Breaking of Coulomb blockade by macrospin-assisted tunneling
BBreaking of Coulomb blockade by macrospin-assisted tunneling
Tim Ludwig , Rembert A. Duine , Institute for Theoretical Physics, Utrecht University,Princetonplein 5, 3584 CC Utrecht, The Netherlands and Department of Applied Physics, Eindhoven University of Technology,P.O. Box 513, 5600 MB Eindhoven, The Netherlands
A magnet with precessing magnetization pumps a spin current into adjacent leads. As a specialcase of this spin pumping, a precessing macrospin (magnetization) can assist electrons in tunneling.In small systems, however, the Coulomb blockade effect can block the transport of electrons. Here,we investigate the competition between macrospin-assisted tunneling and Coulomb blockade for thesimplest system where both effects meet; namely, for a single tunnel junction between a normalmetal and a metallic ferromagnet with precessing magnetization. By combining Fermi’s golden rulewith magnetization dynamics and charging effects, we show that the macrospin-assisted tunnelingcan soften or even break the Coulomb blockade. The details of these effects—softening and breakingof Coulomb blockade—depend on the macrospin dynamics. This allows, for example, to measurethe macrospin dynamics via a system’s current-voltage characteristics. It also allows to control aspin current electrically. From a general perspective, our results provide a platform for the interplaybetween spintronics and electronics on the mesoscopic scale. We expect our work to provide a basisfor the study of Coulomb blockade in more complicated spintronic systems.
Introduction.—
To compete with modern electronics,systems of spintronics—the spin analog of electronics—are becoming smaller. In turn, mesoscopic effects be-come more important. On the one hand, this complicatesthe description of spintronic effects. On the other hand,however, it opens up new ways to investigate and ma-nipulate spintronic systems. Here, we demonstrate howthe Coulomb blockade—a prominent effect of mesoscopicphysics—can be used to measure magnetization dynam-ics via a system’s current-voltage characteristics.We consider a single tunnel junction between a nor-mal metal and a metallic ferromagnet; see Fig. 1. Atunnel junction is a thin insulating layer which separatestwo metallic systems (leads) from each other. Classi-cally, a tunnel junction forms a capacitor, as the insulat-ing layer separates the two metallic systems by a smalldistance. Quantum mechanically—as the name ”tunneljunction” suggests—electrons can tunnel through the in-sulating layer. The classical and quantum perspectivesare related: when an electron tunnels through the junc-tion, it changes the charge on the capacitor and, in turn,it changes the electrostatic Coulomb energy stored in thecapacitor [1, 2]. If an electron does not have enough en-ergy to compensate the cost in Coulomb energy, then thetunneling is blocked; this is the Coulomb blockade [1, 2].The energy to overcome Coulomb blockade can comefrom thermal fluctuations, from the voltage source, or—in the present case—from the magnetization dynamics.To focus on the role of the magnetization dynamics, weconsider the limit of zero temperature. For simplicity,we assume the magnetization to precess in a steady stateand we use the macrospin approximation; that is, we de-scribe the magnetization as a single vector M . A precess-ing macrospin (magnetization) acts as a time-dependentmagnetic field for electrons and—as a special case of adi- abatic pumping [3, 4]—pumps spin-polarized electronsinto adjacent leads [5–7]. For a tunnel junction, thismeans that a precessing macrospin can assist electronsin tunneling [8]; see also [9, 10]. FIG. 1: We consider a single tunnel junction between a nor-mal metal (left lead) and a metallic ferromagnet (right lead)with a magnetization in a steady state precession. Separatingtwo metallic systems, the tunnel junction forms a capacitor(capacitance C ). For R (cid:29) R K = h/e , an electron, when tun-neling, changes the charge on the capacitor by e and, in turn,it looses the charging energy ( E c = e / C ) to its electrostaticenvironment [1, 2]. For low voltage ( eV < E c ) and low tem-perature ( k b T (cid:28) E c ), this energy loss usually forbids chargetransport (Coulomb blockade) [1, 2]. However, we show thatthe precessing magnetization can break the Coulomb blockadeby assisting electrons in tunneling. In this letter, we study the competition betweenCoulomb blockade and macrospin-assisted tunneling.This places our work into the emerging field of meso-scopic spintronics. Other topics in this field include, forexample, the study of noise in spintronics [8, 11–17], themesoscopic Stoner instability [18–20], or spintronics withquantum dots [21–24] which is closely related to our work. a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Here, we show that the macrospin-assisted tunneling canprovide enough energy to overcome the Coulomb block-ade. Thereby, the macrospin-assisted tunneling softensor even breaks the Coulomb blockade and, as a result,the system’s current-voltage characteristics reveals themacrospin dynamics.For slow precession, the macrospin-assisted tunnelingsoftens the Coulomb blockade; see Fig. 2. For strongprecession, the macrospin-assisted tunneling breaks theCoulomb blockade; see Fig. 3. In both cases, themacrospin dynamics governs the flow of charge current.Read in reverse: the charge current can be used to mea-sure the magnetization dynamics; see Figs. 2 and 3. Tobe more explicit, we choose the macrospin’s precessionaxis as the z − direction. The precessing macrospin is thendescribed by M = M (sin θ cos φ, sin θ sin φ, cos θ ), wherepolar angle θ and precession frequency ˙ φ are constant.Roughly speaking: the polar angle θ can be inferred fromthe tunnel junction’s differential conductance; while theprecession frequency ˙ φ can be inferred from the combinedinformation of conductance and differential conductance.Alternatively, one can infer θ and ˙ φ from the position ofkinks in the current-voltage characteristics; see Figs. 2and 3. The Model. —The breaking of Coulomb blockade bymacrospin-assisted tunneling can be found within a sim-ple model: the electrons are described quantum me-chanically by single-particle Hamiltonians; whereas, theCoulomb energy is taken into account on the classicallevel.The magnetic right lead is described by H r = (cid:80) ρσσ (cid:48) | ρσ (cid:105) h σσ (cid:48) r,ρ (cid:104) ρσ (cid:48) | ; where σ and σ (cid:48) denote the spinin z -direction (the magnetization’s precession axis), ρ denotes the right-lead states with corresponding energy (cid:15) ρ , and h r,ρ = (cid:15) ρ − M σ / σ and for simpler notation, the magnetiza-tion length M includes all constants [33]. The left leadis described by H l = (cid:80) λσ | λσ (cid:105) (cid:15) λ (cid:104) λσ | ; where λ de-notes the left-lead states with the corresponding energy (cid:15) λ . We assume spin-conserved tunneling, described by H t = (cid:80) λρσ | ρσ (cid:105) t (cid:104) λσ | + h.c. , where t are the tunnel-ing amplitudes between states ρ and λ ; for simplicity, wedisregard the state-dependence of t .In addition to the Hamiltonian, we need to spec-ify the distribution functions. We assume the tunnel-ing events to be rare, such that local equilibrium is re-established before each tunneling event. Then, the elec-trons are distributed according to the Fermi distribution f ( (cid:15) ) = 1 / [exp[( (cid:15) − µ ) /T ]+1], where we assume the chem-ical potential µ and the temperature T to be equal inboth leads. We emphasize, however, that µ is only the chemical potential—not the electrochemical potential.The (electrostatic) Coulomb energy is taken into ac-count in addition to the single-particle contributions.We assume the environmental resistance R to be large R (cid:29) R K , where R K = h/e is the resistance quantum (von Klitzing constant). Under this assumption, the tun-neling is governed by the electrostatic energy stored inthe capacitor Q / (2 C ), where C is the capacitance of thetunnel junction and Q is its charge [1]. In the following,we focus on the limit of zero temperature T = 0, whichputs the system into the Coulomb blockade regime. The Coulomb blockade regime. —An electron can onlytunnel if it has enough energy available. When one elec-tron tunnels, the charge on the capacitor Q is changedby − e for left-to-right