Journal of Magnetism and Magnetic Materials 240 (2002) 143–145

Spin accumulation and cotunneling effects in ferromagneticsingle-electron transistors

Jan Martineka,b,*, J !ozef Barna!sa,c, Sadamichi Maekawab, Herbert Schoellerd,e,Gerd Sch .ond,f

a Institute of Molecular Physics, Polish Academy of Sciences, ul. Smoluchowskiego 17, 60-179 Pozna !n, Polandb Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan

cDepartment of Physics, Adam Mickiewicz University, ul. Umultowska 85, 61-614 Pozna !n, PolanddForschungszentrum Karlsruhe, Institut f .ur Nanotechnologie, 76021 Karlsruhe, Germany

e Institut f .ur Theoretische Physik A, RWTH Aachen, 52056 Aachen, Germanyf Institut f .ur Theoretische Festk .operphysik, Universit .at Karlsruhe, Germany

Abstract

Electron tunneling through small metallic particles (islands) coupled to two ferromagnetic electrodes is studiedtheoretically in the Coulomb blockade regime, where higher order tunneling processes play a significant role. Transportcharacteristics of the system are analyzed by the real-time diagrammatic technique. It is shown that the spin splitting of

the electrochemical potential due to spin accumulation on the island should be detectable from the spacing between tworesonances in the current–voltage characteristics. r 2002 Elsevier Science B.V. All rights reserved.

Keywords: Tunnel magnetoresistance; Single-electron tunneling; Coulomb blockade

Non-equilibrium spin accumulation [1–3] due to spin-polarized electronic transport occurs in inhomogeneousspin-polarized electron systems and is related to a

difference in local electrochemical potentials for elec-trons with opposite spin orientations. Such a differencemay be created, e.g., by spin injection from ferromag-

netic to normal metals, as predicted theoretically [1] andalso observed experimentally [2,3].

There are several experimental techniques by whichthe spin accumulation can be detected indirectly [2,3].

The question whether spin splitting of the electrochemi-cal potential can be observed directly, for instance byspectroscopic methods analogous to tunneling spectro-

scopy for superconducting gap, is still open. In thispaper, we show how this spin splitting can be evaluatedfrom some peculiarities in the transport characteristics

of ferromagnetic single-electron transistors (FM SETs)with a normal metallic island.

Spin-dependent transport in ferromagnetic double-barrier tunnel junctions was recently studied bothexperimentally [4] and theoretically in the sequential

[5,6], and cotunneling [7,8] regimes. We use thetechnique which is an extension of the real-timediagrammatic formalism developed for nonmagnetic

junctions [9,10], and which is applicable for arbitrarytemperature and arbitrary transport voltage.

We will treat all electrons on the island as twoindependent (in the absence of spin-flip transition)

reservoirs of electrons with opposite spin orientationsand assume the Fermi distribution for each reservoirseparately. The corresponding electrochemical poten-

tials mm and mk can be different in a general case and weassume mm ¼ �mk:

The electric current I in the sequential tunneling limit

is given by I ð1Þ ¼P

s Ið1ÞLs ¼ �

Ps I

ð1ÞRs; with [11]

I ð1Þrs ¼4p2e

h

Xn

pð0Þn þ pð0Þnþ1

h i

�a�ðDnÞ aþrsðDnÞ � aþðDnÞ a�rsðDnÞ

aðDnÞ; ð1Þ

*Corresponding author. Institute for Materials Research,

Tohoku University, Sendai 980-8577, Japan. Fax: +81-22-215-

2006.

E-mail address: martinek@ifmpan.poznan.pl (J. Martinek).

0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 7 6 6 - 1

where Dn ¼ Echðnþ 1Þ � EchðnÞ; r ¼ R;L; and the para-meters a7rsðeÞ are the forward and backward propagators

on the Keldysh contour in the Fourier space

a7rsðeÞ ¼ 7a0rs

e� Dmrs

exp½7bðe� DmrsÞ� � 1; ð2Þ

where a0rs ¼ h=ð4p2e2RrsÞ is the dimensionless conduc-

tance of the junction r for spin s; e is the energy of thetunneling electron, and Dmrs ¼ mr � ms; with mr denotingthe electrochemical potential of the rth electrode.

The parameters a7ðeÞ and aðeÞ in Eq. (1) are definedas a7ðeÞ ¼

Prs a

7r;sðeÞ and aðeÞ ¼ aþðeÞ þ a�ðeÞ;

whereas the probability pð0Þn obeys the equationpð0Þn aþðDnÞ � pð0Þnþ1a

�ðDnÞ ¼ 0 withP

n pð0Þn ¼ 1:

The dominant second-order (cotunneling) contribu-tion to electric current can be divided into three parts,I ð2Þ ¼

P3i¼1 I

ð2Þi ; with I ð2Þi ¼

Ps I

ð2ÞiLs ¼ �

Ps I

ð2ÞiRs: The

three terms describe, respectively, the processes inwhich one electron enters and the other one leavesthe island coherently, the renormalization of the

tunneling conductance, and the renormalization of theenergy gap.

