Spin diffusion and magnetoresistance in ferromagnet/topological-insulator junctions
Takehito Yokoyama1 and Yaroslav Tserkovnyak2
1Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan2Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA
(Received 11 October 2013; revised manuscript received 14 December 2013; published 8 January 2014)
We study spin and charge diffusion in metallic-ferromagnet/topological-insulator junctions. The diffusivetheory is constructed for the coupled transport of the spin-dependent electron densities in the ferromagnetand the charge density in the topological insulator. The diffusion equations for the coupled transport are derivedperturbatively with respect to the strength of the interlayer tunneling. We analytically calculate spin accumulationin the ferromagnet and junction magnetoresistance associated with a current bias along the interface. It isfound that due to the helical spin texture of the surface Dirac fermion, the spin accumulation and the junctionmagnetoresistance depend on the angle between the magnetization and the current-induced spin polarization onthe surface of the topological insulator.
The recent discovery of three-dimensional (3D) topologicalinsulators (TIs) offers a new state of matter that is topologicallydistinct from the conventional band insulators [1–6]. Surfacestates of the strong TI are topologically protected and immuneto small perturbations that respect time-reversal symmetry. Atlow energies, these states are described by Dirac fermionsand hence exhibit strong spin-orbit interactions and intriguingBerry-phase phenomena.
Since the electron’s spin and momentum on the surface ofa TI are essentially interlocked, TIs offer a promising arenafor developing spintronics. In particular, there has been anintense interest in coupling TIs with ferromagnets (FMs). Inprevious works, the focus has been on the surface propertiesof the TIs that are stand alone or exchange coupled to theinsulating FMs [7–9], such as magnetoelectric effect [4,10,11],spin torque and magnetization dynamics [12–18], magneticdomain walls [19–21], charge pumping [22,23], magneticheat transport [24], spin rotation [25,26], spin and chargedynamics [27–29], magnetotransport [30–37], and Majoranafermions (in superconducting heterostructures) [38–45]. Thesephenomena are affected or effected by the exchange fieldof the FM: The in-plane exchange field acts like a vectorpotential while the out-of-plane exchange field makes theDirac fermions massive. On the other hand, the coupled spinand charge transport in metallic FMs interfaces with TIsremains largely unexplored [46]. We study this transport herein the weak-tunneling regime through the FM/TI junction.
Motivated by theoretical predictions, there has also beenintense interest on the magnetic proximity effect [47–53]and observation of Dirac fermions coupled with exchangefield. Recently, signatures of Dirac fermions coupled tomagnetization have been observed by doping Fe, Mn, Cr, Gd,or ferrocene into TIs [54–66], depositing Fe or Co on thesurface of TIs [67–70], making a junction of a TI with Fe,GdN, or EuS [71–74], or intergrowth with Fe7Se8 [75].
In this paper, we construct the diffusive theory for thecoupled transport of the spin-dependent electron densities inthe FM and the charge density in the TI. We derive the coupledspin and charge diffusion equations in layered metallic FM/TIjunctions, under current bias along the TI surface, which are
perturbed by the tunneling across the interface. We analyticallysolve the diffusion equations to calculate the current-inducedspin accumulation in the FM and junction magnetoresistancedue to the FM/TI interlayer tunneling. It is found that due tothe helical spin texture of the surface Dirac fermion, the spinaccumulation and the junction magnetoresistance depend onthe angle between the magnetization and the current-inducedspin polarization on the surface of the TI.
II. MODEL
We consider an FM/TI junction sketched in Fig. 1(a). TheHamiltonian of the system reads H = HF + HT + H ′, where
HF =∑
k
a†k
(�
2k2
2m+ h · σ − εF
)ak (1)
is the Hamiltonian of the FM. Here k denotes the 3D wavevector, h = h(sin θ cos ϕ, sin θ sin ϕ, cos θ ) is the exchangefield, σ = (σx,σy,σz) is a vector of Pauli matrices, and a
†k =
(a†k↑,a
†k↓) is the spin- 1
2 creation operator. We suppose that themagnetization is uniform. The Hamiltonian on the unperturbedsurface of the TI reads
HT =∑
k
b†k[�v(kyσx − kxσy) − εT ]bk, (2)
where the 2D wave vector k is parallel to the interface. Thedispersions of the TI and FM are shown in Fig. 1(b), where thehorizontal line denotes their common equilibrium Fermi level.
