PHYSICAL REVIEW B, VOLUME 63, 184418
Mesoscopic tunneling magnetoresistance
Gonzalo Usaj and Harold U. Baranger
Department of Physics, Duke University, Box 90305, Durham, North Carolina 27708-0305
共Received 28 June 2000; published 18 April 2001兲
We study spin-dependent transport through ferromagnet/normal-metal/ferromagnet double tunnel junctions
in the mesoscopic Coulomb-blockade regime. We calculate the conductance in the absence or presence of
spin-orbit interaction and for arbitrary orientation of the lead magnetizations. The tunneling magnetoresistance
共TMR兲, defined at the Coulomb-blockade conductance peaks, is calculated and its probability distribution
presented. We show that mesoscopic fluctuations can lead to the optimal value of the TMR and that the
conductance in noncollinear configurations gives information about how the spin rotates inside the grain.
DOI: 10.1103/PhysRevB.63.184418
PACS number共s兲: 75.70.Pa, 73.23.Hk, 73.40.Gk, 73.23.⫺b
Spin-polarized electron tunneling has become a very active area of research, motivated by its potential role in new
electronic devices1 as well as by the possibility of observing
novel effects.2–5 In ferromagnet/ferromagnet single tunnel
junctions 共separated by an insulating barrier兲, the subject has
been investigated for some time.6 Their main feature is that
the resistance depends on the relative orientation of the magnetization in the ferromagnets. This is characterized by the
tunneling magnetoresistance 共TMR兲, defined as the relative
change of the resistance when the magnetizations of the ferromagnets rotate from being parallel to antiparallel.
A richer variety of effects are observed in ferromagnet/
normal-metal/ferromagnet 共F-N-F兲 double tunnel junctions
as well as in completely ferromagnetic 共F-F-F兲 double junctions, due to the interplay between charge and spin. These
systems consist of two ferromagnetic leads separated by a
metallic grain. If the capacitance of the grain is such that the
thermal energy k B T is smaller than the charging energy E C ,
then the Coulomb blockade 共CB兲 of electron tunneling occurs. Specifically, the conductance shows pronounced peaks
as a function of an external gate controlling the electrostatic
potential on the grain. How this charging effect modifies the
TMR has been the subject of recent experimental3,4 and
theoretical7–12 work. In F-F-F systems, the TMR is enhanced
in the Coulomb-blockade regime by higher-order tunneling
processes such as cotunneling3,7–9 while in F-N-F systems
the TMR is reduced.10 The effects of quantum interference
within the Coulomb blockade have not been considered in
these systems as of yet.12
On the other hand, the quantum Coulomb blockade has
been extensively investigated in the absence of
ferromagnets.13,14 Quantum interference in these structures
necessarily entails mesoscopic fluctuations as the detailed
shape of the grains cannot be controlled. Such effects become important for sufficiently low temperature or, at room
temperature, for sufficiently small systems. In such a quantum regime (k B TⰆ⌬), the CB conductance peak height
G peak fluctuates strongly as a function of the gate voltage.13,14
Here we study a problem at the intersection of these two
fields, quantum mesoscopics and spintronics. Specifically,
we study quantum effects in the TMR at the CB peaks—the
mesoscopic tunneling magnetoresistance—in F-N-F systems.
In the case of zero spin-orbit 共SO兲 tunneling, we show that
there are CB peaks with the maximum possible value of
0163-1829/2001/63共18兲/184418共5兲/$20.00
TMR; in contrast to the classical case, no special tuning is
required. For strong spin-orbit coupling, the TMR can be
large despite the SO coupling. As for G peak , the mesoscopic
TMR is characterized by a probability distribution rather
than by a single number, and we obtain this distribution in
both cases above.
Since G peak depends on the properties of a single wave
function, one can legitimately wonder if it is possible to
probe its spin structure 共a very active subject5,15,16兲 through
the conductance in noncollinear configurations. For such a
configuration, we calculate G peak as a function of the angle
between the magnetizations in the two leads. In this case as
well as for a strong SO, the usual description in terms of a
rate equation is not possible, and a more general equation for
the conductance is presented. Finally, we study the case of
broken spin degeneracy in the grain, which is equivalent to
having a half-metallic lead.17,18
Figure 1 shows a schematic of the system: a normal-metal
grain is weakly coupled to two ferromagnetic reservoirs by
tunnel junctions. The potential of the grain is controlled by
gate voltage V g . We consider ⌫Ⰶk B TⰆ⌬ⰆE C , where ⌫ is
the total width of the resonant levels in the dot. In this regime, only a single energy level contributes to the conductance. Because of time-reversal symmetry, each level is doubly degenerate 共Kramers doublet兲. Within a Hartree-Fock
共HF兲 treatment of the electron-electron interactions, the
transport is described in terms of the self-consistent singleelectron wave functions; for the resonant doublet, these are
FIG. 1. Schematic picture of the F-N-F double junction. A
normal-metal grain is coupled to ferromagnetic leads by tunnel
junctions and is small enough for charging to be important. When
the magnetization of one lead is flipped, the conductance changes:
the relative change with respect to the parallel configuration defines
the TMR.
