Charge pumping in magnetic tunnel junctions: Scattering theory
Jiang Xiao,1 Gerrit E. W. Bauer,1and Arne Brataas21Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands 2Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
共Received 14 January 2008; revised manuscript received 18 March 2008; published 22 May 2008兲
We study theoretically the charge transport pumped by magnetization dynamics through epitaxial FIF and FNIF magnetic tunnel junctions共F, ferromagnet; I, insulator; N, normal metal兲. We predict a small but mea-surable dc pumping voltage under ferromagnetic resonance conditions for collinear magnetization configura-tions, which may change sign as a function of barrier parameters. A much larger ac pumping voltage is expected when the magnetizations are at right angles. Quantum size effects are predicted for an FNIF structure as a function of the normal layer thickness.
DOI:10.1103/PhysRevB.77.180407 PACS number共s兲: 85.75.⫺d, 72.25.⫺b, 73.40.Gk
A magnetic tunnel junction共MTJ兲 consists of a thin insu-lating tunnel barrier共I兲 that separates two ferromagnetic con-ducting layers 共F兲 with variable magnetization direction.1 With a thin normal metal layer inserted next to the barrier, the MTJ is the only magnetoelectronic structure in which quantum size effects on electron transport have been detected experimentally.2 More importantly, MTJs based on transition-metal alloys and epitaxial MgO barriers3,4 are the core elements of the magnetic random-access memory 共MRAM兲 devices5 that are operated by the current-induced spin-transfer torque.6,7
It is known that a moving magnetization of a ferromagnet pumps a spin current into an attached conductor.8 Spin pumping can be observed indirectly as increased broadening of ferromagnetic resonance 共FMR兲 spectra.9The spin accu-mulation created by spin pumping can be converted into a voltage signal by an analyzing ferromagnetic contact.10This process can be divided into two steps:共i兲 the dynamical mag-netization pumps out a spin current with zero net charge current,共ii兲 the static magnetization 共of the analyzing layer兲 filters the pumped spin current and gives a charge current. In the presence of spin-flip scattering, the spin-pumping magnet can generate a voltage even in an FN bilayer.11,12 Spin-pumping by a time-dependent bulk magnetization texture such as a moving domain wall is also transformed into an electromotive force.13 Other experiments on spin-pumping induced voltages have also been reported.14,15
Here we present a model study of spin-pumping induced voltages共charge pumping兲 in MTJs. Since the ferromagnets are separated by tunnel barrier, we cannot use the semiclas-sical approximations appropriate for metallic struc-tures.10,11,16,17Instead, we present a full quantum-mechanical treatment of the currents in the tunnel barrier by scattering theory. The high quality of MgO tunnel junctions and the prominence of quantum oscillations observed in FNIF struc-tures 共even for alumina barriers兲 provide the motivation to concentrate on ballistic structures in which the transverse Bloch vector is conserved during transport. For a typical MTJ under FMR with cone angle = 5° at frequency f = 20 GHz, we find a dc pumping voltage of兩Vcp兩 ⯝20 nV for
collinear magnetization configurations or ac voltage with amplitude V˜cp⯝0.25V for perpendicular configurations.
The magnetization dynamics-induced voltages could give
simple and direct access to transport parameters of high-quality MTJs such as barrier height, magnetization anisotro-pies, and damping parameters in a nondestructive way. The polarity of the pumping voltage can be changed by engineer-ing the device parameters, etc. An oscillatengineer-ing signal as a function of the thickness of the N spacer leads to Fermi surface calipers that are in tunnel junctions not accessible via the exchange coupling.