tunneling or by + e for right-to-left tunneling. So, the change in electrostatic en-ergy is ∆ E el = Q / C − ( Q ∓ e ) / C . Assuming tun-neling events to be rare, the capacitor is recharged to Q = CV before each tunneling event. In turn, we find∆ E el = ± eV − E c with the charging energy E c = e / C .If the applied voltage is too small (∆ E el < T = 0, the missing energycannot come from thermal activation. However, the pre-cessing macrospin can assist electrons in tunneling [8]and, thereby, it provides the missing energy. Macrospin-assisted tunneling. —As a first step, we de-termine the tunneling rate between states in the left leadand states in the right lead. We assume the tunnel cou-pling to be a weak perturbation, such that we can useFermi’s golden rule.Before Fermi’s golden rule can be applied, we have tochange to the magnetization’s rotating frame of reference,such that the leads’ Hamiltonians become time indepen-dent. So, following Ref. [8], we apply a transforma-tion U ( t ) = (cid:80) ρσ | ρσ ; M ( t ) (cid:105)(cid:104) ρσ | + (cid:80) λσ | λσ (cid:105)(cid:104) λσ | , where | ρσ ; M ( t ) (cid:105) is an instantaneous eigenstate of M ( t ) ˆ σ ; for-mally, M ( t ) ˆ σ | ρσ ; M ( t ) (cid:105) = σ | ρσ ; M ( t ) (cid:105) with M = | M | .This transformation does not affect the left lead’s Hamil-tonian ˜ H l := U H l U † = H l . But it diagonalizes the rightlead’s Hamiltonian ˜ H r := U H r U † = (cid:80) ρσ | ρσ (cid:105) ξ ρσ (cid:104) ρσ | ,where ξ ρσ = (cid:15) ρ − M σ/
2. The magnetization’s time-dependence is transferred to the tunneling Hamilto-nian ˜ H t := U H t U † = (cid:80) ρλσσ (cid:48) | ρσ (cid:105) [ R † ( t )] σσ (cid:48) t (cid:104) λσ (cid:48) | + h.c. . That is, the tunneling amplitudes become time-dependent and non-trivial in spin space, t → R † ( t ) t . (1)The spin-space rotation R ( t ) is defined by its elements[ R ( t )] σσ (cid:48) := (cid:104) ρσ ; M ( t ) | ρσ (cid:48) (cid:105) . Note, however, that thisdefinition is not unique, as a rotation around the magne-tization direction (spin quantization axis) has no physicaleffect. This gives rise to a gauge freedom which can beused to simplify the calculation [15].Due to its time dependence, the transformation notonly rotates the Hamiltonian but it also generates anew term, − iU ˙ U † = − i (cid:80) ρσσ (cid:48) | ρσ (cid:105) [ R † ˙ R ] σσ (cid:48) (cid:104) ρσ (cid:48) | , inthe rotating-frame Hamiltonian. The spin-off-diagonalpart of − iR † ˙ R induces transitions—also known asLandau-Zener-transitions—between spin-up and spin-down states. However, we assume a large magnetiza-tion length M , such that we can disregard these tran-sitions [34]; this is also known as adiabatic approxima-tion [15]. The remaining spin-diagonal part of − iR † ˙ R gives an additional time-evolution phase—also knownas Berry-phase—which is different for spin-up and spin-down states. However, we eliminate the spin-diagonalpart of − iR † ˙ R by fixing the gauge analog to Ref. [15];that is, we explicitly choose R ( t ) = (cid:18) cos θ e − iω − t − sin θ e − iω + t sin θ e iω + t cos θ e iω − t (cid:19) , (2)with ω ± = ˙ φ (1 ± cos θ ) /
2, where ω − is the rate at whichthe Berry-phase is acquired. Even though this choiceeliminates the spin-diagonal part of − iR † ˙ R , it does noteliminate the Berry-phase. Instead, the Berry-phase istransferred to the tunneling elements, Eq. (1). To sum-marize: for the specific choice, Eq. (2), the newly gen-erated term − iR † ˙ R can be disregarded in an adiabaticapproximation.Now, in the rotating frame, it is straightforward to ap-ply Fermi’s golden rule. Treating the tunneling Hamilto-nian ˜ H t as perturbation, we obtain the golden-rule rateΓ λσ (cid:48) (cid:29) ρσ = 2 π (cid:126) | t | σσ (cid:48) cos θ × δ ( ξ ρσ − (cid:15) λ + σ (cid:48) (cid:126) ω σσ (cid:48) − eV ± E c ) , (3)where + E c and − E c correspond to left-to-right tunnel-ing λσ (cid:48) → ρσ and right-to-left tunneling ρσ → λσ (cid:48) re-spectively. The macrospin orientation governs the spin-projection factor (1 + σσ (cid:48) cos θ ) /
2, which is cos θ forequal spins σ = σ (cid:48) and sin θ for opposite spins σ (cid:54) = σ (cid:48) .The macrospin dynamics enters through the frequency ω σσ (cid:48) = ˙ φ (1 − σσ (cid:48) cos θ ) /