63 184418-1
©2001 The American Physical Society
GONZALO USAJ AND HAROLD U. BARANGER
PHYSICAL REVIEW B 63 184418
the
spinors
⌿ 1 (r)⫽ 关 ␾ (r), ␹ (r) 兴 T
and
⌿ 2 (r)
⫽ 关 ⫺ ␹ * (r), ␾ * (r) 兴 T . The mesoscopic tunneling magnetoresistance is defined as
G P ⫺G A P
␩⫽
GAP
共1兲
where G P(A P) is the conductance at the CB peak in the parallel 共antiparallel兲 configuration.
Absence of spin orbit. Consider the simplest case of no
SO interaction and collinear magnetizations. Then, the
Kramers doublet corresponds to twofold spin degeneracy
␹ (r)⬅0, assuming the Zeeman energy associated with any
magnetic field present is much smaller than k B T. Linearresponse theory yields for the conductance19
e 2␭
G peak⫽
បk B T
兺
m⫽1,2
L R
⌫m
⌫m
L
R
⌫m
⫹⌫ m
共2兲
,
where the sum is over the spin degenerate states and ␭ is a
q
, q⫽L(R), due
numerical factor.19–21 The partial width ⌫ m
q
to tunneling to the left 共right兲 electrode is ⌫ m
q 2
q
q
⫽2 ␲ 兺 ␣ ␳ ␣ 兩 V ␣ m 兩 , with V ␣ m the matrix element between
兩 m 典 in the grain and the channels ␣ ⫽↑,↓ in lead q, and ␳ ␣q
the local density of states of those channels. For pointq
q
⫽ 兩 ␾ q 兩 2 t ↑(↓)
, where 兩 ␾ q 兩 2 is the wave
contact leads, ⌫ 1(2)
function of the resonant level at the contact point rq , and t ␣q
is a factor that depends on both ␳ ␣q and the tunnel barrier.
Notice we assume that the tunneling through the barriers
conserves spin. Letting D q (d q ) denote t ␣q for the majority
共minority兲 spins, we find
␩⫽
共 D L ⫺d L 兲共 D R ⫺d R 兲 兩 ␾ L 兩 2 兩 ␾ R 兩 2
共 D L 兩 ␾ L 兩 2 ⫹D R 兩 ␾ R 兩 2 兲共 d L 兩 ␾ L 兩 2 ⫹d R 兩 ␾ R 兩 2 兲
.
共3兲
Since metallic grains are generally irregular in shape, one
expects the chaotic nature of the single-particle classical dynamics to produce random matrix statistics for the wave
functions. Following Ref. 13, we use random matrix theory
to describe the fluctuations of 兩 ␾ q 兩 2 : the Gaussian orthogonal
ensemble 共GOE兲 for the Hamiltonian implies the PorterThomas distribution, P( 兩 ␾ q 兩 2 )⬀ 兩 ␾ q 兩 ⫺1 exp(⫺兩␾q兩2). Assuming no correlation between 兩 ␾ L 兩 2 and 兩 ␾ R 兩 2 because the contact points are far apart, we obtain
P GOE共 ␩ 兲 ⫽
a
␲ 兩 ␩ m兩
⌰ 共 1⫺ ␩ / ␩ m 兲
冑1⫺ ␩ / ␩ m 冑␩ / ␩ m 关 1⫹ ␩ / ␩ m 共 a 2 ⫺1 兲兴
共4兲
with
a⫽ 共 ␨ 1/4⫹ ␨ ⫺1/4兲 / 共 ␦ 1/4⫹ ␦ ⫺1/4兲 ,
␨ ⫽D R d R /D L d L ,
␦ ⫽D R d L /D L d R ,
␩ m⫽2 P L P R / 兵 1⫺ P L P R ⫹ 关共 1⫺ P L2 兲共 1⫺ P R2 兲兴 1/2其 .