We consider a structure shown in Fig. 1共a兲, where two semi-infinite F leads 关F共L兲 and F共R兲兴 are connected by an insulating layer共I兲 of width d and a nonmagnetic metal layer 共N兲 of width a. The magnetization direction of F共L兲 F共R兲,
m1m2共兩m1兩 =兩m2兩 =1兲, is treated as fixed 共free兲. We
disre-gard any spin accumulation in F, thus treating them as ideal reservoirs in thermal equilibrium. This is allowed when the spin pumping current is much smaller than the spin-flip rate in the ferromagnet, which is usually a good approximation. The structure reduces to an FIF MTJ when a = 0. Let
A , B , . . . , F be the spin-dependent amplitudes关A†=共A↑†, A↓†兲兴 at specific points共see Fig.1兲 of flux-normalized spinor wave
functions. The scattering states can be expressed in terms of the incoming waves A and F, such as
A B E F C C' D D' EF Ub 0 x (a) (b) F(L) N I F(R) S1 θ χ S2 tb -a 0 d m1 m2
FIG. 1. 共Color online兲 共a兲 FNIF heterostructure, in which S1,2
indicate two different scattering regions. 共b兲 Potential profiles for majority and minority spins in F are shown by solid and dashed lines. The exchange splitting is⌬ and the tunnel barrier has height
Ubrelative to the Fermi energy EF. PHYSICAL REVIEW B 77, 180407共R兲 共2008兲
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E = sˆEAA + sˆEFF, 共1兲
where sˆEAand sˆEFare 2⫻2 matrices in spin space and can be
calculated by concatenating the scattering matrices of region
S1,2and of the bulk layer I. To first order of the transmission
共tb兲 through the bulk I,
sˆEA= tˆ2关共1 − rb
⬘
rˆ2兲−1tb共1 − rˆ1⬘
rb兲−1兴tˆ1, 共2兲where tˆ1,2共rˆ1,2兲 is the 2⫻2 transmission 共reflection兲 matrix
for S1,2 共see Fig. 1兲, and the hatless tb共rb兲 is the spin-independent transmission共reflection兲 coefficient for the insu-lating bulk I. The primed and unprimed versions specify the scattering of electrons emitted coming from the left and right, respectively. The reflection coefficient rbis due to the impurity scattering inside the bulk I, and its magnitude de-pends mainly on the impurity density in I, especially near the interfaces. All scattering coefficients are matrices in the space of transport channels at the Fermi energy that are la-beled by the transverse wave vectors in the leads: q , q
⬘
共the band index is suppressed兲.The response to a small applied bias voltage can be writ-ten as Jc= GcV with conductance Gc,
Gc=
兺
q,q⬘ gc共q,q⬘
兲 with gc= e2 h Tr关sˆEAsˆEA † 兴, 共3兲where Tr关¯兴 denotes the spin trace and the summation is over all transverse modes in the leads at the Fermi level.
When the structure is unbiased but the magnetic configu-ration is time-dependent, a spin current is pumped through the structure.8When the dynamics is slow, m˙
iⰆEF/ប, it can be treated by the theory of adiabatic quantum pumping.18We consider a situation in which the magnetization共m2兲 of one
layer precesses with velocity˙ around the z axis with con-stant cone angle, whereas the other magnetization共m1兲 is constant 共see Fig. 1兲. We focus on the charge current that
accompanies the spin pumping,
Jcp=
兺
q,q⬘ jcp共q,q⬘
兲, jcp= e˙ 2Tr兵Im关共sˆEA兲sˆEA † +共sˆEF兲sˆEF † 兴其. 共4兲When a dc current is blocked 共open circuit兲, a voltage bias
Vcpbuilds up,
Vcp= Gc−1Jcp. 共5兲
The discussion above is valid for general scattering ma-trices that include, e.g., bulk and interface disorders. In order to derive analytical results, we shall make some approxima-tions. First of all, we assume that spin is conserved during the scattering; tˆifor Si共i=1,2, similar for rˆi兲 is collinear with
mi:8 Expanded in Pauli matrices ˆ =共ˆx,ˆy,ˆz兲,
tˆi= ti
+
+ ti
−ˆ · m
i, with ti⫾=共ti↑⫾ti↓兲/2. ti共=↑ , ↓兲 is the trans-mission amplitude for spin-up 共down兲 electrons with spin quantization axes mi in the scattering region Si. In the ab-sence of impurities共rb= rb
⬘
= 0兲, Eq. 共2兲 becomessˆEA=共t2 + tbt1 + + t2−tbt1 − m1· m2兲 +ˆ ·共t2+tbt1−m1+ t2−tbt1+m2− it2−tbt1−m1⫻ m2兲. 共6兲
Since all hatless quantities in this equation are still matrices in k space, such as t2+= t2+共q,q
⬘
兲, the order of t2, tb, t1 as inEq.共2兲 should be maintained. The sˆEFterm in Eq.共4兲 may be
disregarded, because only the part of sˆEF that depends on both m1 and m2contributes to jcp, and that part is in higher
order of tb.
