Spin-accumulation capacitance and its application to magnetoimpedance
SSpin-accumulation capacitance and its application to magnetoimpedance
Yao-Hui Zhu, a) Xiao-Xue Zhang, Jian Liu, and Pei-Song He
Physics Department, Beijing Technology and Business University, Beijing 100048,China (Dated: 5 September 2018)
It has been known that spin-dependent capacitances usually coexist with geometric capacitances in a magneticmultilayer. However, the charge and energy storage of the capacitance due to spin accumulation has not beenfully understood. Here, we resolve this problem starting from the charge storage in the spin degree of freedom:spin accumulation manifests itself as an excess of electrons in one spin channel and an equal deficiency in theother under the quasi-neutrality condition. This enables us to model the two spin channels as the two platesof a capacitor. Taking a ferromagnet/nonmagnet junction as an example and using a method similar to thatfor treating quantum capacitance, we find that a spin-accumulation (SA) capacitance can be introduced foreach layer to measure its ability to store spins. A spatial charge storage is not essential for the SA capacitorand the energy stored in it is the splitting energy of the spin-dependent chemical potentials instead of theelectrostatic energy. The SA capacitance is essentially a quantum capacitance due to spin accumulation onthe scale of the spin-diffusion length. The SA capacitances can be used to reinterpret the imaginary part ofthe low-frequency magnetoimpedance.
I. INTRODUCTION
In many applications, capacitances have a remarkableinfluence on the speed and power dissipation of the de-vices, and also set an upper frequency limit (cutoff fre-quency) for their correct operations.
The effects of thecapacitances on spintronic devices have also been ob-served and analyzed recently. The magnetocapacitanceor magnetoimpedance has been studied, for example, inmagnetic tunnel junctions (MTJs) and in a singleelectron transistor. Using a time-dependent approach, Rashba has studiedthe frequency-dependent impedance of a junction com-posed of a ferromagnetic (FM) conductor, a spin-selectivetunnel or Schottky contact, and a nonmagnetic (NM)conductor. The imaginary part of the impedance wasattributed to a diffusion capacitance C diff , which canbe compared to (but is certainly different from) that ofa p-n junction. However, Rashba’s derivation involvedthe quasi-neutrality condition, which assumes that thecharge accumulation is negligible everywhere. Thus itseems difficult to reconcile this condition with the chargestorage that is usually associated with the diffusion ca-pacitance. In the present paper, we try to resolve thisproblem by introducing a spin-accumulation (SA) capac-itance for each layer. The two plates of the SA capac-itor model the two spin channels, since spin accumula-tion manifests itself as an excess of electrons in one spinchannel and an equal deficiency in the other under thequasi-neutrality condition. Thus the charge storage ofthe SA capacitor happens in the spin degree of freedomrather than the coordinate space, and a spatial chargestorage is not essential to it. We also prove that the SAcapacitances lead to the same low-frequency reactance as C diff . Then the conflict mentioned above is resolved. a) Electronic mail:[email protected]
The form of energy storage associated with the dif-fusion capacitance is another basic question, because itindicates the type and origin of the capacitance. How-ever, this issue has not been addressed to our knowl-edge.
On the other hand, most studies on the mag-netoimpedance of the MTJs assumed that the energy isstored in an extra electrostatic field like usual geomet-ric capacitances.
Here, we also show that the energystorage associated with the magnetoimpedance of theFM/NM junction is equal to that stored in the SA ca-pacitors. It is the splitting energy of the spin-dependentchemical potentials instead of the electrostatic energy.This characteristic is similar to that of a quantum ca-pacitance, which stores energy in forms other than theelectrostatic field.
We study the SA capacitance using a method simi-lar to that for dealing with quantum capacitance. Thismethod is quite different from Rashba’s time-dependentapproach. It enables us to find the connections betweenthe SA capacitance, the diffusion capacitance, and thequantum capacitance. Moreover, this method has thepotential to be used in more complicated situations.This paper is organized as follows. In section II,we introduce the SA capacitance for the FM/NM junc-tion. Then the model is used to reinterpret the mag-netoimpedance in section III. The main conclusions aregiven in section IV.
