Dynamical spin-blockade in a quantum dot with paramagnetic leads

Europhys. Lett., 66(3), pp. 405–411 (2004) DOI:10.1209/epl/i2004-10009-9

Dynamical spin-blockade in a quantum dot with paramagnetic leads

A. Cottetand W. Belzig

Departement Physik und Astronomie, Universit¨at Basel Klingelbergstrasse 82, CH-4056 Basel, Switzerland

(received 26 January 2004; accepted 20 February 2004) PACS.73.23.-b – Electronic transport in mesoscopic systems.

PACS.72.70.+m – Noise processes and phenomena.

PACS.72.25.Rb – Spin relaxation and scattering.

Abstract. – We investigate current fluctuations in a three-terminal quantum dot in the se- quential tunneling regime. Dynamical spin-blockade can be induced when the spin-degeneracy of the dot states is lifted by a magnetic field. This results in super-Poissonian shot noise and positive zero-frequency cross-correlations. Our proposed setup can be realized with semicon- ductor quantum dots.

Introduction. – Non-equilibrium current noise in mesoscopic structures is a consequence of the discreteness of the charge carriers (for reviews, see refs. [1,2]). For conductors with open channels the fermionic statistics of electrons results in a suppression of shot noise below the classical Schottky limit [3]. This was first noted by Khlus [4] and Lesovik [5] for single-channel conductors. Subsequently, B¨uttiker generalized this suppression for many-channel conduc- tors [6]. Mesoscopic conductors are often probed by two or more leads. The quantum statis- tics induces cross-correlations between the currents in different terminals. Since these cross- correlations vanish in the classical limit, even their sign is not obviousa priori. Using only the unitarity of the scattering matrix, B¨uttiker proved that cross-correlations for non-interacting fermions are always negative for circuits with leads maintained at constant potentials [7].

Note that this also holds in the presence of a magnetic field. It has also been found that an interacting paramagnetic dot shows negative cross-correlations in the absence of a magnetic field [8]. Spin-dependent cross-correlations in a non-interacting 4-terminal spin valve were studied [9] and found to be negative. On the experimental side, negative cross-correlations were measured by Hennyet al.[10, 11] and Oliver et al.[12] in mesoscopic beam splitters.

Several ways to produce positive cross-correlations in fermionic systems have been pro- posed (see, e.g., [13] for a recent review). Among these possibilities are sources which inject correlated electrons [14–27] and finite-frequency voltage noise [13, 28]. The question of the existence of intrinsic mechanisms,i.e.due to interactions occurring in the beam splitter device itself, has been answered positively by us [29]. Surprisingly, a simple quantum dot connected to ferromagnetic contacts can lead to positive cross-correlations due to the so-calleddynami- cal spin-blockade. Simply speaking, up-and down-spins tunnel through the dot with different

c EDP Sciences Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2007/3329/

URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-33290

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rates. In the limit where the Coulomb interaction prevents a double occupancy of the dot, the spins which tunnel with a lower rate modulate the tunneling of the other spin direction, which leads to an effective bunching of tunneling events. In a three-terminal geometry with one input and two outputs, this results in positive cross-correlation between the two out- put currents. Independently, Sauret and Feinberg proposed a slightly different setup of a ferromagnetic quantum dot, which also produces positive cross-correlations [30].

Experimentally, it is more difficult to fabricate quantum dots with ferromagnetic leads.

However, quantum dots with paramagnetic leads have shown to exhibit spin-dependent trans- port. A magnetic field lifts the spin-degeneracy and a spin-polarized current with nearly 100%

efficiency can be created [31, 32]. In this letter, we will address the current correlations in a few-electron quantum dot connected to three paramagnetic leads. We will show below that positive cross-correlations can be produced in this device simply by applying amagnetic field.

Furthermore, this system also shows a super-Poissonian shot noise.

To arrive at these conclusions we consider a quantum dot with one orbital energy level E₀ connected to three terminals by tunnel contacts. The junctions are characterized by bare tunneling rates γ_i (i = 1,2,3) and capacitances C_i. We assume that a magnetic field B is applied to the dot, which leads to a Zeeman splitting of the level according to E_↓(↑) = E₀+ (−)gµ_BB/2, whereµ_B =e/2m is the Bohr magneton. The double occupancy of the dot costs the charging energyE_c=e²/2C, withC =

iC_i. The energy spacing to the next orbital is ∆. We will assume

k_BT, eV, µ_BBE_c,∆. (1) According to these inequalities, the dot can be only singly occupied and we have to take into account only one orbital level.