Another approximation is the free-electron approximation tailored for transition-metal-based ferromagnets.19 We as-sume spherical Fermi surfaces for spin-up and spin-down electrons 关in both F共L兲 and F共R兲兴 with Fermi wave vectors
kF↑=
冑
2mEF/ប2 and k F↓=
冑
2m共EF−⌬兲/ប2, with an effective
electron mass m in F. Electrons in N are assumed to be ideally matched with the majority electrons in F共kF= kF↑, mN= m兲. Let Uband mb=m be the barrier height of and effective mass in the tunnel barrier. The adopted po-tential profile is shown in Fig. 1共b兲. We assume the trans-verse wave vector q to be conserved共q=q
⬘
兲 by disregarding any impurity or interface roughness scattering, which means the scattering matrices共t1,2 , t1,2⫾ , tb兲 are diagonal in k space. With these approximations, the double summation in Eqs.共3兲and 共4兲 is replaced by a single integration over transverse
wave vectors. The scattering amplitudes tiand rican be cal-culated by matching the flux-normalized wave functions at the interfaces. The transmission coefficient in the barrier bulk is the exponential decay: tb= e−d with =
冑
2mbUb/ប2+ q2. Then we obtain our main result from Eq. 共6兲, gc= e2 2he −2d共T 1 +T 2 ++ T 1 −T 2 −m 1· m2兲, 共7a兲 jcp= e 2e −2dT 1 −m 1⫻ 关兩t2−兩2共m2⫻ m˙2兲 + Im共t2 +* t2−兲m˙2兴, 共7b兲 where Ti+=兩ti↑兩2+兩ti↓兩2is the total transmission probability for scattering region Si, and Ti−
= piTi
+
=兩ti↑兩2−兩ti↓兩2 with polariza-tion pi= Ti−/Ti+. In Eq.共7b兲, the term in the square brackets is the transmitted spin pumping current, and T1−m1 represents the filtering by the static layer that converts the spin into a charge current.
For an Fe/MgO/Fe MTJ: kF↑= 1.09 Å−1 and
kF↓= 0.42 Å−1 for Fe,19 and Ub⯝1 eV and= mb/m=0.4 for MgO.3,20 This implies E
F⯝4.5 eV, ⌬⯝3.8 eV⬇0.85EF, and Ub⬇0.25EF 共tbⰆ1 when Ub⬎0.1EF and d⬎0.5 nm兲. For an FIF structure 共a=0兲, both S1 and S2 contain only a
single F共L兲/I 共for S1兲 or I/F共R兲 共for S2兲 interface. From the potential profile in Fig.1共b兲,
t1= t2=2
冑
ikx/
kx+ i/
共8兲 for kx2= kF2− q2⬎0 and zero otherwise.
If m2 precesses about an axis that is parallel to m1 共= 0° or 180°, see Fig.1兲, m1· m˙2= 0 and the second term in
Eq. 共7b兲 vanishes. The dot product 兩m1·共m2⫻m˙2兲兩
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= 2f sin2is time-independent, thus it generates a dc signal.
