Single-spin polaron memory effect in quantum dots and single molecules

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Single-spin polaron memory effect in quantum dots and single molecules

Dmitry A. Ryndyk, Pino D’Amico, and Klaus Richter

Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany 共Received 25 September 2009; published 31 March 2010兲

We propose theoretically a spin memory effect in quantum dots and single molecules with strong electron-vibron interaction and coupled to ferromagnetic leads. The controlled electrical switching between spin states is achieved due to an interplay between Franck-Condon blockade of electron transport at low voltages and spin-dependent tunneling at high voltages. Spin lifetimes, currents, and spin polarizations are calculated as a function of the bias voltage by the master-equation method. We further propose to use a third ferromagnetic tunneling contact to probe and readout the spin state.

DOI:10.1103/PhysRevB.81.115333 PACS number共s兲: 85.65.⫹h, 72.25.⫺b, 73.23.⫺b, 85.75.⫺d

I. INTRODUCTION

One of the most promising directions in the fields of mo-lecular electronics and spintronics is the experimental and theoretical investigation of spin manipulation in quantum dots and single molecules. In particular, new methods have been recently developed to investigate spin states of single atoms and molecules, e.g., using spin-polarized scanning-tunneling spectroscopy.1,2 _{Motivated by such achievements} the promising question arises whether a single-spin memory effect共namely, the controlled switching of the spin state of a single electron by applying an external voltage to a system兲 is possible. The main challenge is to combine long spin life-times and fast switching.

One of the ways for single-spin manipulation is based on the interplay of charge, spin, and vibron degrees of freedom in molecular junctions. In various experiments signatures of the electron-vibron 共e-v兲 interaction have been observed in atomic-scale structures.3–5_{In the case of strong e-v} interac-tion the formainterac-tion of a local polaron can lead to a charge-memory effect, which was first predicted long time ago,6_and has been recently reconsidered theoretically in more detail:7–12_{neutral and charged}_{共polaron兲 states correspond to} different local minima of an effective energy surface and are metastable if the e-v interaction is strong enough. By apply-ing an external voltage, one can change the charge state of this bistable system, an effect that is accompanied by hyster-etic charge-voltage and current-voltage curves. A similar memory effect was found in recent scanning tunnel micro-scope experiments13 _{as a multistability of neutral and} charged states of single metallic atoms coupled to a metallic substrate through a thin insulating ionic film; the correspond-ing scorrespond-ingle-level polaron model was discussed in Refs.11and

12.

In this paper we propose an approach to observe a single-spin memory effect by combining the polaron-memory mechanism and spin-dependent tunneling. To this end we consider a single-level and single-vibron quantum system be-tween magnetic leads共Fig.1兲. We study the case of a

sym-metric junction with antiparallel magnetizations of left and right leads; besides a third electrode can be used as a gate or to probe the spin state. In the case of parallel orientation of magnetization in the leads similar effects are observed but spin-up and spin-down states are not equivalent and the

memory effect is hidden. The problem can be solved with well-controlled approximation in the limit of weak coupling to the leads, where the master equation for sequential tunnel-ing can be used. Thus we focus our major discussion on this limit.

This paper is organized as follows: in Sec.IIwe introduce the electron-vibron model and describe the master equation method used for calculations. In Sec. III we consider spin lifetimes and show the existence of metastable spin states. In the Sec. IV bias-voltage-controlled switching between memory states is discussed and we conclude in Sec. V.

II. MODEL AND METHOD

The Hamiltonian of the single-level polaron 共Anderson-Holstein兲 model is Hˆ =

兺

␴ ⑀ ˜_␴d_␴†d_␴+␻₀a†_{a +}_␭共a†_{+ a}_{兲nˆ + Unˆ} ↑nˆ↓ +

兺

ik␴ 关共⑀ik␴+ e␸i兲cik␴ † cik␴+共Vik␴cik␴ † d_␴+ H.c.兲兴. 共1兲 Here the first line describes the free electron states with energies ˜⑀_␴, the free vibron of frequency ␻0, and the electron-vibron and Coulomb interactions with coupling strength ␭ and U, respectively; ␴ is the spin index, and nˆ_␴= d_␴†d_␴, nˆ = nˆ_↑+ nˆ_↓. The other terms in Eq. 共1兲 are the

Hamiltonians of the leads and the tunneling coupling 共i=L,R is the lead index, k labels the electronic states兲. The bias voltage V is introduced through the left and right

elec-L

Γ

L

ε

0

Γ

R

0

ω

L

_R

L

Γ

R G

V

Gate L

ϕ

_ϕ

_R

FIG. 1. 共Color online兲 Schematic of the considered system: a gated single-level quantum dot interacting with a vibron and coupled to ferromagnetic leads. The gate electrode can be consid-ered also as a third test electrode coupled weakly to the system.