II. SPIN-ACCUMULATION CAPACITANCE
To illustrate the concept of the SA capacitance, weconsider a magnetic multilayer with current perpendic-ular to the plane, which is the well-known configurationgiving rise to spin accumulation. To be specific, we con-sider the same FM/NM junction studied by Rashba. The junction is composed of a ferromagnetic layer occu-pying z <
0, a spin-selective contact at z = 0, and a a r X i v : . [ c ond - m a t . m t r l - s c i ] J u l nonmagnetic layer occupying z >
0. The z axis is set tobe perpendicular to the layer plane as shown in Fig. 1.Without loss of generality, the magnetization of the FMis set to be “up” and the current is flowing in the positivedirection of the z axis.Although the importance of capacitance usually showsup in time-dependent transport, it is also present in theconstant DC situation, which is much easier to deal with.The spin accumulation is usually described by the split-ting of the spin-dependent chemical potentials µ m ( z ) = 2∆ µ ( z ) = µ + ( z ) − µ − ( z ) , (1)where the subscripts “+” and “ − ” stand for the absolutespin directions “up” and “down”, respectively. In eachlayer, ∆ µ ( z ) satisfies the well-established spin diffusionequation ∂ ∆ µ ( z ) ∂z = ∆ µ ( z ) l , (2)where l sf is the spin-diffusion length (see Appendix A).The solution to Eq. (2) in each layer is of exponentialform as shown by Eqs. (A9) and (A10). Introducing theaverage chemical potential µ ( z ) = [ µ + ( z ) + µ − ( z )] / µ ± ( z ) = µ ( z ) ± ∆ µ ( z ) . (3)Since we are concerned with capacitance, it is more con-venient to use the charge density in the two spin channels ρ ± ( z ) = − eN s (cid:2) µ ± ( z ) − µ (cid:3) , (4)where − e is the charge of an electron and N s the den-sity of states at the Fermi level µ . According to theValet-Fert theory, N s is assumed to be the same forthe two spin directions and also for both layers. We have µ ( z ) = µ in the NM layer, whereas µ ( z ) is unequal to µ in general for the FM layer. However, it can be justifiedfor our problem to assume µ ( z ) = µ (5)even in the FM layer using the quasi-neutrality approxi-mation (see Appendix B). Then substituting Eqs. (3) and(5) into Eq. (4), we have ρ ± ( z ) = ∓ eN s ∆ µ ( z ) , (6)which is valid in both FM and NM layers. Therefore, theelectron excess in one spin channel cancels the deficiencyin the other, ρ + ( z ) + ρ − ( z ) = 0, in both layers.The basic idea of the SA capacitance arises initiallyfrom the resemblance of charge storage between the twospin channels and the two plates of a normal capacitor,as shown in Fig. 1. Further analysis shows that the spinaccumulation also bears similarity to a capacitor in otheraspects, including energy storage, leakage current, andheat generation. NMFM
FIG. 1. Sketch of the spin-dependent charge density ρ ± ( z )in the FM/NM junction. We plot the curves using Eq. (6)but without realistic parameters, because a demonstration ofthe basic profile is enough for our current purpose. The unitsfor ρ ± ( z ) and z are arbitrary. The constant DC current has adensity of J . The solid curves on the left and right sides of theinterface at z = 0 stand for ρ + ( z ) in the FM and NM layers,respectively. Similarly, the dashed curves are for ρ − ( z ). Thevertical dotted lines indicate the scale of the spin-diffusionlengths, l Fsf and l Nsf , in the FM and NM layers, respectively.The capacitances C Fsa and C Nsa are defined by Eq. (10), andwill be connected in the circuit shown by Fig. 2(b).
A. Charge storage
When the two spin channels are modeled as the twoplates of the SA capacitor, the charge storage of the ca-pacitor can be defined as the absolute value of the chargeaccumulation in either spin channel. Treating the FMand NM layers separately, we can write the charge stor-age as Q Fsa = (cid:90) −∞ | ρ ± ( z ) | d z = (cid:90) −∞ eN s | ∆ µ ( z ) | d z, (7) Q Nsa = (cid:90) ∞ | ρ ± ( z ) | d z = (cid:90) ∞ eN s | ∆ µ ( z ) | d z, (8)where Eq. (6) has been used. Following the proceduredetailed in Appendix A, we can write Q Fsa and Q Nsa in aform resembling a normal capacitor Q Fsa = C Fsa V Fc , Q Nsa = C Nsa V Nc , (9)where we have introduced the SA capacitances C Fsa = T F1 r F , C Nsa = T N1 r N , (10)and the corresponding effective voltages V Fc = | ∆ µ (0 − ) | e , V Nc = | ∆ µ (0 + ) | e , (11)for the semi-infinite FM and NM layers, respectively. Inthe expression of C F(N)sa given in Eq. (10), T F1 ( T N1 ) isthe spin-relaxation time in the FM (NM) layer. Theresistances r F and r N are defined as r F = ρ ∗ F l Fsf , r N = ρ ∗ N l Nsf , (12)where ρ ∗ F ( ρ ∗ N ) is the FM (NM) resistivity. Using theexpressions of ∆ µ (0 − ) and ∆ µ (0 + ) in Eq. (A11), we canrewrite Eq. (11) in a form similar to Ohm’s law V Fc = r sfF | α F | J , V Nc = r sfN | α N | J , (13)where we have defined the dimensionless parameters α F = β ( r N + r ∗ b ) − γr ∗ b r F + r ∗ b + r N , α N = βr F + γr ∗ b r F + r ∗ b + r N , (14)and the characteristic (spin-flip) resistances r sfF = 2 r F , r sfN = 2 r N , (15)for the FM