Here
共5兲
P q⫽
D q ⫺d q
,
D q ⫹d q
共6兲
q⫽L,R
is the spin polarization of the electrons coming from the
leads.22 Note that in the case of symmetric barriers and the
same ferromagnets—and only in this case—P GOE( ␩ ) factorizes into a term depending on the polarizations P q and one
depending on the properties of the grain.
The maximum value of the TMR, ␩ m , is also the upper
limit for the TMR when k B TⰇE C . In that regime, it is
reached when ␨ ⫽1 共optimal asymmetry兲 while in our case
the condition is 兩 ␾ L 兩 2 / 兩 ␾ R 兩 2 ⫽ ␨ . If the two leads are made of
different materials, the former condition requires that the
tunneling barriers be expressly designed while in the mesoscopic regime there are always fluctuations that satisfy the
latter condition. Indeed, note that the probability of the TMR
being close to the maximum value is not small. In this case,
then, the mesoscopic fluctuations lead to the optimal value of
the TMR.
Broken spin degeneracy. The spin degeneracy can be
removed by applying a magnetic field such that the Zeeman
energy exceeds k B T. Then, only one channel contributes to
Eq. 共2兲; the resonant level acts as an ideal spin filter rendering the polarization of the left lead irrelevant. In this way an
effective half-metallic17 injector, an object of considerable
interest in the spintronics community, can be constructed
from conventional materials.18 Proceeding as before, we find
for the TMR
␩⫽
␩ m兩 ␾ L兩 2
兩 ␾ L 兩 2 ⫹ 兩 ␾ R 兩 2 D R /D L
with ␩ m ⫽
2 PR
.
1⫺ P R
共7兲
Results for this case can be obtained from the spin degenerate results by formally taking d L →0. This yields the distribution in Eq. 共4兲 and the mean value 具 ␩ 典 ⫽ ␩ m /(1⫹a), as
before. Note that for D R ⰆD L , the average TMR is much
bigger than the symmetric single tunnel-junction result:6
具 ␩ 典 ⯝ ␩ m compared to 2 P R2 /(1⫺ P R2 ), a factor (1⫹1/P R )
larger. For ferromagnetic leads made of Co ( P⯝0.35) this
means an enhancement of 300%. The magnetic field here
must, of course, be smaller than the coercive field of the
electrode 共in the A P configuration兲, making low temperatures advantageous in observing this effect 共typically, B
ⱗ500 G and Tⱗ70 mK兲.
Spin-orbit coupling. SO coupling causes the direction of
the spin of an electron to rotate while in the grain, so that the
natural axis of quantization on the left is not the same as on
the right. Time-reversal symmetry still applies but now neither component of the spinor 关 ␾ (r), ␹ (r) 兴 T vanishes. Since
the quantization axes in the leads are fixed by the bulk magnetization of the ferromagnets, the electrons tunnel into a
superposition of ‘‘up’’ and ‘‘down’’ states. This coherence
cannot be included in a simple rate equation—a more general
approach is required. Technically, the mixing of the spin
channels means that ⌫ L and ⌫ R , which in the general case
q
⫽2 ␲ 兺 ␣ ␳ ␣q V ␣q *n V ␣q m , cannot be diagonalized
are matrices ⌫nm
in the same basis. In the relevant 2⫻2 subspace, ⌫q takes
the form
184418-2
MESOSCOPIC TUNNELING MAGNETORESISTANCE
⌫q ⫽
冉
兩 ␾ q 兩 2 t q↑ ⫹ 兩 ␹ q 兩 2 t q↓
␹ q ␾ q 共 t q↓ ⫺t q↑ 兲
q
q
␹*
q ␾*
q 共 t ↓ ⫺t ↑ 兲
冊
兩 ␹ q 兩 2 t q↑ ⫹ 兩 ␾ q 兩 2 t q↓ .
PHYSICAL REVIEW B 63 184418
共8兲
Note that 关 ⌫L ,⌫R 兴 ⫽0 only when both leads are ferromagnetic and SO coupling is present.