The FM and TI are coupled by the tunneling Hamiltonian
H ′ =√
�γ
π
∑k,k′
a†kbk′ + H.c., (3)
where√
�γ
πdenotes the tunneling amplitude. It is assumed
that the interface between FM and TI is rough, such that,upon transmission through the interface, the momentum ofthe electron is scrambled while the spin and the energy areconserved. This spin-conserving form of tunneling wouldgenerally need to be extended to account for details of atomisticmatrix elements for the states composing the Dirac electronband, in order to construct a more quantitative theory. We
TAKEHITO YOKOYAMA AND YAROSLAV TSERKOVNYAK PHYSICAL REVIEW B 89, 035408 (2014)
FIG. 1. (Color online) (a) Schematic picture of the model.(b) Dispersions of the TI and FM. The horizontal line denotes theFermi level.
believe, however, that our simple treatment gives rise tothe appropriate symmetry-based phenomenological structureof the final diffusion equations. We assume the ambienttemperature T to be low, kBT � εF ,εT > 0, and focus onsmall deviations from an equilibrium state.
Spin-resolved electron-number density (measured relativeto the equilibrium state) in the FM, nσ with σ =↑ and ↓ forminority and majority spins, respectively, obeys the diffusionequation [76–79]
∂nσ
∂t= Dσ∇2nσ −
(nσ
τσ
− nσ
τσ
)− i ′σ , (4)
where i ′σ is the spin-σ particle flux from the FM into theTI across the FM/TI interface and σ = −σ . According toequilibrium considerations, the spin-flip rates are related byρ↑/τ↑ = ρ↓/τ↓ ≡ �, where ρσ is the Fermi-level spin-σdensity of states in the FM. Electron density on the surfaceof the TI, n, in general, obeys the continuity equation:
∂n
∂t= −∇ · j + (i ′↑ + i ′↓). (5)
Due to the strong spin-orbit coupling in the TI, we do notseparate diffusion into two spin components. In lieu of thediffusive relations for n↑ and n↓ in the FM, we have those forn and j = (jx,jy) in the TI. The nonequilibrium spin density,as a hydrodynamic quantity, is thus replaced in the TI by thecurrent density (which is physically motivated by the helicityof electron transport).
To the lowest order in tunneling, the spin-dependentinterlayer currents can be calculated using the Fermi’s goldenrule:
i ′σ = 2π
�
∑k,k′,σ
|〈bk′H ′a†kσ 〉0|2(fF,kσ − fT,k′ )δ(εkσ − εk′).
(6)
Here fT and fF are the associated distribution functions; εk =�vk − εT and εkσ = (�k)2/2m + σh − εF are the correspond-ing energy eigenvalues; and the expectation value 〈· · · 〉0 istaken with respect to the absolute vacuum. In the state of globalthermal equilibrium, the electron distributions are given bythe Fermi-Dirac distribution function: f0(ε) = [eε/kBT + 1]−1.The nonequilibrium particle densities in the FM and TI canbe accounted for by simply shifting their chemical potentialsby μ = n/ρ and μσ = nσ /ρσ , respectively, where ρ is theFermi-level density of states in the TI.
In the dilute limit of scattering impurities, the particle-current density j in the xy plane on the surface of the TIresults in a shift δk of the electron distribution function in themomentum space:
j = vk
4πδk , fT = f0 − ∂f0
∂k· δk. (7)
Here k is the Fermi wave number in the TI.