and NM layers, respectively. In Eq. (14),the bulk spin asymmetry coefficient β in the FM layer isdefined by the relation1 /σ ↑ ( ↓ ) = 2 ρ ∗ F [1 − (+) β ] , (16)where σ ↑ ( ↓ ) is the conductivity for the majority (minor-ity) spin direction. Similarly, we have 1 /σ ± = 2 ρ ∗ N inthe NM layer. The interfacial resistance r ∗ b and its spinasymmetry coefficient γ are defined by r ↑ ( ↓ ) = 2 r ∗ b [1 − (+) γ ] , (17)where r ↑ ( ↓ ) is the resistance of the majority (minority)spin channel. In general, the effective voltages V Fc and V Nc are not equal to each other due to the interface resis-tance. In the configuration specified above (“up” magne-tization and positive DC current), α N is always positive,whereas α F may be negative. Their interpretation willbe given by Eq. (25) in combination with the effectivecircuit shown in Fig. 2(b).The expression of C F(N)sa in Eq. (10) is similar to thatof the diffusion capacitance in a p-n junction. However, C F(N)sa does not require a spatial charge storage, which isessential for the diffusion capacitance in a p-n junction.Therefore, C F(N)sa measures the ability of a material tostore spins instead of charge. Spin accumulating and dis-sipating correspond to charging and discharging of theSA capacitor, respectively. Note that C F(N)sa cannot becompared directly with the diffusion capacitance C diff inEq. (C5), which was introduced by Rashba. One dis-tinction between them is the independence of C F(N)sa onthe spin asymmetry coefficient β or γ . B. Energy storage
Since µ m ( z ) is the splitting between the chemical po-tentials of the two spin channels, spin accumulation alsoaccompanies an increase in energy in comparison to theequilibrium state. The energy stored in the differ-ential d z is equivalent to the energy required to shift anumber of electrons, N s | ∆ µ ( z ) | d z , from the spin channelwith lower chemical potential to the other. Because theaverage energy increase per electron is just | ∆ µ ( z ) | , theenergy storage in d z is N s [∆ µ ( z )] d z . The total energystorage is the sum of the following two terms W Fsa = (cid:90) −∞ N s [∆ µ ( z )] d z, (18) W Nsa = (cid:90) ∞ N s [∆ µ ( z )] d z, (19)where W Fsa and W Fsa are the contributions from the FMand NM layers, respectively. Using Eqs. (A9) and (A10),we can write them in a form resembling the energy storedin a capacitor W Fsa = 12 C Fsa (cid:0) V Fc (cid:1) , W Nsa = 12 C Nsa (cid:0) V Nc (cid:1) , (20)where Eqs. (10) and (11) have also been used.The energy stored in an SA capacitor is essentially thesplitting energy of the spin-dependent chemical poten-tials instead of the electrostatic energy associated with ageometric capacitance. This character is similar to thequantum capacitance, which stores energy in the form ofthe Fermi degeneracy energy. In general, quantum ca-pacitance appears when the spatial charge accumulationon at least one plate of the capacitor induces a changein chemical potential. This is also true for the SA ca-pacitance according to the discussion above. Moreover,substituting Eq. (A5) into Eq. (10), we can also rewritethe SA capacitance as C F(N)sa = C Q l F(N)sf , (21)where we define C Q as the quantum capacitance per unitvolume and spin channel C Q = e N s , (22)following Ref. 13. Thus we can also interpret the SAcapacitance as the quantum capacitance due to spin ac-cumulation on the scale of the spin-diffusion length. Ithas the characteristics of the diffusion capacitance as wellas the quantum capacitance as shown by Eqs. (10) and(21). However, the SA capacitance is different from theusual quantum capacitance: its change in chemical po-tential happens in the spin degree of freedom instead ofthe coordinate space owing to the relation µ ( z ) = µ . C. Leakage current
Spin accumulation coexists with spin relaxation, inwhich electrons undergo transitions from the spin channelwith higher chemical potential to the other via spin-flipscattering. This process can be modeled as the leakageof the SA capacitor (spin flux in Ref. 22). If we set thepositive direction of leakage current to be from the spin-down channel to the spin-up channel, the leakage currentcan be written as J Fsf = (cid:90) −∞ eN s µ m ( z ) τ Fsf d z, (23) J Nsf = (cid:90) ∞ eN s µ m ( z ) τ Nsf d z, (24)for the FM and NM layers, respectively. Here τ sf = 2 T is the spin-flip scattering time. Using Eqs. (A9) and(A10), we can write J Fsf and J Nsf in a form resemblingOhm’s law J Fsf = V Fc r sfF = α F J , J Nsf = V Nc r sfN = α N J r sfF ( r sfN ) as the resistor in parallelwith the SA capacitor C Fsa ( C Nsa ) as shown in Fig. 2(b).Using α F + α N = β , we have β J J Fsf + J Nsf , (26)which can also be derived directly by integrating Eq. (A4)from −∞ to ∞ . In fact, βJ / J m ( −∞ ), whichis the spin current density produced by the bulk FMlayer. It decreases to zero exponentially at z = ∞ af-ter the spin relaxation around the interface. During thisprocess, half of J m ( −∞ ), that is βJ /
2, leaks from spin-down channel to spin-up channel in the FM and NMlayers.