Let us consider an FIF MTJ with barrier width d = 0.8 nm, with m2 precessing around the z axis at frequency
f = 20 GHz with cone angle = 5°. We find a dc charge pumping voltage over the F leads Vcp⯝15 nV when m1 is parallel to the precession axis 共= 0 °兲 and Vcp⯝−19 nV when antiparallel共= 180°兲.21兩V
cp兩 is higher for the
antipar-allel configuration simply because its resistance is higher. When the precession cone angle = 10°,22 the dc voltage 兩Vcp兩 ⯝60 nV, similar to a previously measured pumping
voltage in a metallic junction.12
Figure2共a兲shows the dc Vcpas a function of the barrier
height Ub for an FIF structure at = 5° and f = 20 GHz.
Vcp= Gc−1Jcpincreases as a function of barrier height mainly because gc
−1
jcp increases as a function of Ub: From Eq. 共7兲, we have gc−1jcp⬇共T1−/T1+兲共兩t2−兩2/T 2 +兲 共assume T 1 −T 2 −ⰆT 1 +T 2 +兲.
The first ratio T1−/T1+= p1⬀关共/兲2− k↑k↓兲/关共/兲2+ k↑k↓兴
in-creases as a function of Ub through共Ub兲, whereas the sec-ond ratio 兩t2−兩2/T
2 +⬀共
冑
k↑−
冑
k↓兲2/共k↑+ k↓兲 is independent of /. The pumping voltage, therefore, increases with Ub共and 1/兲. We also see that Vcpdecreases when d increases, whichcan be understood by the following: The effect of the tunnel barrier is to focus the transmission electrons on small q’s due to the exponential decay factor exp关−2共q兲d兴. Smaller q im-plies larger kinetic energy normal to the barrier and therefore reduced sensitivity to the spin-dependent potentials. Hence,
Vcp decreases with barrier width. The lowest curve in Fig. 2共a兲is approximately Vcp= gc
−1共q兲j
cp共q兲兩q=0, because for large
d the electrons near q = 0 completely dominate the
transmis-sion. The negative value of Vcpin Fig.2共a兲is caused by the
negative polarization 共p1⬍0兲 at low barrier height Ub for electrons with small q. Vcpremains finite for infinitely high or wide barrier, however the time to build up this voltage, the RC time共RC兲, goes to infinity due to the exponential growth
of the resistance.
When m1 is perpendicular to the precession axis of m2, i.e., = 90°, the charge pumping voltage oscillates around zero because both dot products in Eq. 共7b兲, m1·共m2⫻m˙2兲
⬇2f sincos共2ft兲 and m1· m˙2= 2f sinsin共2ft兲, give
rise to an ac signal. With Vcp= am1·共m2⫻m˙2兲+bm1· m˙2, where the two components are out of phase by /2, the amplitude is given by V˜cp= 2f sin
冑
a2+ b2. An FMR with= 5° and f = 20 GHz then gives an ac pumping voltage with
amplitude as large as V˜cp⯝0.25V. Figure 2共b兲 shows the
barrier height dependence of amplitude V˜cp quite similar to
the dc case in Fig. 2共a兲and for similar reasons. For a half-metallic junction, the magnitude of the dc pumping voltage
Vcpcan be shown to be bounded by共ប/2e兲sin2, and the ac
pumping voltage amplitude V˜cp must be smaller than 共ប/2e兲sin, where= 2f.
When an N layer of thickness a is inserted, it is interesting to inspect the two different modes: mode 1, FNIF˜ ; and mode 2, F˜ NIF, where F˜ indicates the F layer under FMR. Equation 共7b兲 applies to mode 1, and it applies to mode 2 with
sub-scripts 1 and 2 swapped. The N layer forms a quantum well for spin-down electrons that causes the oscillation in the charge pumping voltage as a function of a as shown in Fig.
3. The period of the quantum oscillation due to the N inser-tion layer is about/kF⬇3 Å. However, due to the aliasing
effect caused by the discrete thickness of the N layer,23 the observed period should be/兩kF−/兩, where is the
thick-ness of a monolayer. In mode 1, the quantum well formed by the N layer can modulate T1−共a兲 such that the electrons con-tributing to the transmission the most have T1−⬎0 共p1⬎0兲 or
T1−⬍0 共p1⬍0兲, and thus change the sign of the pumping
voltage Vcp. On the other hand, there is no sign change in
mode 2 because T2−is independent of a. Similar oscillations could also be found for the amplitude of the ac pumping voltage.