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trical potentials, V =␸L−␸R. The energy˜⑀␴=⑀␴+ e␸0includes the bare level energies ⑀_↑, ⑀_↓ and the electrical potential ␸0 describing the shift of the central level by the gate voltage VGand by the bias voltage drop between the left and right

lead: ␸₀=␸R+␩共␸L−␸R兲+␣VG, where 0⬍␩⬍1 describes

the symmetry of the voltage drop across the junction, ␩= 0.5 stands for the symmetric case considered below. We assume the simplest linear dependence of the molecular po-tential 共␩= const兲 but its nonlinear dependence14,15 _{can be} easily included in our model. We further choose ⑀_↑=⑀_↓=⑀₀ and as reference energy⑀0= 0 for VG= 0 and assume a linear

capacitive coupling,⑀0=␣eVG, putting ␣= 1.

The coupling to the leads is incorporated in the level-width function

⌫i␴共⑀兲 = 2␲

兺

k

兩Vik␴兩2␦共⑀−⑀ik␴兲. 共2兲

The spin-dependent densities of states in the leads and the tunneling matrix elements are assumed to be energy-independent 共wide-band limit兲 so that ⌫L_␴ and⌫R_␴are

con-stants. The full level broadening is⌫_␴=⌫L␴+⌫R␴. Below we

consider a symmetric junction with antiparallel magnetiza-tion of the leads and use the notamagnetiza-tion⌫L↓=⌫R↑=⌫ for

major-ity spins and⌫L_↑=⌫R_↓=␬⌫ for minority spins,␬Ⰶ1.

The spin effects addressed are particularly pronounced in the limit of large U, i.e., we neglect the doubly occupied state so that only three states in the charge sector should be considered: neutral兩0典, charged spin up 兩↑典 and charged spin down 兩↓典. Employing the polaron canonical transformation,16,17 _{the corresponding eigenstates of the} iso-lated system共⌫=0兲 read

兩␺0q典 = 共a†_兲q

冑q!

兩0典, 共3兲兩␺␴q典 = e−␭/␻0共a†−a兲d␴†d_␴ d_␴†共a †_兲q

冑q!

兩0典共4兲

with the eigenenergies

E0q=␻0q, E␴q=˜⑀␴

⬘

+␻0q, ˜⑀␴

⬘

=˜⑀␴− ␭ 2 ␻0

. 共5兲

Here the quantum number q characterizes vibronic eigen-states, which are superpositions of states with different num-ber of bare vibrons.

␭ ␻0 共a†_{− a}_兲

册

共a †_兲q⬘

冑q

_⬘

!

冏

0

n_{共t兲,n=0, ↑ ,↓, to find the system in one of}

the polaron eigenstates, Eqs. 共3兲 and 共4兲, can be written

as18–22_{共see also the master equation approach for a quantum} dot between ferromagnetic leads in Refs. 24 and 25, and references therein兲 dPq n dt =_n

兺

_⬘_q_⬘⌫qq⬘ nn⬘ P_qn_⬘⬘−

兺

n⬘q⬘ ⌫q⬘q n⬘n Pq n + IV关P兴. 共10兲 Here the first term represents the tunneling transition into the state 兩nq典 and the second term the transition out of the state 兩nq典. IV_{关P兴 is the vibron scattering integral describing the}

relaxation of the vibrons to the thermal equilibrium. Finally, the average charge and the spin polarization are

Q = e

兺

q

共Pq↑+ Pq↓兲, S =

兺

q

共Pq↑− Pq↓兲, 共11兲

respectively, and the average current 共from the left or right lead兲 reads Ji=L,R= e

兺

␴qq⬘ 共⌫iqq⬘ ␴0 _P q⬘ 0 −⌫_iqq0␴_⬘P_q␴_⬘兲共12兲 with⌫_iqq␴0_⬘and⌫_iqq0␴_⬘defined in Eqs.共6兲 and 共7兲.