In the sequential-tunneling limitγ_jk_BT, the time evolution of the occupation proba- bilitiesp_ψ(t) of states ψ∈ {↑,↓,0} is described by the master equation

d

dtp_ψ =M_ψϕp_ϕ, (2)

where

Mˆ =





−Γ⁻_↑ −Γ_↓↑ Γ_↑↓ Γ⁺_↑ Γ_↓↑ −Γ⁻_↓ −Γ_↑↓ Γ⁺_↓ Γ⁻_↑ Γ⁻_↓ −Γ⁺_↑ −Γ⁺_↓



. (3)

The rate for an electron to tunnel on/off the dot (= +/−) through junction j is given by Γ_jσ =γ_j/(1 + exp[(E_σ−eV_j)/k_BT]), whereV₁ =V₃ =−C₂V/C and V₂ = (C₁+C₃)V/C.

Here, we took the Fermi energyE_F= 0 for lead 2 as a reference. The total tunneling rates are Γ_σ=

jΓ_jσ andγ=

jγ_j. Spin flips on the dot are described by rates Γ_{↓↑(↑↓)}, which obey the detailed balance rule Γ_↑↓/Γ_↓↑= exp[gµ_BB/k_BT]. From eq. (2), the stationary occupation probabilities ¯p_σ are

p¯_σ= Γ⁺_σΓ⁻_−σ+ Γ_σ,−σ

Γ⁺_↑ + Γ⁺_↓ γ²−Γ⁺_↑Γ⁺_↓ +

γ+ Γ⁺_↓ Γ_↑↓+

γ+ Γ⁺_↑ Γ_↓↑ , (4) and ¯p₀ = 1−p¯_↑−p¯_↓. These probabilities can be used to calculate the average value I_j of the tunneling currentI_j(t) through junctionj as

I_j =e

,σ

Γ_jσp¯_A(σ,−), (5)

Fig. 1 – Current-voltage characteristic of the circuit shown in the inset forE₀ <0,C₁ =C₂ =C₃, γ₁ =γ₃,k_BT/|E0| = 0.05, gµ_BB/|E0|= 1, and different values ofγ₂/γ. The average currentI2 through lead 2 is plotted in units ofeγ_p= 2eγ₂γ/(˜˜ γ+2γ₂); the voltage is in units ofE₀. The positions ofV₊ andV₋are indicated in dotted lines.

where A(σ, ) is the state of the dot after the tunneling of an electron with spin σ in the direction , i.e., A(σ,−1) = 0 and A(σ,+1) = σ. The frequency spectrum of the noise correlations can be defined as

S_ij(ω) = 2 _+∞

−∞

dtexp[iωt]

∆I_i(t)∆I_j(0)

, (6)

where ∆I_i(t) =I_i(t)− I_i is the deviation from the average current in terminali. It can be calculated using the method developed in refs. [33–35] as

S_ij(ω)

2e² =

,σ

Γ_jσp¯_A(−,σ)δ_ij+

σσ

S_iσjσ (ω), (7)

where the first term is the Schottky noise produced by tunneling through junctionj, and S_iσjσ (ω) = Γ_iσG_A(σ,−),A(σ,)(ω)Γ_jσp¯_A(σ,−)+ Γ_jσG_A(σ,−),A(σ,)(−ω)Γ_iσp¯_A(σ,−), (8) withG_ψ,ϕ(ω) = ¯p_ψ/iω−(iω+M)⁻¹_ψ,ϕ.

In the following we study the dot in a beam-splitter configuration, in which a bias voltage V is applied between terminal 2 and terminals 1 and 3. We consider the caseV >0, so that it is energetically more favorable for electrons to go from lead 2 to leads 1 and 3. We will limit our discussion to the case in which the two Zeeman sublevels are below the Fermi energy at equilibrium (i.e. E₀±gµ_BB/2<0). The opposite case was discussed in ref. [36] for a two- terminal dot. We are mostly interested in the total zero-frequency current noiseS₂₂=S₂₂(0) and the cross-correlationsS₁₃ =S₁₃(0) between the two output leads. It is useful to define the Fano factorF₂=S₂₂/2eI₂ and, correspondingly,F₁₃=S₁₃/2eI₂ .