III. RESULTS
A. Diffusion equations
The tunneling matrix elements entering in Eq. (6) aredetermined by the spinor eigenfunctions
|k〉 = 1√2
(i
eiφ
),
|↑〉 =(
cos θ2
eiϕ sin θ2
), and |↓〉 =
(− sin θ
2
eiϕ cos θ2
), (8)
in the TI and FM, respectively, where φ = tan−1(ky/kx) is thepolar angle of k in the TI. We find (absorbing the FMs volumeinto γ )
i ′σ = γ
∫k2dk
2π2
k′dk′dφ
(2π )2[1 + σ sin θ sin(φ − ϕ)]
× (fF − fT )δ(εkσ − εk′)
= γ
[σρσ sin θ
veϕ · j + ρnσ − ρσ n
], (9)
where eϕ = (sin ϕ, − cos ϕ).Substituting spin-dependent tunneling current density (9)
into Eq. (4) complements the diffusion equation in the FMwith the tunneling leakage current, which is itself expressed interms of the consistent hydrodynamics quantities. In order tocomplete our diffusion theory, we, however, still need to relatethe current on the surface of the TI, j, to the out-of-equilibriumdensities, n and nσ , which we proceed to do in the following.
Driving the TI electrons by a spatially homogeneousin-plane electric field E, in the presence of the interlayertunneling, the current j on the surface of the TI obeys thescattering relation,
∂j∂t
= − j − gE/e
τ+ ∂j
∂t
∣∣∣∣t
, (10)
where τ is the transport mean free time, e < 0 is the electroncharge, and g is the electrical conductivity, which is given,according to the Einstein relation, by g = e2Dρ, in termsof the diffusion coefficient D = v2τ/2. The last term on the
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right-hand side represents the tunneling contribution to thetime derivative of the current. At frequencies ω � 1/τ , thisreduces to
j = eDρE + τ∂j∂t
∣∣∣∣t
≡ jd + jt , (11)
where the second term stands for the tunneling contributionto the planar TI current. In general, −eE should be replacedwith the electrochemical-potential gradient, −eE → ∇μ,which also captures the diffusive contribution to the current:jd = −D∇n. The tunneling contribution to the current is alsoevaluated using Fermi’s golden rule, giving
∂j∂t
∣∣∣∣t
= 2π
�
∑k,k′,σ
〈bk′vb†k′ 〉0|〈bk′H ′a†
kσ 〉0|2
× (fF,kσ − fT,k′ )δ(εkσ − εk′ )
= −γ
[ρ+j + v sin θ
2(ρn− − ρ−n)eϕ
]. (12)
where vi = ∂HT
�∂ki(i = x,y), ρ± ≡ ρ↑ ± ρ↓, and n± ≡ n↑ ± n↓.
(We later also benefit from the definition D± ≡ D↑ ± D↓.)The first term gives a correction of orderO(γ τ ) to the diffusioncoefficient D, which can be absorbed by a redefinition of D.Note that in the absence of spin-orbit interactions in the FM,∂jσ /∂t |t = 0 there, which allowed us to use simple diffusiverelations in Eq. (4).
Substituting the current Eq. (11) into Eq. (5), we obtain thediffusion equation in the TI (to the lowest order in tunneling),
∂n
∂t= −∇ · jd − ∇ · jt + γ
[ρ− sin θ
veϕ · jd + ρn+ − ρ+n
],
(13)
which becomes, upon substituting the above explicit expres-sions for jd and jt ,
∂n
∂t= D∇2n + γ
[ρD sin θ
veϕ · ∇n−
− 2ρ−D sin θ
veϕ · ∇n + ρn+ − ρ+n
]. (14)
In the most general case, our final diffusion equations for theTI and FM are written in terms of the full electrochemicalpotentials (including electrostatic contributions), μ and μσ ,
∂n
∂t= Dρ∇2μ + γρ
[D sin θ
veϕ · (ρ↑∇μ↑ − ρ↓∇μ↓)
− 2ρ−D sin θ
veϕ · ∇μ + ρ↑μ↑ + ρ↓μ↓ − ρ+μ
](15)
and∂nσ
∂t= Dσρσ∇2μσ − �(μσ − μσ )
+ γρρσ
[σD sin θ
veϕ · ∇μ + μ − μσ
]. (16)
These equations constitute the central result of this paper.
B. Static 1D case
As an example of FM/TI junctions, consider the static 1Dcase. The diffusion equations in the FM then become
0 = Dσρσ ∂2xμσ − σ�(μ↑ − μ↓)
+ γρρσ
[σD sin θ sin ϕ
v∂xμ + μ − μσ
], (17)
and that on the surface of the TI is
0 = D∂2xμ + γ
[D sin θ sin ϕ
v(ρ↑∂xμ↑ − ρ↓∂xμ↓)
− 2ρ−D sin θ sin ϕ
v∂xμ + ρ↑μ↑ + ρ↓μ↓ − ρ+μ
].