2, which is ω − for equal spins σ = σ (cid:48) and ω + for opposite spins σ (cid:54) = σ (cid:48) . In the rotatingframe, the macrospin dynamics translates into the time-dependence of the perturbation (tunneling Hamiltonian);see Eqs. (1) and (2). Consequently, the macrospin dy-namics induces the energy shift, σ (cid:48) (cid:126) ω σσ (cid:48) , in the golden-rule rate. In other words, the precessing macrospin givesenergy to—or takes energy from—the tunneling elec-trons; that is, it can assist electrons in tunneling [8].Now, knowing the golden-rule rate, Eq. (3), we candetermine the charge current. Charge current. —The net charge current I = I l → r − I r → l is the difference between the left-to-right current I l → r and the right-to-left current I r → l .At first, we focus on the left-to-right current I l → r ; thatis, we consider only electrons that are tunneling fromthe left lead to the right lead. Formally, it is givenby I l → r = e (cid:80) ρλσσ (cid:48) Γ λσ (cid:48) → ρσ f ( (cid:15) λ )[1 − f ( ξ ρσ )], where thegolden-rule rate, Eq. (3), is summed over all states and—since electrons can only tunnel from filled states into FIG. 2: This figure shows the softening of Coulomb blockadefor slow precession ( (cid:126) | ˙ φ | < E c ) with 0 < θ < π/ φ < eV > E c + (cid:126) ω + . The details of the current flow depend onthe macrospin dynamics. Thus, the macrospin dynamics canbe measured (indirectly) by the charge current. The standardCoulomb blockade is included as reference (blue-dashed). empty states—it is weighted by the filling factor f ( (cid:15) λ )and the Pauli-blocking factor [1 − f ( ξ ρσ )]. More explic-itly, I l → r = g t e (cid:88) σσ (cid:48) σσ (cid:48) cos θ × (cid:90) d(cid:15) l (cid:90) d(cid:15) r f ( (cid:15) l − eV )[1 − f ( (cid:15) r )] δ ( (cid:15) r − (cid:15) l + E c + σ (cid:48) (cid:126) ω σσ (cid:48) ) , where we shifted the integrals (cid:15) l → (cid:15) l − eV and (cid:15) r → (cid:15) r + M σ/
2. Furthermore, we assumed the densities ofstates ρ l and ρ r to be constant on all scales smaller than M and to be independent of the spin [35]. The tunnel-ing conductance g t is defined by g t = 8 π | t | ρ l ρ r e /h .From Eq. (4), it becomes clear that eV is just the elec-trical part of the electrochemical potential: f ( (cid:15) l − eV ) =1 / [exp[( (cid:15) l − µ l ) /T ] + 1], where µ l = µ + eV is the elec-trochemical potential of the left lead.For infinite capacitance ( E c = 0) and without mag-netization dynamics ( ω σσ (cid:48) = 0), the δ -function in Eq.(4) ensures the conservation of energy for the tunnelingelectrons. For finite capacitance, however, a tunnelingelectron loses the charging energy E c to the electrostaticenvironment (capacitor) [1, 2]; see Fig. 1. The energyshift σ (cid:48) (cid:126) ω σσ (cid:48) accounts for the effect of the macrospin dy-namics onto the tunneling electron; namely, it describesthe energy gain or loss due to the macrospin precession.Performing the integrals in Eq. (4), we obtain I l → r = g t e (cid:104) cos θ eV − E c − (cid:126) ω − )+ sin θ eV − E c + (cid:126) ω + )+ sin θ eV − E c − (cid:126) ω + )+ cos θ eV − E c + (cid:126) ω − ) (cid:105) , (5)where Π( x ) is the ramp function; that is, Π( x ) = 0 for x ≤ x ) = x for x >
0. The four terms in I l → r arise from the different combinations of spins in left-to-right tunneling.The right-to-left current I r → l can be found analogouslyto the left-to-right current I l → r ; only the roles of theleads are exchanged [36]. Combining both, we find thatthe charge current, I = I l → r − I r → l , is antisymmetricin the voltage; that is, I ( − V ) = − I ( V ). This is a con-sequence of assuming the densities of states to be spin-independent.To gain a better understanding of the charge cur-rent, let’s consider a situation with static macrospin( ˙ φ = 0) at first. In the limit of infinite capacitance( E c = 0), the tunnel junction behaves as a resistor; thatis, the current-voltage relation is described by Ohm’slaw I = g t V . For finite capacitance ( E c > E c tothe electrostatic environment. This loss effectively re-duces the voltage by E c /e . Consequently, we obtain I = g t [( V − E c /e ) Θ( V − E c /e ) − ( − V − E c /e ) Θ( − V − E c /e )],which is the standard Coulomb blockade result [1, 2]: ifthe voltage is too low ( | eV | < E c ), the charge transport isblocked ( I = 0). However, when the macrospin precesses( ˙ φ (cid:54) = 0), it can assist electrons in tunneling; thereby,it softens the Coulomb blockade or—if the precession isstrong enough—it can even break the Coulomb blockade. Breaking of Coulomb blockade —When the macrospinprecesses slowly ( (cid:126) | ˙ φ | < E c ), the Coulomb blockade issoftened: electrons can tunnel through the junction, evenif the applied voltage is smaller than—but close enoughto—the charging energy; see Fig. 2. The missing en-ergy is provided by the precessing macrospin. So, themacrospin dynamics governs the softening of Coulombblockade. In turn, a measurement of the charge currentcan reveal the macrospin dynamics. For example in thevoltage regime E c + (cid:126) ω + < eV < E c + (cid:126) ω − , compareFig. 2, the current is given by I = g t sin ( θ/ eV − E c − (cid:126) ˙ φ cos ( θ/ / e . Thus, a measurement of the differen-tial conductance dIdV = sin ( θ/ g t / θ . Then, knowing θ , the precession frequency ˙ φ can be inferred from the current I itself. A shortcomingof this method is that one has to know in which regimethe voltage is. A simpler way would be to measure thecurrent-voltage characteristics, Fig. 2, and determine themagnetization dynamics from the position of the kinks at FIG. 3: This figure shows the breaking of Coulomb blockadefor strong macrospin precession ( (cid:126) | ω + | > E c ) with 0 < θ <π/ φ <
0. The standard Coulomb blockade is includedas reference (blue-dashed). Because of the macrospin-assistedtunneling, the Coulomb blockade disappears; that is, a chargecurrent flows at arbitrary low (but nonzero) voltages. Yet,the details depend on the macrospin dynamics. In turn, ameasurement of charge current can reveal the macrospin dy-namics. E c ± (cid:126) ω + and E c ± (cid:126) ω − —or analogously from the positionof jumps in the differential conductance.While only softened for slow precession, the Coulombblockade is completely broken for strong macrospin pre-cession ( (cid:126) | ω + | > E c and/or (cid:126) | ω − | > E c ). In this case, theprecessing macrospin gives enough energy to the tunnel-ing electrons, such that tunneling is possible even if thereis no other source of energy. In turn, even at low voltages,we find a linear relation between current and voltage; seeFig. 3. So, the macrospin dynamics governs the breakingof Coulomb blockade. And again, a measurement of thecharge current can reveal the macrospin dynamics. How-ever, in contrast to the softening of Coulomb blockade,the (differential) conductance can reveal the polar angle θ even at zero voltage. For example in the low voltageregime ( E c + (cid:126) ω + < eV < − E c − (cid:126) ω + ), as shown in Fig.3, the current is given by I = g t V sin ( θ/ θ but not the precession frequency˙ φ [37]. To determine the precession frequency, one hasto go to higher voltages again. Discussion. —We have found that macrospin-assistedtunneling can break the Coulomb blockade. More explic-itly, we considered a tunnel junction between a normalmetal and a metallic ferromagnet where the macrospin(magnetization) is in a steady state precession. The pre-cessing macrospin creates a time-dependent field for elec-trons, which can assist them in tunneling [8]. As wehave shown, this macrospin-assisted tunneling shrinksthe regime of Coulomb blockade; see Fig. 2. Whenthe macrospin precession is strong enough, the regimeof Coulomb blockade vanishes completely; see Fig. 3. Inother words, the macrospin-assisted tunneling can softenor even break the Coulomb blockade. The details of thesoftening or breaking of Coulomb blockade depend on themacrospin dynamics. Thus, a measurement of the chargecurrent can reveal the macrospin dynamics.To get a better understanding of the scales involved,let’s consider a specific system. For example in Ref. [25],they report on a magnetic tunnel junction with ellipticalshape (minor axis 40 nm; major axis 80 nm) and a MgOtunnel barrier with thickness 0 . C ≈ .