III. SPIN LIFETIMES

Let us first calculate the characteristic lifetimes of the neutral, spin-up, and spin-down states being in the vibronic ground state q = 0. We define the switching rates␥␴0from the neutral to the charged state with spin␴and vice-versa as the sum of the rates of all possible processes which change these states

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␥␴0₌

兺

q

⌫q0␴0, ␥0␴=

兺

q

⌫q00␴. 共13兲

In the sequential tunneling approximation the spin lifetime ␶␴is determined by the switching rate from the charged state ␥0␴ _{through Eqs.} _共₇_{兲 and 共}₁₃_{兲. It reads 共setting the Fermi} level to zero兲11 ␶␴−1=␥0␴=共1 +␬兲⌫

兺

q e−ggq q! f 0_共−_˜_⑀ ␴

⬘

+␻0q兲. 共14兲

At large g the sequential tunneling rates are exponentially suppressed 共Franck-Condon blockade兲 and the cotunneling contribution to ␶_␴−1 may become dominant. It is particularly important for the spin lifetime because cotunneling does not change charge but may change spin. The cotunneling rate can be estimated as22

␶_␴−1共ct兲_⬇␬⌫2T␻02

␭4 . 共15兲

Although the cotunneling contribution is not suppressed ex-ponentially, it is of second order in the tunneling coupling and suppressed additionally by the small polarization param-eter ␬ and large␭. At typical parameters, considered in this paper 共weak coupling to the leads, low temperatures兲, the cotunneling contribution can be neglected, but it can be es-sential at larger tunneling couplings and larger temperatures. Figure 2 shows the dependence of the inverse spin life-time␶_␴−1 and the inverse lifetime of the neutral state ␥␴0on the scaled electron-vibron interaction constant

冑

g =␭/␻0. The transition rates from the neutral state to the charged states and vice-versa are suppressed compared to the bare tunneling rate⌫ if g is large. Thus all states are metastable at low temperatures and zero voltage. Moreover, the lifetime of the charged states can be much larger than that of the neutral state. The last condition is important because then a pure spin-memory effect can be observed without charge memory, see below.

IV. SWITCHING

In the following we analyze whether fast switching be-tween the two spin states is feasible. To this end we consider

voltage sweeps with different velocities, ␶exp is the

charac-teristic time of the voltage change. At this point an assump-tion about the relaxaassump-tion time ␶V of the vibrons without

change in the charge state is due. We assume that the relax-ation is fast, ␶VⰆ␶_␴,␶exp so that after an electron tunneling

event the system relaxes rapidly into the vibronic ground state 兩␴0典 or 兩00典. In this case the probabilities P␴=兺qPq␴of

the charged states and P0₌_兺

qPq0 of the neutral state are

de-termined from dP0 dt =

兺

_␴ 共␥ 0␴_P␴₋_␥␴0_P0_兲, _共16兲 dP␴ dt =␥ ␴0_P0₋_␥0␴_P␴_, _共17兲 where the switching rates␥␴0,␥0␴, at finite voltage are cal-culated from Eqs. 共6兲, 共7兲, and 共13兲

␥␴0_{共V兲 =}

兺

q e−ggq q! 兵⌫L␴f 0_关_˜_⑀ ␴

⬘

+␻₀q −共1 −␩兲eV兴 +⌫R␴f0共˜⑀_␴

⬘

+␻0q +␩eV兲其, 共18兲 0 1 2 3 4 5 6

λ/ω

0 0.0 0.2 0.4 0.6

(

τ

σ

Γ)

−1 σ→0 0→σ

FIG. 2. Inverse lifetime ␥␴0/⌫ of the neutral state 共thin black lines兲 and the inverse spin lifetime 共␶_␴⌫兲−1共thick gray lines兲 as a function of the scaled electron-vibron coupling