Because the ac voltage is proportional to sin and dc voltage is proportional to sin2, the ac pumping voltage is
much larger than the dc counterpart at small. However, in order to observe an ac pumping voltage, the time to build up the voltage, the RC timeRC= RC, has to be shorter than the
pumping period, i.e., RC⬍1/ f. Approximately RC
⬃共0h/2e2兲e2冑2mbUb/ប 2d
/d, where and 0are the dielectric
constant and electric constant, respectively. A more accurate estimation of the RC time for a typical structure is as fol-lows: the resistance-area共RA兲 value of the MTJ in our cal-culation is ⬇3 ⍀m2 for d = 0.8 nm 共and ⬇70 ⍀m2 for
d = 1.2 nm, which is consistent with experimental values3兲. The capacitance of an MgO tunnel barrier with d = 0.8 nm is calculated by C/A=0/d⬇0.1 F/m2 共⬇9.7 for MgO兲.
Therefore RC=共RA兲共C/A兲⬇0.3 psⰆ1/ f ⬃102ps. The
electromagnetic response is therefore sufficiently fast to fol-low the ac pumping signal.
0 0.25 0.5 0.75 1 −15 0 15 30 45 DC Voltage barrier height: U b/EF DC Voltage (nV) d = 0.8 nm d = 1.2 nm d = 3.0 nm 0 0.25 0.5 0.75 1 0 0.25 0.5 AC Voltage barrier height: U b/EF AC Voltage (µ V)
FIG. 2.共Color online兲 Barrier height 共Ub兲 and width 共d兲 depen-dence of the pumping voltage. Left: dc. Right: Maximum amplitude of ac voltage. −30 0 30 0 0.5 1 1.5 2 −30 0 30
insertion layer width: a (nm)
D C Voltage (nV) mode 1 mode 2 d = 1.2 nm d = 0.8 nm
FIG. 3. 共Color online兲 Vcp vs N-layer thickness a for FNIF 共Ub= 0.25EF兲.
CHARGE PUMPING IN MAGNETIC TUNNEL JUNCTIONS:… PHYSICAL REVIEW B 77, 180407共R兲 共2008兲
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We ignored interface roughness and barrier disorder in the calculation of the pumping voltage. This may be justified by the high quality of the epitaxial MgO tunnel barrier.3,4 Fur-thermore, the geometric interface roughness reduces mainly the nominal thickness of the barrier,24 which can be taken care of by an effective thickness parameter. Impurity states in the barrier open additional tunneling channels with Ub
⬘
⬍Ub, which generally increases tunneling but also reduces the spin-dependent effects when spin-flip is involved. In general, interface roughness and disorder can be important quantita-tively, but have been shown not to qualitatively affect the features predicted by a ballistic model.25In order to be quan-titatively reliable, the real electronic structure has to be taken into account as well. Both band structure and disorder effects can be taken into account by first-principles electronic-structure calculation as demonstrated for metallic struc-tures.26Recently, a magnetization-induced electrical voltage of the order of V was measured for an FIN structure by
Moriyama et al.27The authors explain their findings by spin pumping, but note that the signal is larger than expected. An FMR generated electric voltage generation up to 100V was theoretically predicted for such FIN structures.28 Sur-prisingly, this voltage is much larger thanប/2e⬃V, the maximum “intrinsic” energy scale in spin-pumping theory.
To summarize, a scattering matrix theory is used to cal-culate charge pumping voltage for a magnetic multilayer structure. An experimentally accessible charge pumping volt-age is found for an FIF MTJ; the pumping voltvolt-age can be either dc or ac depending on the magnetization configura-tions. In FNIF structure, we find on top of the previously reported oscillating TMR共Ref.2兲 a charge pumping voltage
that oscillates and may change sign with the N-layer thick-ness.
This work has been supported by EC Contract No. IST-033749 “DynaMax.”
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