25 fF for the tunnel junction [38].In turn, we find a charging energy of E c ≈ .
32 meVwhich corresponds to a temperature of T c = E c /k B ≈ . f c = E c /h ≈
78 GHz. To enterthe regime of Coulomb blockade, the temperature mustbe well below T c . Then, the precessing macrospin couldbreak the Coulomb blockade, if it precesses at frequenciesabove f c . While the precession frequency reported in Ref.[25] is only of the order of 10 GHz, it is still close enoughto the critical frequency f c , such that one can expect aclear softening of the Coulomb blockade; analog to Fig.2. For a tunnel junction of larger dimensions and witha thinner barrier, the critical frequency f c can fall below10 GHz such that one might also observe the breaking ofCoulomb blockade; analogous to Fig. 3.Also beyond the specific setup considered here, thebreaking of Coulomb blockade by macrospin-assisted tun-neling might be interesting; for example, for scanningtunneling microscope (STM) setups. In STM setups, thecapacitance is harder to estimate; see Ref. [26] for ex-ample. However, in Ref. [27], where they also use theCoulomb blockade to investigate a system in a scanningtunneling spectroscopy (STS) setup, they find a junc-tion capacitance of C = 21 . E c ≈ . µ eV, a tem-perature of T c = E c /k B ≈
42 mK, and a frequency of f c = E c /h ≈ . f c , such that the macrospin assisted tunneling can easilybreak the Coulomb blockade. This effect might be par-ticularly interesting for resonant-state-STM setups [28]—where the charging energy E c can be tuned, because ofa large variability in the distance between STM-tip andprobe material.While we focused on a passive use (indirect measure-ment of magnetization dynamics), we can also think ofmore active uses of the interplay between Coulomb block-ade and macrospin-assisted tunneling. It could be used tocontrol a spin current electrically [39]. Or, when the mag-net’s density of states is spin-dependent, it can be usedto pump a charge current [40]. 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In morephysical terms, the magnetization must be slow com-pared to its length.[35] Formally, the spin independence means ρ r ( (cid:15) − M/
2) = ρ r ( (cid:15) + M/
2) = ρ r [36] Formally, we have I r → l = e (cid:80) ρλσσ (cid:48) Γ ρσ → λσ (cid:48) f ( ξ ρσ )[1 − f ( (cid:15) λ )]. Explicitly, we obtain I r → l = g t e (cid:2) cos θ Π( − eV − E c + (cid:126) ω − ) + sin θ Π( − eV − E c − (cid:126) ω + ) + sin θ Π( − eV − E c + (cid:126) ω + ) + cos θ Π( − eV − E c − (cid:126) ω − ) (cid:3) .[37] Note, however, that this only works if | (cid:126) ω + | > E c and | (cid:126) ω − | < E c . If | (cid:126) ω + | > E c and | (cid:126) ω − | > E c , then wewould simply find Ohm’s law I = g t V at low voltages.[38] To estimate the capacitance, we used the formula for par-allel plates C = (cid:15) (cid:15) r A/d ; with the dielectric constant (cid:15) ≈ · −
12 Fm , the relative permittivity for MgO whichis (cid:15) r ≈
10, the ellipsis area A = π · , and thebarrier thickness d = 0 . E c + (cid:126) ω + < eV < E c + (cid:126) ω −−