D. Heat generation
The spin relaxation also generates heat as the spin-flipscattering causes dissipation of the energy stored in thechemical-potential splitting. The heat generation ratedue to spin-flip scattering can be written asΣ F , sfheat = (cid:90) −∞ N s [ µ m ( z )] τ Fsf d z, (27)Σ N , sfheat = (cid:90) ∞ N s [ µ m ( z )] τ Nsf d z, (28)for the FM and NM layers, respectively. Using Eqs. (A9)and (A10), we can write Σ F , sfheat and Σ N , sfheat formally asΣ F , sfheat = r sfF (cid:0) J Fsf (cid:1) = 12 r F α J , (29)Σ N , sfheat = r sfN (cid:0) J Nsf (cid:1) = 12 r N α J , (30)where Eq. (25) has been used. Thus the heat generationdue to spin relaxation can be modeled as the Joule heatof the resistors in parallel with the SA capacitors. (a)(b) NMFMECST: FIG. 2. (a) Sketch of the FM/NM junction with the spin-selective contact, which is signified by the thick vertical lineat z = 0. The vertical dotted lines indicate the scales of spin-diffusion length in the two layers. (b) An effective circuitof the spin transport (ECST) in the junction shown by (a).The SA capacitors in Fig. 1 are connected in parallel withtwo characteristic resistors to model the leakage due to spinrelaxation. The resistances labeled by r sfF ( r sfN ) and r sdF ( r sdN )have the same value 2 r F (2 r N ). The superscripts “sf” and“sd” stand for spin flip and spin diffusion [see the text rightbefore Eq. (32)], respectively. The current densities J Fsf , J Nsf ,and J Csd are given by Eqs. (25) and (35), respectively. Thecurrent densities βJ / γJ / E. An effective circuit for spin transport
We have constructed an effective circuit, shown inFig. 2(b), to model the spin transport, which takes placeon the scales of spin-diffusion length around the inter-face. We will explain briefly how we build the circuiton the basis of the physical analysis above. Meanwhile,we will also point out the differences between this circuitand those in the Valet-Fert theory. In Fig. 2(b), the SA capacitors ( C Fsa and C Nsa ) are con-nected in parallel with the resistors ( r sfF and r sfN ), respec-tively. The two capacitors model the charge storage inthe two spin channels due to spin accumulation as shownby Eq. (9). The leakage current of the SA capacitorsmodels the electron flow from one spin channel to theother due to spin relaxation as shown by Eqs. (23) and(24). Comparing Eqs. (9) and (25), one can see that thetwo capacitors have the same voltages as the two resis-tors, respectively. Therefore, the leakage can be repre-sented equivalently by connecting the two finite resistorsin parallel with the SA capacitors, respectively. The volt-ages here are defined by Eq. (11) and stand for the ef-fective voltages of the chemical-potential splitting, whichmay be measured by some spin-resolved optical method.This circuit has one obvious difference from those in theValet-Fert theory: the capacitors and resistors here areconnected between the effective electrodes for the twospin channels instead of the real electrodes.The current sources of this circuit supply only the spin-polarized part of the current in each spin channel. Thisfeature also makes the circuit different from those in theValet-Fert theory. There are two sources of the spin-polarized current in the junction: the FM layer with bulkspin asymmetry coefficient β (cid:54) = 0 and the contact withinterfacial spin asymmetry coefficient γ (cid:54) = 0. They arerepresented by two independent current sources in thecircuit: βJ / γJ /
2. The current source βJ / γJ / J ef ± ( z C ) = (1 ∓ γ ) J / , (31)for the two spin channels, where J flows through r + =2 r ∗ b (1 + γ ) and r − = 2 r ∗ b (1 − γ ) in parallel. Here thespin-up electrons are assumed to be in the minority chan-nel without loss of generality. The spin-selective contactmakes J ef ± ( z C ) deviate from its unpolarized value J / γJ /
