microwave-induced current as a function of the Zeeman splitting D2 for dB ¼ 0:85 mT, dg ¼ 0, when ‘ o0 ¼ 10 meV (solid line) and ‘ o0 ¼ 12 meV (dashed line). (b) The same as in (a) but for dB ¼ 0 and dg ¼ 0:02. (c) Relative current of the right-hand-side peak when dB ¼ 0:85 mT, dg ¼ 0 (solid line) and dB ¼ 0, dg ¼ 0:02 (dashed line).

1000

1000 1000 750 Ir (fA)

800

600

800

γ=0.03 ΜΗz

0

400

10

20 ν (µeV)

30

40

400 T=60 K T=12 K T=1.2 K

200

γ=3 ΜΗz

Ir (fA)

0

γ=0.3 ΜΗz

600

250

Ir (fA)

γ=0

500

200

0

0 0

1

2 ν (µeV)

3

4

0

1

2 ν (µeV)

3

4

Fig. 2. Relative current on resonance as a function of the Zeeman asymmetry for different temperatures and the parameters t ¼ 0:3 meV, ‘ o0 ¼ 10 meV, O1;2 ¼ 90 MHz. The inset is for t ¼ 3 meV.

Fig. 3. Relative current on resonance as a function of the Zeeman asymmetry for different spin relaxation rates and the parameters t ¼ 0:3 meV, T ¼ 1:2 K, ‘ o0 ¼ 10 meV, O1;2 ¼ 90 MHz.

states acquire a ﬁnite population and have to be included in the dynamics of the density matrix as can be shown from the steadystate occupations. As a result the spin-blockade is gradually lifted and Ip I0. Already, by kBT U/5 the relative current is less than about 80 fA (see Fig. 2). On the other hand, for low enough temperatures (kB T t U=80) only states which are relevant in the spin-blockade regime are involved and the relative current is temperature independent. The variation of the current with respect to the temperature depends on the coupling to the

leads, the spin relaxation rate, and the applied source-drain bias. The variation may not be monotonic though for high enough temperatures (kB T \U) Ir 0 and hence the resonances cannot be clearly resolved. The relative current also drops when we take into account spin relaxation. This is because spin-ﬂip events enable incoherent singlet-triplet transitions and these lead to a leakage current in the spin-blockade regime even at low temperatures and n ¼ 0. In Fig. 3 we show a typical case for the relative current for various

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spin relaxation rates, g, which can be estimated as g ¼ pjLj2 DðdeÞ½2nðde; TÞ þ 1=‘ with the Bose function nðde; TÞ ¼ ½expðde=kB TÞ11 , de ¼ ‘ o0 , D the density of states for the bath that we assume constant and L ¼ L1;j ¼ L2;j in Eq. (4). This expression is derived explicitly from a master equation for a single spin with a Zeeman splitting de coupled to a bosonic bath at temperature T. In this work we consider only the case where the spin relaxation rate is much smaller than the tunnelling rate through the DD which is the most interesting experimental regime. The leakage current increases with g and this makes it difﬁcult to resolve the resonances. In GaAs typical relaxation rates can be on the order of 103 MHz and dephasing rates on the order of MHz. For a good resolution g has to be smaller than the microwave-induced Rabi frequency and the DD-lead tunnelling rates which are proportional to jtL j2 , (jtR j2 ) for the left(right) lead as can be shown from our general master equation. On the other hand, the Rabi frequency has to be larger than the tunnelling rates to allow for complete spin rotations while the electrons are on the dots.

4. Discussion and conclusion A driven single quantum dot can be used to detect the state of a nearby spin which interacts via an Ising interaction with the spins of the dot [4]. The current that ﬂows through the dot displays resonant behaviour as a function of the external ﬁeld, due to coherent spin rotations, that enables single-shot read-out of the state of the nearby spin. This idea can in principle be extended to a DD system for which one expects the interaction between the nearby spin and the dot spins to induce an effective Zeeman asymmetry. The successful detection of the spin depends on the strength of the interaction and the minimum electrical current that can be measured. Moreover, since the DD spin detector will depend on a lifting of a Pauli spin blockade it is likely to work at higher temperatures than the single dot. The properties of this system will be presented in detail elsewhere. We have shown that a microwave-driven DD can be used to detect a magnetic ﬁeld gradient. Provided that the Zeeman splitting in the two dots is different the source–drain current as a function of a static magnetic ﬁeld exhibits resonances due to coherent intradot spin rotations which are induced by an

oscillating magnetic ﬁeld. The magnitude of the induced Zeeman asymmetry that can be probed is related to the interdot tunnel coupling which is tunable with electrostatic gates. A difference in the g-factors between the two dots can also be identiﬁed and discriminated from a magnetic ﬁeld gradient through its dependence on microwave frequency. The resonances survive to temperatures of a few Kelvin for a charging energy of 10 meV, much higher than the energy scale set by the Zeeman splitting, signiﬁcantly extending the temperature window of a single quantum dot detector. We showed that the sensitivity of the DD detector is limited by spin relaxation. For an appreciable current the tunnelling rates between dots and leads as well as the Rabi frequency have to be much larger than spin relaxation rates.

Acknowledgments The work is part of the UK QIP/IRC (GR/S82176/01). G.G. is supported by QIP/IRC. J.W. thanks the Wenner-Gren Foundations for ﬁnancial support. B.W.L. thanks the Royal Society for a University Research Fellowship. J.H.J. acknowledges support from the UK MoD. G.A.D.B. is supported by an EPSRC Professorial Research Fellowship (GR/S15808/01).

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