From the Keldysh formalism, the current through an arbitrary system connected to two leads is23–25
J⫽
e
2h
冕
d␧ Tr兵 共 ⌫L ⫺⌫R 兲 iG⬍ ⫹ 共 ⌫L f L ⫺⌫R f R 兲 A其 , 共9兲
where A⫽i(Gr ⫺Ga ) is the spectral function, Gr(a) is the
retarded 共advanced兲 many-body Green function of the full
system, and f L(R) is the Fermi distribution in the left 共right兲
lead. It is convenient to express G⬍ , the lesser Green function, in terms of the operators Iq ⬅⌫q (G⬍ ⫺i f q A): G⬍
⫽(⌫L ⫹⌫R ) ⫺1 关 IL ⫹IR ⫹i(⌫L f L ⫹⌫R f R )A兴 . For elastic transport and within the HF approximation, (IL ⫹IR ) nn ⫽0, so that
Eq. 共9兲 becomes
J⫽
e
h
冕
d␧ Tr兵 ⌫R 共 ⌫L ⫹⌫R 兲 ⫺1 ⌫L A其 共 f L ⫺ f R 兲 .
FIG. 2. Probability density of TMR for symmetric barriers and
the same ferromagnets with P⫽0.35, 0.5, 0.8 in the symplectic
ensemble 共GSE兲. The TMR is scaled with ␩ m ⫽ P 2 /(1⫺ P 2 ). The
distribution for P⫽0.35 is multiplied by 2 for purposes of comparison. The functional form of the distribution depends strongly on the
value of P. In all cases, there is a cusp at ␩ ⫽0. Note that ␩ can be
both positive and negative and significantly different from zero.
共10兲
再冉
A共 ␩ 兲 ,
This equation simplifies for linear-response and weak coupling. A is then evaluated in equilibrium: it depends on the
equilibrium populations of the grain eigenstates 共which depend only on the energy兲 and the eigenfunctions of the isolated grain.23 Therefore, it can be shown that within the resonant subspace, A is proportional to the identity. For the
conductance, we finally get
G peak⫽
e 2␭ ⬘
Tr兵 ⌫R 共 ⌫L ⫹⌫R 兲 ⫺1 ⌫L 其 ,
បk B T
共11兲
where the numerical factor ␭ ⬘ has the same origin as ␭ in
Eq. 共2兲. Notice that Eq. 共11兲 contains Eq. 共2兲 as a special
limit.
Since G peak can be calculated in any basis, we choose one
where ␹ L ⫽0. Then, for symmetric leads and barriers ( P q
⫽ P, ␨ ⫽1) the TMR from Eqs. 共8兲 and 共11兲 is
␩⫽
2 P 2 cos ␪ g
1⫹ 共 1⫺ P 2 兲 ␰ ⫺ P 2 cos ␪ g
,
共12兲
where ␪ g is the azimuthal angle of the spinor ⌿ 1 (rR ) and
␰⫽
兩 ⌿ L兩 4⫹ 兩 ⌿ R兩 4
2 兩 ⌿ L兩 2兩 ⌿ R兩 2
共13兲
with 兩 ⌿ q 兩 ⫽ 兩 ⌿ 1 (rq ) 兩 . Note that ␩ ⫽0 if the spin at the right
contact is perpendicular to the polarization in the leads. For
simplicity we will assume that the SO is strong and so describe the statistics by the Gaussian symplectic ensemble
共GSE兲. The intermediate case is straightforward numerically.
Then, ␾ and ␹ are uncorrelated complex random variables.
Using their Gaussian distributions we find P( ␰ ) and show
that P( ␰ ,cos ␪g)⫽P( ␰ )/2. Therefore, the distribution of the
TMR is
P GSE共 ␩ 兲 ⫽
1
3/2
2␩m
共 2⫹ ␩ 兲 2
⫻
A ⫺
0⭐ ␩ ⭐ ␩ m
␩
1⫹ ␩
冊
,
⫺ P 2 ⭐ ␩ ⭐0
,
共14兲
where A(x)⫽ 冑␩ m ⫺x(6⫹2 ␩ m ⫹x) and ␩ m ⫽ P 2 /(1⫺ P 2 ).
The upper and lower limits correspond to the electron tunneling out, either conserving its original spin ( ␩ m ) or with
opposite spin (⫺ P 2 ). Figure 2 shows the distribution for
three different values of P. In the asymmetric case, the shape
of the distribution is strongly dependent on the ratio D L /D R
共not shown兲.