2, and thus it can be modeled as aspin-current source.The capacitors are charged by the two current sources.This models the formation of spin accumulation underthe drive of the spin current. If the interface resistance isnegligible, the two capacitors will be charged simultane-ously by the current source βJ / C Nsa . Thus it isreasonable to connect it in the way shown in Fig. 2(b).Two additional resistances, r sdF and r sdN , need to beintroduced in the effective circuit to model the heat gen-eration due to the spin diffusion in the FM and NM lay-ers, denoted by Σ F , sdheat and Σ N , sdheat , respectively. Using amacroscopic approach based on the Boltzmann equation,we can prove in general that the spin diffusion leads tothe same heat generation as the spin relaxation (or thespin-flip scattering) in semi-infinite layers. To meet thisrequirement, we connect r sdF and r sdN in series with r sfF and r sfN , respectively, as shown in Fig. 2(b). Thus r sdF and r sdN should be defined as r sdF = r sfF , r sdN = r sfN , (32)to satisfy Σ F , sdheat = Σ F , sfheat and Σ N , sdheat = Σ N , sfheat . Then usingEqs. (29) and (30), we can write the spin-dependent heatgeneration asΣ Fheat = Σ F , sdheat + Σ F , sfheat = r F α J , (33)Σ Nheat = Σ N , sdheat + Σ N , sfheat = r N α J , (34)for the FM and NM layers, respectively. The currents J Fsf and J Nsf given by Eq. (25) can also bederived by applying the superposition theorem of elec-trical circuits to the circuit in Fig. 2(b). Both currentsources drive current to flow from the spin-down to thespin-up channel in the NM layer. However, they driveopposite currents in the FM layer and the net current J Fsf can be negative in some cases. The effective voltage onthe resistance r sdF ( r sdN ) is also equal to that on r sfF ( r sfN ),which is consistent with Eq. (11). Moreover, using Kirch-hoff’s current law, one can calculate the current densityflowing through the resistance 4 r ∗ b J Csd = γ J − J Nsf = α C J , (35)where we have introduced the dimensionless parameter α C = γ − α N = γ ( r F + r N ) − βr F r F + r ∗ b + r N . (36)Then the effective voltage on the resistance 4 r ∗ b can bewritten as 2( V Nc − V Fc ) or [ µ m (0 + ) − µ m (0 − )] /e . UsingJoule’s law, we can write the heat generation of the re-sistance 4 r ∗ b asΣ Cheat = 4 r ∗ b (cid:0) J Csd (cid:1) = (cid:0) V Nc − V Fc (cid:1) r ∗ b = r ∗ b α J , (37)which is equal to the result derived by using a more mi-croscopic approach. It has been shown, in Ref. 25, thatΣ
Cheat results only from the spin diffusion across the inter-face because the spin relaxation at the interface is neg-ligible. Combining Eqs. (33), (34), and (37), we canwrite the total heat generation due to the spin relaxationand the spin diffusion asΣ
FNheat = Σ
Fheat + Σ
Nheat + Σ
Cheat = r ∗ FN J , (38)where r ∗ FN is defined as r ∗ FN = r F α + r N α + r ∗ b α . Onecan verify that r ∗ FN is equal to the spin-coupled interfaceresistance of the FM/NM junction. F. An overall equivalent circuit
To make our results comparable with Rashba’s the-ory and experimental results, we have also constructedan overall equivalent circuit, which includes the ECSTshown in Fig. 2(b). In the equivalent circuit shown inFig. 3, the current source is the total charge current J instead of the spin current sources βJ / γJ /
2. Thismakes the circuit similar to those in the Valet-Fert the-ory. However, we have combine the two spin channels tomake the circuit more comparable to the practical situa-tion. The two terminals in Fig. 3(b) are supposed to beconnected to the real electrodes, which are not shown forsimplicity. Thus its voltage can be measured directly inexperiments.We require that the equivalent circuit has the sameenergy storage W F(N)sa and heat generation Σ
F(N)sa as the (a)(b) ECUECU:
FIG. 3. (a) An equivalent circuit unit (ECU) for the ECSTshown in Fig. 2(b). In the ECU, the current flowing throughthe various resistances is the total charge current J , insteadof the spin currents in ECST. The equivalent resistances r ∗ F , r ∗ N , and r ∗ C are given in Eqs. (39) and (40), respectively. TheESA capacitances C F , ∗ sa and C N , ∗ sa are defined in Eq. (43). (b)An overall equivalent circuit of the FM/NM junction shown inFig. 2(a). The block labeled by “ECU” stands for the circuitunit in Fig. 3(a). The FM (NM) layer thickness t F ( t N ) ismuch larger than the spin diffusion length l Fsf ( l Nsf ). ECST, which in turn has the same W F(N)sa and Σ