(18)
The diffusion equations in the FM can be combined to obtainthe equation for the spin accumulation μs ≡ μ↑ − μ↓,
0 = ∂2xμs − μs
λ2+ γρ
[ (1
D↑+ 1
D↓
)D sin θ sin ϕ
v∂xμ
+(
1
D↑− 1
D↓
)μ − μ↑
D↑+ μ↓
D↓
], (19)
where λ is the spin-diffusion length given by
1
λ2= �
(1
ρ↑D↑+ 1
ρ↓D↓
). (20)
For the internal consistency of our quasi-1D treatment of thediffusion in the FM, we need the FM layer to be thinner thanλ. The charge transport in the FM is governed by the followingrelation:
0 = (D↑ρ↑∂2
xμ↑ + D↓ρ↓∂2xμ↓
)+ γρ
[Dρ− sin θ sin ϕ
v∂xμ + ρ+μ − ρ↑μ↑ − ρ↓μ↓
].
(21)
We in the following solve the static 1D equations, Eqs. (18),(19), and (21), iteratively, to first order in γ . In the absence oftunneling, i.e., γ = 0, the general solutions are
μ0 = Ax + B, μ0s = aex/λ + be−x/λ,
(22)
μ0c = μ0 + cx + d − Pμ0
s
2,
where P ≡ (D↑ρ↑ − D↓ρ↓)/(D↑ρ↑ + D↓ρ↓) is the FM con-ductivity polarization (according to the Einstein’s relation)and μc ≡ (μ↑ + μ↓)/2 is the spin-averaged chemical poten-tial. Here, a, b, c, d, A, and B are yet unknown constants,which should be determined by the boundary conditions. The
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TAKEHITO YOKOYAMA AND YAROSLAV TSERKOVNYAK PHYSICAL REVIEW B 89, 035408 (2014)
solutions to the first order in γ are then obtained by inserting these zeroth-order solutions into the terms proportional to γ in thediffusion equations, which produces O(γ ) source terms. After some algebra, we thus find solutions to the first order in γ :
μs = aex/λ + be−x/λ + γρλ2
D↑D↓
[f ex/λ + ge−x/λ + DD+ sin θ sin ϕ
vA + D−(cx + d) + D+ + PD−
4λx(aex/λ − be−x/λ)
],
μc = μ0 + cx + d − γρ
2(D↑ρ↑ + D↓ρ↓)
[ux + v + Dρ− sin θ sin ϕ
vAx2 − ρ+
(c
3x3 + dx2
)
+ (Pρ+ − ρ−)λ2(aex/λ + be−x/λ)
]− Pμs
2, (23)
μ = Ax + B + γ
2D
[Cx + E − D sin θ sin ϕ
v{(ρ+ − Pρ−)λ(aex/λ − be−x/λ) + ρ−(c − A)x2} − ρ+
(c
3x3 + dx2
)
+ (Pρ+ − ρ−)λ2(aex/λ + be−x/λ)
].
Here, f, g, u, v, C, and E are to be determined by the boundary conditions (at order γ ).
In order to impose specific boundary conditions, supposenow that a ferromagnetic layer is attached to the TI along−W/2 < x < W/2. Since the FM is terminated at x = −W/2and W/2, we require vanishing of the diffusive spin-dependentplanar fluxes normal to the boundaries, such that ∂xμc =∂xμs = 0 at x = ±W/2. Applying a uniform 2D particle fluxj to the TI surface in the x direction, we have j = −Dρ∂xμ,in the absence of tunneling, which determines A = −j/Dρ.We can, furthermore, set B = 0 without loss of generality(absorbing it by a gauge potential shift, if necessary). Solvingthe first two of Eqs. (23) then gives the spin accumulation inthe FM of the form
μs = −4γ λ2j
{sin θ sin ϕ
v(D+ − PD−)− D−
D(D2+ − D2−)
×[x − λ sinh(x/λ)
cosh(W/2λ)
]}. (24)
The spin accumulation, which is a nonequilibrium quantity, isproportional to the current j , as expected. The first term in thecurly brackets is proportional to the y projection of the FMexchange field. This can be interpreted to result directly fromthe current-induced spin accumulation in the TI, which pointsalong the −y direction (when the current is flowing along thex axis). Since the TI current is spin polarized helically evenwithout the FM, this contribution is position-independent andpersists even when D− → 0. The other contribution to μs isindependent of the direction of the exchange field, and requiresD− �= 0, which means that it is due to nonequilibrium transportthat is spin polarized by the FM. This contribution is positiondependent, vanishing when W � λ and growing linearly withx when W λ.