In the following, we will first assume that k_BT gµ_BB. Transport through the down level is energetically allowed forV V₋= (−E₀−gµ_BB/2)C/eC₂. However, forV V₊ = (−E₀+gµ_BB/2)C/eC₂, the dot is blocked by an up spin, thus down spins cannot cross the

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dot. AroundV V₊, the lower Zeeman level is close to the Fermi level of leads 1 and 3, as represented by the level diagram in the lower right inset of fig. 1. The blockade of the dot is then partially lifted and transport through both levels is allowed. In this regime, we can write the tunneling rates as Γ⁺_2σ =γ₂, Γ⁻_2σ = 0, Γ⁻_1(3)↑ =xγ₁₍₃₎, Γ⁺_1(3)↑ = (1−x)γ₁₍₃₎, Γ⁺_1(3)↓ = 0, and Γ⁻_1(3)↓=γ₁₍₃₎, wherex= 1/(1 + exp[−(E0−gµ_BB−eV₁)/k_BT]) ranges from 0 to 1 with increasing voltage. Furthermore, takingγ_sf = 0 and ˜γ=γ₁+γ₃, we find for the current

I₂ = 2exγ₂γ˜

γ˜+γ₂(1 +x), (9)

for the Fano factor

F₂= 1 + 2γ₂

˜γ(1−3x) + (1−x)²γ₂

γ˜+γ₂(1 +x) ² , (10) and for the cross-correlations

F₁₃=γ₁γ₃ γ˜²

2(1−x)²γ₂³+ (1−7x+x²+x³)γ₂²˜γ−2(1−x²)γ₂γ˜²−(1−x)˜γ³

γ₂(˜γ+ (1 +x)γ₂)² . (11)

We observe that the current increases with voltage (i.e. with x) around the voltage step V₊. Note that this current is not spin-polarized because up and down spin have the same probability to enter the dot, regardless of what happens at the output. The Fano factor F₂ and the cross-correlations F₁₃ deviate from their Poissonian values depending on the applied voltage. Our main results are that F₂ can be super-Poissonian andF₁₃positive forx <1, as can be cleary seen from (10) or (11) in the limitγ₂γ. These features are a consequence of˜ dynamical spin-blockade: up spins leave the dot with a rate smaller than down spins, leading to a bunching of tunneling events [29]. In the limit x → 1, the Fano factor is always sub- Poissonian and the cross-correlations always negative. This is due to the fact that the tunnel rates of up and down spins are equal, thus the Zeeman splitting plays no role and the dot is equivalent to a simple quantum dot with a spin-degenerate level. In the limit x→0, one could also expect the super-Poissonian nature of F₂ and the positivity ofF₁₃to be lost since the transport is enabled only by thermally activated processes. However, below the voltage thresholdV₊, the Fano factor tends to

F₂= 1 + 2γ₂

γ₂+ ˜γ, (12)

which is always super-Poissonian. If the coupling to terminal 2 dominates, i.e. γ₂ ˜γ, the Fano factor takes a maximal value of 3. In the opposite limit, ˜γ γ₂, F₂ approaches the Poisson limit of uncorrelated single charge transfer. It is interesting to note that a symmetric junction ˜γ = γ₂ produces twice the Poisson noise level. The cross-correlations in the same limit have the form

F₁₃=γ₁γ₃ γ˜²

(2γ₂+ ˜γ)(γ₂−˜γ)

γ₂(γ₂+ ˜γ) . (13)

In the three cases discussed above, the cross-correlations thus take the limiting values F₁₃=

−γ₁γ₃/γ₂˜γfor ˜γ γ₂,F₁₃= 0 for ˜γ=γ₂ andF₁₃= 2γ₁γ₃/˜γ² forγ₂˜γ. Hence, both the super-Poissonian nature of F₂ and the positivity of F₁₃ can persist for x →0. Even if the transport is enabled only by thermally activated processes, dynamical spin-blockade already results in a correlated transfer of electrons.

We now turn to the discussion of the general results displayed in figs. 1-3, obtained from an exact treatment of the full master equation. Figure 1 shows the full voltage dependence

Fig. 2 – Fano factorF₂=S₂₂/2eI2of the total current as a function of voltage, for the same circuit parameters as in fig. 1. Left panel: data for different values ofγ₂/˜γandγ_sf = 0. Right panel: effect of spin-flip scattering forγ₂/γ = 5 and different values ofγ_sf. The cu rves displayed in both panels are independent of the asymmetry between the output leads.

of the average current. As expected, the current shows a single step atV ≈V₊ [37–41]. The step width is about 10k_BT C/C₂, whereas its position varies only slightly with the asymmetry of the junctions (the maximal variation is about 0.7k_BT C/C₂).

The left panel of fig. 2 shows the voltage dependence of the Fano factor in the absence of spin-flip scattering, for some values ofγ₂/˜γ. The divergence 2k_BT/eV of the Fano factor at zero voltage is simply a result of the dominating thermal noise in the limit k_BT > eV. Note that, similarly to I₂ , the Fano factor F₂ shows one single step at V ∼ V₊. The right panel of fig. 2 shows the effect of spin-flip scattering onF₂, for the caseγ₂ = 5˜γ. For V₋ < V < V₊, spin-flips become effective when Γ_↑↓ =γ_sfexp[gµ_BB/2k_BT]∼γ_i, see eq. (3).