F(N)sa asthe real physical system. We first determine the equiv-alent resistances by requiring the same heat generation.Using Eqs. (33) and (34), we can set the equivalent re-sistances to be r ∗ F = r F α , r ∗ N = r N α , (39)for the FM and NM layers, respectively. Moreover, usingEq. (37), we can set the equivalent resistance for 4 r ∗ b inFig. 2(b) to be r ∗ C = r ∗ b α , (40)which satisfies r ∗ FN = r ∗ F + r ∗ N + r ∗ C . Then the equivalentvoltages on them can be written as V ∗ F = r ∗ F J , V ∗ N = r ∗ N J . (41)For simplicity, we assume that the equivalent spin-accumulation (ESA) capacitance C F , ∗ sa ( C N , ∗ sa ) is con-nected in parallel with the equivalent resistance r ∗ F ( r ∗ N ).Then, V ∗ F ( V ∗ N ) is also the voltage on C F , ∗ sa ( C N , ∗ sa ). Werequire that C F , ∗ sa and C N , ∗ sa have the same energy storageas C Fsa and C Nsa W Fsa = 12 C F , ∗ sa ( V ∗ F ) , W Nsa = 12 C N , ∗ sa ( V ∗ N ) . (42)Substituting Eqs. (20) and (41) into Eq. (42), we can find C F , ∗ sa = T F1 r ∗ F = C Fsa α , C N , ∗ sa = T N1 r ∗ N = C Nsa α , (43)for the FM and NM layers, respectively. Finally, we canconstruct an overall equivalent circuit shown in Fig. 3(b),which also include the normal resistances (1 − β ) ρ ∗ F t F ,(1 − γ ) r ∗ b , and ρ ∗ N t N . III. REINTERPRETATION OF THEMAGNETOIMPEDANCE
As an application of the SA or ESA capacitances, werevisit the magnetoimpedance of the FM/NM junctionby using the equivalent circuit shown in Fig. 3. Thereactance in this configuration has been interpreted byRashba in terms of the diffusion capacitance C diff . The SA capacitance is approximately a constant fortime-varying DC and (sinusoidal) AC signals in theregime for the quasi-static approximation to be valid.When the junction is driven by a constant DC currentas discussed in the previous section, the spin accumu-lation is an exponential function in each layer and theSA capacitance is a constant independent on the currentdensity. In general, the SA capacitance will change if thedriving current becomes time-dependent, for example, atime-varying DC or AC signal. However, if the chargecurrent J ( t ) changes with time slowly enough that thesolutions to the time-dependent equations are still ex-ponential functions approximately at each moment, theycan be replaced by the static solutions for the DC cur-rent of the same density J ( t ). Then the SA capacitancecan be assumed to be a constant for simplicity. This isthe quasi-static approximation, which has been widelyused in circuit analysis and also applied to the diffusioncapacitance of a p-n junction. The valid range for the quasi-static approximation canbe determined as follows. Since we are considering theimpedance, it is enough to focus on the sinusoidal AC sig-nal. The characteristic time scale of the spin dynamics isthe spin relaxation time T N1 , if we have the relation T N1 >T F1 as usual. Therefore, the quasi-static approxima-tion is valid if the frequency of the driving current sat-isfies ω (cid:28) /T N1 or equivalently ωC N , ∗ sa (cid:28) / (2 r ∗ N ). Inpractice, most experiments on time-dependent transportuse slowly-varying or low-frequency AC signals. In thehigh-frequency regime, the quasi-static approximationbreaks down because the spin dynamics cannot followthe variation of the driving signal and the spin accumu-lation deviates from the exponential form severely. Inthis case, the spin wave-diffusion theory should be usedinstead. Consequently, the inductive terms becomes re-markable and may be dominant over the capacitive termsif the frequency is high enough. If the AC current density is set to be J ( t ) = J exp ( − i ωt ), the impedances of C F , ∗ sa and C N , ∗ sa inFig. 3(a) can be written, respectively, as i / (cid:0) ωC F , ∗ sa (cid:1) andi / (cid:0) ωC N , ∗ sa (cid:1) using the quasi-static approximation. Usingusual circuit theorems, we can calculate the impedance, Z ECU , of the ECU in Fig. 3(a). Then expanding Z ECU in terms of ω and keeping up to the first-order terms, wehave Z ECU = r ∗ FN + i X FNsa , (44)where r ∗ FN is defined in Eq. (38) and the reactance X FNsa can be written as X FNsa = ωC F , ∗ sa ( r ∗ F ) + ωC N , ∗ sa ( r ∗ N ) . (45)We will prove that the impedance given in Eq. (44)is equal to that derived by Rashba ( Z n − eq ) in the low-frequency regime. Rashba’s basic results are outlinedwith our notations in Appendix C. Using Eqs. (14) and(36) in the present paper, we can rewrite the real part ofthe impedance, R n − eq given in Eq. (C3), in a form withmore transparent physical interpretation R n − eq = r ∗ F + r ∗ N + r ∗ C = r ∗ FN , (46)which is equal to the real part of Z ECU in Eq. (44). Theimaginary part of the impedance, Im( Z n − eq ) in