冑

g =␭/␻0 at

⑀0=␭2/2␻0 共solid lines兲 and at ⑀0= 0.1␭2/␻0 共dashed lines兲,

T = 0.1␻0. -100 -75 -50 -25 0 25 50 75 100

eV

/ω

₀ 0.0 0.2 0.4 0.6 0.8 1.0

(

τ

σ

Γ)

−1

FIG. 3. 共Color online兲 Inverse spin lifetime ␶_␴−1=␥0␴ as a function of normalized bias voltage eV/␻0 at ␭/␻0= 3 ,⑀0

=␭2_/2␻

0, T = 0.1␻0, for the spin-up state 共thin red lines兲 and the

spin-down state共thick blue lines兲 for two different polarizations in the leads: strongly polarized共␬=0.01, solid lines兲 and the same for a less polarized junction共␬=0.1, dashed lines兲.

-100 -50 0 50 100

eV

␥0␴_{共V兲 =}

兺

q e−g_gq q! 兵⌫L␴f 0_关−_˜_⑀ ␴

⬘

+␻₀q +共1 −␩兲eV兴 +⌫R␴f0共−˜⑀_␴

⬘

+␻0q −␩eV兲其. 共19兲 The voltage dependence of the inverse spin lifetimes is de-picted in Fig. 3. If the voltage is large enough, the Franck-Condon blockade is overcome and the system is switched into the spin-up 共spin-down兲 state at positive 共negative兲 voltage. If the bias voltage is swept fast enough, i.e., faster than the long spin lifetime at zero voltage,␶expⰆ␶_␴共0兲, both

spin states can be considered as stable at zero voltage 共spin-memory effect兲 and hysteresis can take place. This is shown in Fig.4where the solid共dashed兲 lines mark the spin popu-lation for increasing 共decreasing兲 bias voltage. Figure 5

shows how the hysteresis behavior depends on the voltage sweep velocity. In the adiabatic limit the voltage change is so slow that the system relaxes into the equilibrium state, and the population-voltage curve is single-valued, without hys-teresis 共middle line in Fig. 5兲. Note that we assume the

re-laxation of vibrons to be fast. The other time scale-spin life-time, is actually very long at strong electron-vibron coupling so that small ␶exp can be large enough in comparison with

vibron lifetime.

Finally, we address the question how to discuss the spin-memory effect proposed. It is quite difficult to measure di-rectly a single-spin polarization in an experiment. A reason-able alternative is to study the signatures of the spin polarization in the current-voltage curves which are most easily accessible experimentally. To this end consider now a three-terminal setup, where the third electrode共which can be the gate electrode with tunneling coupling to the system兲 is ferromagnetic. The tunneling coupling to this electrode is

very weak so that it does not perturb the state. The test tun-neling current is calculated simultaneously with the current between left and right electrodes. In Fig. 6 we show the result of this calculation at large negative constant voltage applied to the test electrode. The bias-voltage dependence of the test current is shown as red and blue lines for different orientations of the test electrode magnetization. Since the test current is sensitive to the magnetization of the test electrode, the spin state can be monitored during the experiment. Ad-ditionally, such a small current can be used to readout the memory element.

V. CONCLUSION

In conclusion, we demonstrated within the framework of a polaron model for a single-level quantum dot coupled to ferromagnetic leads that, owing to Franck-Condon blockade, quantum switching between the two spin states of the central system is strongly suppressed at small bias, but applying a finite bias, the population of spin-up and spin-down states can be controlled. By taking into account nonstationary ef-fects, in particular, the interplay between the time scales of bias sweeping and spin switching, we demonstrate electri-cally controlled hysteretic behavior and bistability in the spin polarization.

ACKNOWLEDGMENTS

This work was funded by the Deutsche Forschungsge-meinschaft within the Priority Program SPP 1243 and Collaborative Research Center SFB 689.

1_{C. Iacovita, M. V. Rastei, B. W. Heinrich, T. Brumme, J. Kortus,}

L. Limot, and J. P. Bucher, Phys. Rev. Lett. 101, 116602共2008兲.

2_{F. Meier, L. Zhou, J. Wiebe, and R. Wiesendanger, Science 320,}

82共2008兲.

3_{X. H. Qiu, G. V. Nazin, and W. Ho, Phys. Rev. Lett. 92, 206102}

共2004兲.