These terms are given by the following expressions

[11]:

I ð2Þ1rs ¼4p2e

h

Xn

pð0Þn

Zdo ½a�ðoÞ aþrsðoÞ

� aþðoÞ a�rsðoÞ� Re RnðoÞ2 ; ð3Þ

Ið2Þ2rs ¼ �

1

2

4p2e

h

Xn

pð0Þn þ pð0Þnþ1

� �

�a�ðDnÞ aþrsðDnÞ � aþðDnÞ a�rsðDnÞ

aðDnÞ

�Z

do aðoÞRe RnðoÞ2 þ Rnþ1ðoÞ

2�

; ð4Þ

Ið2Þ3rs ¼ �

1

2

4p2e

h

Xn

pð0Þn þ pð0Þnþ1

� �

�q

qDn

a�ðDnÞ aþrsðDnÞ � aþðDnÞ a�rsðDnÞaðDnÞ

�

�Z

do aðoÞRe RnðoÞ � Rnþ1ðoÞ½ � ; ð5Þ

where RnðoÞ ¼ 1=ðo� Dn þ i0þÞ � 1=ðo� Dn�1 þ i0þÞ :The spin accumulation on the island can be deter-

mined from the condition of spin current conservation,with the spin relaxation term taken into account in a

general case,Xr

ðI ð1Þrs þ I ð2Þrs Þ � emsDI=tsf ¼ 0 ; ð6Þ

where tsf is the spin relaxation time and DI is the densityof states on the island.

Numerical calculations were performed for both

parallel (P) and antiparallel (AP) alignment of theelectrode magnetization. For the parallel alignment we

assumed RRm ¼ RLm ¼ aRQ and RRk ¼ RLk ¼ð1 � PÞ=ð1 þ PÞRRm; where RQ ¼ h=e2; P is the spin

polarization of the electrodes (0:35 for Co), and a ¼ 5:

Fig. 1. Differential conductance vs. (a) bias voltage V (at

Vg ¼ 0) and (b) gate voltage Vg (at eV=2EC ¼ 1:45) in both

magnetic configurations, calculated for kBT=EC ¼ 0:03; a ¼ 5;and for spin polarization of the electrodes as in Co:

Fig. 2. Energy diagrams for a symmetric magnetic double

tunnel junction with normal metallic island in the Coulomb

blockade regime and in the AP configuration.

J. Martinek et al. / Journal of Magnetism and Magnetic Materials 240 (2002) 143–145144

For the antiparallel alignment one then finds RRm ¼RLk ¼ aRQ and RRk ¼ RLm ¼ ð1 � PÞ=ð1 þ PÞRRm:

In Fig. 1 we show the differential conductance for Coelectrodes in both magnetic configurations, calculatedfor kBT=EC ¼ 0:03 and for no spin-flip processes, whereEC ¼ e2=2C is the charging energy. We find well

resolved splitting of the conductance peak in the AP

configuration, which is a result of the spin splitting ofthe corresponding electrochemical potential of the island

(see the diagram in Fig. 2). Fig. 1(b) also shows that thesplitting of the conductance peaks can be observed whenthe gate voltage Vg is varied and the bias voltage V is

fixed. The possibility to observe the resonances is relatedto the fact that the electrochemical potentials areeffectively shifted due to the Coulomb energy, as shownschematically by the diagram in Fig. 2.

In Fig. 3 we show the differential conductance versusgate Vg and transport V voltages in a gray-scalerepresentation for both P and AP magnetic configura-

tions. The splitting of the resonance in the APconfiguration (shown in Fig. 1(a) for Vg ¼ 0 andFig. 1(b) for eV=2EC ¼ 1:45) is clearly visible in

Fig. 3(b).

JM and JB acknowledge support by the Polish StateCommittee for Scientific Research through the ProjectNo. 5 P03B 091 20 and EU-COST program action P5.

One of us (JM) would like to acknowledge thehospitality of the University and ForschungszentrumKarlsruhe, where part of this work was performed. One

of us (SM) acknowledges support by the HumboldtFoundation.

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Fig. 3. Differential conductance (the units as in Fig. 1) versus

gate Vg and transport V voltages in a gray-scale representation

in the (a) parallel and (b) antiparallel configurations, calculated

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J. Martinek et al. / Journal of Magnetism and Magnetic Materials 240 (2002) 143–145 145