Eq. (C4),can be rewritten asIm( Z n − eq ) = ω T F1 r F ( r F α F ) + ω T N1 r N ( r N α N ) , (47)where we have used Eqs. (14) and (36). Then by usingthe SA capacitances defined in Eq. (10), we can rewriteEq. (47) asIm( Z n − eq ) = ωC Fsa ( r F α F ) + ωC Nsa ( r N α N ) . (48)Finally, by using Eqs. (39) and (43), we can prove thatIm( Z n − eq ) is equal to X FNsa in Eq. (45). Therefore, theimpedance Z ECU is the same as Z n − eq derived by Rashbaand the ECU shown in Fig. 3(a) can also be used tointerpret Z n − eq .In comparison to Rashba’s theory, we wrote the reac-tance of the FM/NM junction in terms of the SA or ESAcapacitances instead of the diffusion capacitance C diff de-fined in Eq. (C5). One benefit is that the form of the SAor ESA capacitances resemble obviously that of the diffu-sion capacitance in a p-n junction. More importantly, ourresults revealed the form of the energy storage associatedwith the reactance. In the low-frequency limit, the en-ergy storage associated with Z n − eq is given by Eq. (20) or(42), which is the splitting energy of the spin-dependentchemical potentials instead of the electrostatic energy.Our results also suggest a way to calculate the reactancewithout using the time-dependent equations.One characteristic of the reactance, X FNsa , is that itis zero if β = 0 and γ = 0. This means that a spin-current source is indispensable to the reactance we con-sider. Multilayers with β = 0 and γ = 0 may still haveparasitic capacitance in parallel with resistance as longas the material resistivity varies from layer to layer. Itresults from the induced charge accumulation at the in-terfaces, which adjusts the electric field in different lay-ers to maintain a homogeneous current across the entiremultilayer. The energy stored in this kind of capacitanceis essentially electrostatic energy and it will not be dis-cussed in this paper.It is worthwhile to compare the magnetoimpedance inthe present paper with that reviewed in Ref. 28. Al-though the same term “magnetoimpedance” is used in the two papers, their configurations are different in thefollowing three ways. First, Ref. 28 discussed the changeof impedance with an applied magnetic field, whereassuch an external field is not essential in our problem andthe change of impedance here depends on the spin polar-ization as shown by Eq. (48). Second, the impedancein Ref. 28 is mostly inductive. On the contrary, ourimpedance is capacitive. Finally, Ref. 28 was mainlyconcerned with the impedance of a single FM conduc-tor. However, we are interested in magnetic multilayers,such as an FM/NM junction. Therefore, the imaginarypart of the magnetoimpedance reviewed in Ref. 28 arisesfrom quite different origin and is beyond the scope of thecurrent paper. IV. CONCLUSIONS
In summary, we model the two spin channels of a metalor degenerate semiconductor as the two plates of a ca-pacitor. By using a method similar to that for studyingquantum capacitance, we show that the SA capacitancecan be introduced for each layer to measure its ability tostore spins rather than charge and a spatial charge stor-age is not essential to it. The energy stored in the SAcapacitors is the splitting energy of the spin-dependentchemical potentials instead of the electrostatic energy.The leakage of the SA capacitor models the spin relax-ation. The heat generation due to the spin relaxation canbe written formally as the Joule heat of the resistance inparallel with the SA capacitor. Equivalent circuits canbe constructed to give a transparent interpretation of themodel for a typical FM/NM junction.As an application of the model, we derive the magne-toimpedance of the FM/NM junction directly from theequivalent circuit using the quasi-static approximation.We rewrite the reactance in terms of the SA capacitances,whose formula is obviously similar to that of the diffusioncapacitance in a p-n junction. Our results also reveal theform of the energy storage associated with the reactance.We expect that the SA capacitance can be generalizedto other structures, such as MTJs, and used to interpretthe ongoing experiments on the magnetoimpedance.
ACKNOWLEDGMENTS
We thank Prof. H. C. Schneider and Y. Suzuki forfruitful discussions. This work was supported by the Na-tional Natural Science Foundation of China under GrantNos 11404013, 11605003, 61405003, and 11474012.