4_{S. W. Wu, G. V. Nazin, X. Chen, X. H. Qiu, and W. Ho, Phys.}

Rev. Lett. 93, 236802共2004兲. -100 -50 0 50 100

eV

/ω

₀ -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2

P

↑—

P

↓

FIG. 5. 共Color online兲 Voltage dependence of the spin polariza-tion for ␭/␻0= 3 , ⑀0=␭2/2␻0, ␬=0.01, T=0.1␻0, for three

dif-ferent sweep velocities ␻0␶exp= 102 共black兲, ␻0␶exp= 103 共green兲,

␻0␶exp= 105共blue兲, solid lines are for increasing voltage and dashed

lines for decreasing voltage, the curve in the adiabatic limit ␶exp→⬁ is not hysteretic 共red dashed line兲.

-100 -50 0 50 100

eV

/ω

₀

C

ur

ren

t

[a

rb

.u

.]

FIG. 6. 共Color online兲 Bias current 共top black兲 and test current for spin-up共middle red兲 and spin-down 共bottom blue兲 test electrode magnetization as a function of normalized voltage eV/␻0 for

␭/␻0= 3 , ⑀0=␭2/2␻0, ␬=0.01, T=0.1␻0. Solid lines denote

in-creasing voltage and dashed lines dein-creasing voltage. The curves are displayed with vertical offset in arbitrary units. The magnitude of the test current being much smaller than the bias current, is plotted enhanced.

(5)

5_{J. Repp, G. Meyer, S. M. Stojković, A. Gourdon, and}

C. Joachim, Phys. Rev. Lett. 94, 026803共2005兲.

6_{A. C. Hewson and D. M. Newns, J. Phys. C 12, 1665}_共1979兲. 7_{A. S. Alexandrov and A. M. Bratkovsky, Phys. Rev. B 67,}

235312共2003兲.

8_{M. Galperin, M. A. Ratner, and A. Nitzan, Nano Lett. 5, 125}

共2005兲.

9_{A. Mitra, I. Aleiner, and A. J. Millis, Phys. Rev. Lett. 94,}

076404共2005兲.

10_{D. Mozyrsky, M. B. Hastings, and I. Martin, Phys. Rev. B 73,}

035104共2006兲.

11_{D. A. Ryndyk, P. D’Amico, G. Cuniberti, and K. Richter, Phys.}

Rev. B 78, 085409共2008兲.

12_{P. D’Amico, D. A. Ryndyk, G. Cuniberti, and K. Richter, New J.}

Phys. 10, 085002共2008兲.

13_{J. Repp, G. Meyer, F. E. Olsson, and M. Persson, Science 305,}

493共2004兲.

14_{S. Datta, W. Tian, S. Hong, R. Reifenberger, J. I. Henderson, and}

C. P. Kubiak, Phys. Rev. Lett. 79, 2530共1997兲.

15_{T. Rakshit, G.-C. Liang, A. W. Ghosh, M. Hersam, and S. Datta,}

Phys. Rev. B 72, 125305共2005兲.

16_{I. G. Lang and Y. A. Firsov, Sov. Phys. JETP 16, 1301}_共1963兲. 17_{A. C. Hewson and D. M. Newns, Jpn. J. Appl. Phys., Suppl. 2,}

121共1974兲.

18_{S. Braig and K. Flensberg, Phys. Rev. B 68, 205324}_共2003兲. 19_{A. Mitra, I. Aleiner, and A. J. Millis, Phys. Rev. B 69, 245302}

共2004兲.

20_{J. Koch and F. von Oppen, Phys. Rev. Lett. 94, 206804}_共2005兲. 21_{J. Koch, M. Semmelhack, F. von Oppen, and A. Nitzan, Phys.}

Rev. B 73, 155306共2006兲.

22_{J. Koch, F. von Oppen, and A. V. Andreev, Phys. Rev. B 74,}

205438共2006兲.

23_{K. Huang and A. Rhys, Proc. R. Soc. London, Ser. A 204, 406}

共1950兲.

24_{W. Rudziński and J. Barnaś, Phys. Rev. B 64, 085318}_共2001兲. 25_{W. Rudziński, J. Barnaś, R. Świrkowicz, and M. Wilczyński,}