We wish to emphasize three aspects of P GSE( ␩ ): 共a兲 The
TMR can be either positive or negative for any polarization
as a consequence of the spin rotation inside the grain—note
that P GSE( ␩ ⬎0)⫽PGSE( ␩ ⬍0). 共b兲 The functional form is
strongly affected by the SO—note in particular the disappearance of the divergence at the maximum value and the
cusp at zero 共which only vanishes for P→1). 共c兲 Even for a
strong SO the TMR can be close to its maximum zero-SO
value; this is completely different from the situation in the
classical regime, where the TMR in the presence of the SO is
always smaller than in its absence10 共note that the GSE represents the strongest SO coupling兲. This robustness of the
mesoscopic TMR against the SO is a consequence of tunneling through a single level.
Noncollinear magnetizations. If the magnetizations of
the two ferromagnetic leads are not collinear, a rate equation
does not apply 共even in the absence of a SO兲 for the same
reason as in the SO case—tunneling involves a coherent
superposition of spin states. Instead, we must use Eq. 共11兲.
We assume the magnetization on the left lead points in the
z direction while on the right lead it is tilted by an angle
2 ␪ . The spin channels on the right lead are then given by
184418-3
GONZALO USAJ AND HAROLD U. BARANGER
兩 n̂,↑ 典 ⫽(cos ␪,sin ␪)T,
兩 n̂,↓ 典 ⫽(⫺sin ␪,cos ␪)T.
⌿ m (rR ) in this basis, e.g.,
PHYSICAL REVIEW B 63 184418
Writing
⌿ 1 共 rR 兲 ⫽ 共 ␾ R cos ␪ ⫹ ␹ R sin ␪ 兲 兩 n̂,↑ 典
⫹ 共 ␹ R cos ␪ ⫺ ␾ R sin ␪ 兲 兩 n̂,↓ 典 ,
we readily obtain ⌫ R and then the conductance. Using the
basis where ␹ L ⫽0 we get
G peak共 2 ␪ 兲 ⫽G peak共 2 ␪ m 兲
1⫹ 共 1⫺ P 2 兲 ␰ ⫺ P 2 B共 2 ␪ m 兲
1⫹ 共 1⫺ P 2 兲 ␰ ⫺ P 2 B共 2 ␪ 兲
共15兲
with B(x)⫽cos x cos ␪g⫹cos ␸g sin x sin ␪g . Here, ␪ g and ␸ g
describe the rotation of the spin inside the grain as going
from left to right. B(2 ␪ ) has a clear physical meaning: it is
the cosine of the angle between the magnetization in the lead
and the spin in the grain at the right contact. The maximum
共minimum兲 conductance occurs when this angle is minimized 共maximized兲. Then, the extreme values of G peak(2 ␪ )
are shifted in the presence of the SO. They occur for
tan(2 ␪ m )⫽tan( ␪ g )cos(␸g). Figure 3 shows G peak(2 ␪ ) for
different polarizations both in the absence and in the presence of the SO. Note that the maximum becomes narrowed
as P→1. It is important to point out that Eq. 共15兲 is different
from the result obtained in the classical regime.26 In particular, G peak(2 ␪ ) can be smaller than the universal lower bound
found in Ref. 26. We believe this difference arises from the
resonant character of the transport considered here.
This angular dependence might be used to study the spin
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1
FIG. 3. Conductance as a function of the angle 2 ␪ between the
magnetization of the leads for different values of P. The solid
共dashed兲 line shows a typical random realization in the GOE 共GSE兲.
The maximum occurs for tan(2 ␪ m )⫽tan( ␪ g )cos(␸g) where ␪ g and
␸ g describe the rotation of the spin inside the grain. Note that ␪ g
⫽ ␸ g ⫽0 for the GOE and that the maximum becomes narrowed as
P→1.
structure of single wave functions. A spin-polarized scanning
tunnel microscope16 could provide a spin-resolved image of
a single wave function on the surface of a small grain, carbon
nanotube, or quantum corral. In principle it might also be
used for imaging ferromagnetic quantum dots. This potential
for revealing the spin structure of a single wave function
deserves further investigation.
We appreciate helpful conversations with I. L. Aleiner,
M. Johnson, K. A. Matveev, and D. C. Ralph. G.U. acknowledges partial support from CONICET 共Argentina兲.
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MESOSCOPIC TUNNELING MAGNETORESISTANCE
21
PHYSICAL REVIEW B 63 184418
This numerical factor takes the Coulomb correlation into account
共no double occupancy兲. It depends on the change of the spin of
the dot; for instance, ␭ 0→1/2⫽(3⫺2 冑2) and ␭ 1/2→1 ⫽( 2
⫺3 冑6).
22
Since P q is a property of the ‘‘lead ⫹ tunnel barrier,’’ it can be
significantly different from the bulk spin polarization; J. M. De
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