It should be remarked that d is given by d =−j (ρ−/vρρ+) sin θ sin ϕ, to the zeroth order in tunneling,reflecting the fact that the average chemical potential in theFM can be shifted by spin injection. The tangential particleflux in the FM is obtained as
∑σ
jσ = −∂x(D↑ρ↑μ↑ + D↓ρ↓μ↓) = γρ+2D
j
(W 2
4− x2
).
(25)
The current in the FM flows in the same direction as that onthe surface of the TI and reaches its maximum value at thecenter of the junction.
The chemical potential on the surface of the TI (settingE = 0) can be expressed as
μ
μ0= 1 − γ
2
[ρC
j− ρ− sin θ sin ϕ
vx − ρ+
3Dx2
], (26)
where μ0 = −jx/Dρ. The continuity of the TI current (11) atjunction boundaries, x = ±W/2, at first order in γ , gives
ρC
j= ρ+W 2
4D− 4Dρ+
v2− 2Dρ2
− sin2 θ sin2 ϕ
v2ρ+. (27)
Only when ρ− is nonzero does the chemical potential on thesurface of the TI depend on the direction of the exchange field.Finally, the additional resistance δR ≡ R − R0 of the TI/FMjunction (normalized to the bare TI resistance R0) is found tobe
δR
R0= δμ(−W/2) − δμ(W/2)
jW/Dρ
= γ
[−ρ+W 2
12D+ 2Dρ+
v2− Dρ2
− sin2 θ sin2 ϕ
v2ρ+
], (28)
where δμ ≡ μ − μ0 is the chemical-potential shift due totunneling. The magnetoresistance ∝−h2
y reaches maximumfor h ⊥ y and minimum for h ‖ y. Note that this resistancerespects the Onsager reciprocity for two-terminal systems.Namely, δR ∝ const − h2
y is invariant under time reversal:h → −h. The normalized magnetoresistance is given by
R(h ⊥ y) − R(h ‖ y)
R0= γρ2
−τ
2ρ+. (29)
For ρ−/ρ+ ∼ 1, �γρ− ∼ 10−3 eV, and τ ∼ 10−13 s, weestimate γρ2
−τ/(2ρ+) ∼ 0.1.
IV. CONCLUSION
In summary, we studied spin and charge diffusion inFM/TI junctions under current bias on the surface of the TI.The FM is assumed to be metallic. The diffusive theory isconstructed for the coupled transport of the spin-dependent
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electron densities in the FM and the charge density in the TI.The diffusion equations which include the interlayer couplingand the current bias are derived based on the perturbativeapproach. By solving the diffusion equations analytically, weobtained spin accumulation in the FM induced by the currentand magnetoresistance due to the interaction between themagnetization and the current flowing on the surface of the TI.The dependencies of the spin accumulation and the junctionmagnetoresistance on the magnetization reflect the helical spintexture of the surface Dirac fermion.
ACKNOWLEDGMENTS
This work is partly supported by a Grant-in-Aid for YoungScientists (B) (Grant No. 23740236) and the “TopologicalQuantum Phenomena” (Grant No. 23103505) Grant-in-Aid forScientific Research on Innovative Areas from the Ministry ofEducation, Culture, Sports, Science, and Technology (MEXT)of Japan, FAME (an SRC STARnet center sponsored byMARCO and DARPA), the NSF under Grant No. DMR-0840965, and Grant No. 228481 from the Simons Foundation.
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