Appendix A: The FM/NM junction with a constant DC
According to the Valet-Fert theory, ∆ µ ( z ) and cur-rent density J ± ( z ) satisfy the following equations eσ ± ∂J ± ∂z = ± µl ± , (A1) J ± = σ ± (cid:18) F ± e ∂ ∆ µ∂z (cid:19) , (A2)where σ ± and l ± denote the conductivity and spin-diffusion length for spin “ ± ” (“up” or “down”), respec-tively. The field F is defined as F = (1 /e ) ∂ ¯ µ/∂z , where¯ µ = (¯ µ + + ¯ µ − ) / ± ” components of Eq. (A1)directly, we have ∂J m ∂z = ∆ µer F(N) l F(N)sf , (A3)where we have introduced the spin current density J m ( z ) = J + ( z ) − J − ( z ). The spin diffusion length isdefined by 1 /l = 1 /l + 1 /l − . The resistance r F ( r N )is defined in Eq. (12). Multiplying σ ± on both sides ofEq. (A1) and subtracting its “ ± ” components, we have ∂J m ∂z = 4 eN s ∆ µτ F(N)sf , (A4)where σ ± = e N s D ± and l ± = D ± τ sf have been used. Moreover, D ± is the diffusion constant for spin “ ± ”.Comparing Eqs. (A3) and (A4), we can find the relation T F(N)1 r F(N) = e N s l F(N)sf , (A5)where τ F(N)sf = 2 T F(N)1 has been used. Dividing both sidesof Eq. (A2) by σ ± and then subtracting its ‘ ± ’ compo-nents, we have J m = ∓ βJ + 1 eρ ∗ F ∂ ∆ µ∂z (A6)for FM layers. Here, “ − ” (“+”) corresponds to “up”(“down”) magnetization of the FM layer. For NM layers,one only needs to set β = 0 and replace ρ ∗ F by ρ ∗ N inEq. (A6). Substituting Eq. (A6) into Eq. (A3), we canderive Eq. (2) in section II.At the interface located at z = 0, the electrochemicalpotential ¯ µ ± ( z ) and J ± ( z ) satisfy the boundary condi-tions J s (cid:0) + (cid:1) − J s (cid:0) − (cid:1) = 0 , (A7)¯ µ s (cid:0) + (cid:1) − ¯ µ s (cid:0) − (cid:1) = er s J s (0) , (A8)where r s and γ are defined in Eq. (17). Then applyingthe Valet-Fert theory to the FM/NM junction considered in section II, we have∆ µ ( z ) = ∆ µ (0 − ) exp( z/l Fsf ) , z < µ ( z ) = ∆ µ (0 + ) exp( − z/l Nsf ) , z > µ (0 − ) = er F α F J , ∆ µ (0 + ) = er N α N J . (A11)The parameters α F and α N are defined in Eq. (14).Substituting Eq. (A9) into Eq. (7), we have Q Fsa = e N s l Fsf V Fc (A12)where V Fc is defined in Eq. (11). Substituting Eq. (A5)into Eq. (A12), we find Q Fsa = C Fsa V Fc (A13)where C Fsa = T F1 / (2 r F ) is the same as that defined inEq. (10). One can also rewrite Q Nsa in a similar way.
Appendix B: Quasi-neutrality approximation
The average chemical potential µ ( z ) can be simpli-fied by using the quasi-neutrality approximation. Thenonequilibrium charge density of each spin channel, ρ ± ( z ), has two exponential terms with different decay-ing lengths: the Thomas-Fermi screening length and thespin diffusion length. The two terms will be discussedseparately in the following.The screening length is on the order of angstrom andthe charge accumulation of this length scale is not spin-polarized. Therefore, the screening charge can be con-sidered as an interfacial charge layer, which gives rise toa constant electric field on each side of the interface. Itleads to a parasitic capacitance that is beyond the scopeof the present paper and thus the screening charge be-comes irrelevant to our problem.On the scale of the spin diffusion length, one spin chan-nel has an excess of electrons while the other is deficientin electrons. The excess and deficiency cancel each otherexactly in the NM layer, but they have slightly differ-ent magnitude in the FM layer, which results in a netcharge accumulation coexisting with the spin accumula-tion. The ratio of the net electron concentration to theexcess of electrons in one spin channel (or deficiency inthe other) satisfies βλ / [( l Fsf ) − λ ] (cid:28) Therefore, the netcharge accumulation is necessary for some problems, suchas the interpretation of the GMR effects, where the extrapotential plays a crucial role. This interpretation is es-sentially the widely-used quasi-neutrality approximation.
Appendix C: Rashba’s results for the magnetoimpedance
According to Eqs. (12) and (13) of Rashba’s paper, the nonequilibrium impedance of the FM/NM junctioncan be rewritten as Z n − eq ( ω ) = 1 r FN ( ω ) (cid:2) r N ( ω ) r ∗ b γ + r N ( ω ) r F ( ω ) β + r ∗ b r F ( ω ) ( γ − β ) (cid:3) (C1)where r FN ( ω ) is defined as r FN ( ω ) = r F ( ω ) + r N ( ω ) + r ∗ b .To simplify the expression, we have substituted γ and β for ∆Σ / Σ and ∆ σ/σ F (used by Rashba ), respec-tively. The resistance r ∗ b is the same as r c in Ref. 12.The frequency-dependent r F(N) ( ω ) is defined as r F ( ω ) = r F (cid:112) − i ωT F1 , r N ( ω ) = r N (cid:112) − i ωT N1 , (C2)where T F1 ( T N1 ) is the same as τ Fs ( τ Ns ) in Ref. 12. Ex-panding Z n − eq ( ω ) in ω and keeping up to the first-orderterms, one can write its real part as R n − eq = r N r ∗ b γ + r N r F β + r ∗ b r F ( γ − β ) r FN , (C3)which is just Eq. (10) of Rashba’s paper. The imaginarypart in the expansion of Z n − eq ( ω ) can be written asIm ( Z n − eq ) = ωR C diff , (C4)where the diffusion capacitance is defined as C diff = 12 R r (cid:8) T N1 r N ( γr ∗ b + βr F ) + T F1 r F [ γr ∗ b − β ( r ∗ b + r N )] (cid:9) (C5)and R defined as R = R n − eq + (1 − γ ) r ∗ b . D. A. Neamen,
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