The sensitivity toγ_sf thus increases withB. BelowV₋, even smaller spin-flip rates suppress the super-Poissonian noise because the dwell time of electrons on the dot is very long. Far aboveV₊, spin-flip scattering has no effect on the sub-Poissonian noise.

The left panel of fig. 3 shows the voltage dependence of the cross-correlations factorF₁₃ between the two output terminals, for the same parameters as in fig. 2. First, around the voltage thresholdV₊, we observe the features discussed above. The cross-correlations develop from a positive or negative level belowV₊ depending on the ratioγ₂/˜γto the usual negative cross-correlations aboveV₊, where the spin-splitting plays no role anymore. Remarkably, in contrast toF₂, the cross-correlations also show a step around the lower voltage thresholdV₋. This illustrates clearly thatF₁₃ and F₂ are qualitatively different. The absence of the lower step forF₂ can be interpreted as a consequence of the unidirectionality of tunneling through junction 2. Indeed, Γ⁻_2σ→0 means that F₂ depends only on ¯p₀ andG_0,↑(↓)(see (5) and (8)).

Now, forV ∼V₋, Γ⁻_1/3↑→0 implies that the contribution of these terms is independent ofV. On the contrary,F₁₃ also depends on ¯p_↑(↓) andG_σ,0 withσ∈ {↑,↓,0}. For Γ⁻_1/3↑→0, these last terms depend strongly on Γ⁻_1/3↓which varies itself significantly with V aroundV₋. Note that the absence of a step in F₂ implies a redistribution of the noise between S₁₁, S₃₃ and S₁₃ when the threshold V₋ is crossed (due to charge conservation,S₂₂ =S₁₁+S₃₃+ 2S₁₃).

The extra step ofF₁₃ disappears forγ₂ γ. In this limit, the cross-correlations display a˜ single low-voltage plateauF₁₃= 2γ₁γ₃/˜γ², which is an upper bound for the two low-voltages plateaux found in the general case. The inset in the left panel of fig. 3 shows the effect of

410 EUROPHYSICS LETTERS

Fig. 3 – Left: cross-Fano factorF₁₃=S₁₃/2eI2 between leads 1 and 3 as a function of voltage, for the same circuit parameters as in fig. 1 and different values ofγ₂/γ. In all curves of the main frame γ_sf = 0. The inset shows the effect of spin-flip scattering for γ₂/γ = 5 and different values of γ_sf. Right: F₁₃ as a function of voltage for different values ofB. The curves are shown forγ₂/γ= 5. A cross indicates the position ofV₋for each case.

spin-flip scattering on the cross-correlations. As expected, they suppress all spin effects and the positive cross-correlations become finally negative. As for the Fano factor, very small spin-flip scattering ratesγ_sf are already sufficient to modifyF₁₃forV < V₊.

Since the positive cross-correlations found in this work are intimately related to the dy- namical spin-blockade, we expect a strong dependence on the magnetic field. The right panel of fig. 3 shows the voltage dependence of F₁₃ around the step V₊, for a fixed temperature and various magnetic fields. Just below V₊ the limiting value of F₁₃ is determined by for- mula (13). Thus, for a constant voltage V ≤ V₊ we predict a crossover from negative to positive cross-correlations with increasing magnetic field. One can see a qualitative change in the curves, which can be understood by a gradual splitting of the voltage steps V₋ and V₊. The lower step is at V₋−V₊ = −gµ_BBC/C₂e. As long as gµ_BB 10k_BT, the two voltage steps are indistinguishable. However, positive cross-correlations are already expected forgµ_BB 2k_BT. For gµ_BB= 6k_BT, the two steps still overlap, resulting in a broad peak, whereas for the higher magnetic fieldgµ_BB= 20k_BT the lower threshold atV₋is outside the plotting region.

The regimeV ∼V₊has the advantage that the current is not exponentially small (cf. fig. 1) and thus observable more easily in an experiment. Forγ₁ =γ₂/5 =γ₃, the maximum value obtained for the cross-correlations is S₁₃ 0.09e²γ_p at x0.17, i.e.V 4.26E₀ in fig. 3.

Withγ_p5 GHz, this corresponds to 10⁻²⁹A²s, a noise level accessible experimentally [42].

In conclusion, we have studied current correlations for a three-terminal quantum dot with unpolarized leads, placed in a magnetic field. Below the voltage threshold V₊, as a result of dynamical spin-blockade, the Fano factor of the input current shows an interesting super- Poissonian behavior and the cross-correlations in the two output leads can be positive. At higher voltages, the Fano factor becomes sub-Poissonian and the cross-correlations negative, as usual. The effect we predict should be observable in semiconductor quantum dots of ref. [41].

∗ ∗ ∗

We acknowledge discussions withC. Bruder, H.-A. EngelandT. Kontos. This work was financially supported by the RTN Spintronics, the Swiss NSF and the NCCR Nanoscience.

REFERENCES

[1] Blanter Ya. M.andB¨uttiker M.,Phys. Rep.,336(2000) 1.

[2] Nazarov Yu. V.(Editor),Quantum Noise in Mesoscopic Physics(Kluwer, Dordrecht) 2003.

[3] Schottky W.,Ann. Phys. (Leipzig),57(1918) 541.

[4] Khlus V. A.,Sov. Phys. JETP,66(1987) 1243.

[5] Lesovik G. B.,JETP Lett.,49(1989) 592.

[6] B¨uttiker M.,Phys. Rev. Lett.,65(1990) 2901.

[7] B¨uttiker M.,Phys. Rev. B,46(1992) 12485.

[8] Bagrets D. A.andNazarov Yu. V.,Phys. Rev. B,67(2003) 085316.

[9] Belzig W.andZareyan M., cond-mat/0307070.

[10] Henny M.et al.,Science,284(1999) 296.

[11] Oberholzer S. et al.,Physica E,6(2000) 314.

[12] Oliver W. D., Kim J., Liu R. C.andYamamoto Y.,Science,284(1999) 299.

[13] See the article of B¨uttiker M.in ref. [2].

[14] Martin T.,Phys. Lett. A,220(1996) 137.

[15] Anantram M. P.andDatta S.,Phys. Rev. B,53(1996) 16390.

[16] Torres J.andMartin T.,Eur. Phys. J. B,12(1999) 319.

[17] Gramespacher T.andB¨uttiker M.,Phys. Rev. B,61(2000) 8125.

[18] Torres J., Martin T.andLesovik G. B.,Phys. Rev. B,63(2001) 134517.

[19] B¨orlin J., Belzig W.andBruder C.,Phys. Rev. Lett.,88(2002) 197001.

[20] Samuelsson P.andB¨uttiker M.,Phys. Rev. Lett.,89(2002) 046601.

[21] Samuelsson P.andB¨uttiker M.,Phys. Rev. B,66(2002) 201306.

[22] Taddei F.andFazio R.,Phys. Rev. B,65(2002) 134522.

[23] Texier C.andB¨uttiker M.,Phys. Rev. B,62(2000) 7454.

[24] Safi I., Devillard P.andMartin T.,Phys. Rev. Lett.,86(2001) 4628.

[25] Crepieux A., Guyon R., Devillard P.andMartin T.,Phys. Rev. B,67(2003) 205408.

[26] S´anchez D., L´opez R., Samuelsson P.andB¨uttiker M.,Phys. Rev. B,68(2003) 214501.

[27] Bignon G., Houzet M., Pistolesi F.andHekking F. W. J., cond-mat/0310349.

[28] Martin A. M.andB¨uttiker M.,Phys. Rev. Lett.,84(2000) 3386.

[29] Cottet A., Belzig W.andBruder C., cond-mat/0308564.

[30] Sauret O.andFeinberg D., cond-mat/0308313.

[31] Hanson R.et al., cond-mat/0311414.

[32] Recher P., Sukhorukov E. V.andLoss D.,Phys. Rev. Lett.,85(2000) 1962.

[33] Korotkov A. N.,Phys. Rev. B,49(1994) 10381.

[34] Hershfield S.et al.,Phys. Rev. B,47(1993) 1967.

[35] Hanke U.et al.,Phys. Rev. B,48(1993) 17209.

[36] Thielmann A., Hettler M. H., K¨onig J¨urgen and Sch¨on G., Phys. Rev. B, 68 (2003) 115105.

[37] Ralph D. C., Black C. T.andTinkham M.,Phys. Rev. Lett.,74(1995) 3241.

[38] Ralph D. C., Black C. T.andTinkham M.,Phys. Rev. Lett.,78(1997) 4087.

[39] Cobden D. H.et al.,Phys. Rev. Lett.,81(1998) 681.

[40] Cobden D. H.andNygard J.,Phys. Rev. Lett.,89(2002) 046803.

[41] Hanson R.et al.,Phys. Rev. Lett.,91(2003) 196802.

[42] Birk H., de Jong M. J. M.andSch¨onenberger C.,Phys. Rev. Lett.,75(1995) 1610.