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Current and Noise in a FM/quantum dot/FM System
F. M. Souza, 1 - 2 J. C. Egues 2 ' 3 and A. P. Jauho 1
1 Mikroelektronik Centret, Danmarks Tekniske Universitet,
DK-2800 Kgs. Lyngby, Denmark
2 Departamento de Fisica e Informdtica,
Institute) de Fisica de Sao Carlos,
Universidade de Sao Paulo,
13560-970 Sao Carlos, Sao Paulo, Brazil.
3 Department of Physics and Astronomy, University of Basel,
Klingelbergstrasse 82, CH-4056 Basel, Switzerland
(Dated: February 1, 2008)
Using the Keldysh nonequilibrium technique we calculate current, noise and Fano factor in a
ferromagnetic(FM)-quantum dot-ferromagnetic(FM) system with Coulomb interaction and spin-flip scattering
in the dot. The lead polarizations are considered in both parallel P and antiparallel AP alignments. We show
that spin-flip can increase both AP-current and AP-noise, while the P-current and P-noise are almost insensible
to it. This fact leads to a suppression of the tunnelling magnetoresistance with increasing spin-flip rate.
I. INTRODUCTION
The emerging field of spintronics nj[]-[p]|, where the electron spin and charge are used to design new devices, has led to
fascinating and novel ideasjsuch as spin filters [^-[qI, spin field effect transistors [Qj, and has offered many proposals for solid
state quantum computing |g] . For example, quantum dot systems are useful in the control of the electron spin and are suitable to
create quantum bits relevant for quantum gate operations
The study of nonequilibrium transport properties of spintronic devices is of great importance to understand basic physical
phenomena and to predict new functionalities. Calculation of the current, for example, can give the conductance/resistance of a
system and its dependence on magnetic field, Coulomb interaction, spin-flip and so on. On the other hand, current fluctuations,
due to the granularity of the charge (shot noise I10|]), are also relevant because their measurements can provide additional
information not contained in the average current jlI|F
Here we apply the Keldysh nonequilibrium tecnnique [n3| to calculate current and its fluctuations (noise) in a quantum dot
coupled to two ferromagnetic leads as a function of the applied voltage for parallel and antiparallel lead-polarization alignments.
We include Coulomb interaction in the Hartree-Fock approximation as well as spin-flip in the dot. We show that spin-flip makes
the alignment of the lead polarizations less important; both P and AP results coincide for large enough spin-flip rates. This fact
gives rise to a reduction of both Fano factor and tunnelling magnetoresistance (TMR) as we show here.
Our paper is organized as follows. In Sec. 2 we describe the system and present its Hamiltonian. In Sec. 3 we apply the
Keldysh technique to determine current and noise in our system. In Sec. 4 we discuss our results for current and noise and Sec.
5 gives our conclusions.
II. SYSTEM
Our system is composed of two ferromagnetic leads coupled to a quantum dot via tunnelling barriers (Fig.[j]). While the
left lead has a fixed polarization (hard lead), the right one can have its polarization switched from parallel P to antiparallel AP
alignment (soft lead). This polarization rotation (TP— *AP) changes the transport properties of the system |]13[] . This effect is
included in our approach.
We model the system with the Hamiltonian H = Hl + Hjj + Hp + Ht, where Hl(r) is tne left (right) lead Hamiltonian,
Ho describes the dot and Hj~ gives the coupling between leads and dot. In our model, Coulomb interaction and spin-flip are
restricted to the dot, while the electrons in the leads are free. The leads are assumed to be in thermal equilibrium with chemical
potential /i£ and ptR for the left and the right leads, respectively. When a voltage V is applied across the system, the chemical
potentials differ by //£ — fin — eV, where e is the electron charge. This difference drives the system out of equilibrium, thus
giving rise to current and noise. More explicitly, the Hamiltonian of the lead 77 (77 = L. R) is
ka
where C)- Va (cjLo-) destroys (creates) an electron into the lead 77 with wave vector k and spin a. The electron energy e^a depends
on 77 and the spin component a because of the applied voltage and the band spin-split, respectively.
The Hamiltonian of the dot is
H D =Y^ e odtd a + Un ]ni + R(d\d l + d{d T ), (2)
cr
where d a (dj.) destroys (creates) an electron in the dot with spin a and the energy eo is spin independent tyM, Jl5|. In addition,
we assume we have a small enough dot in order to have only one active level eo. In the presence of a voltageme level shifts
by eo = e<j — where e^ is the dot level for zero bias (for numerical convenience we use e^ = y). This assumption does
2
Left Lead
Right Lead
polarization
DOT
parallel case
►
antiparallel case
1 1 1
V
FIG. 1 : Schematic of the system. It is composed of two ferromagnetic leads and a quantum dot as a spacer. The electrons are allowed to tunnel
through the left and right barriers (with tunnelling rate Tl and Tr, respectively) in order to generate a tunnelling current when a voltage V is
applied. The left lead has a fixed polarization (hard side) while the right lead switches its polarization from parallel (up arrow) to antiparallel
(down arrow) alignment. This polarization rotation changes the majority/minority spin population, leading to a variation in the resistance of
the system, which is reflected in both current and noise.
not take into account charge accumulation in the dot, which tends to wash out this linear_drop. A nwe sophisticated approach,
which includes charging effects in a self-consistent way, will be discussed elsewhere Q161. In Eq.(B) the Coulomb interaction
is taken into account via the Hubbard term with a correlation parameter U > and spin-nip scattering is described by the last
term, where B is the spin-flip scattering amplitude.
The tunnelling Hamiltonian is
Ht = ^2{t* km d\c kria + t kria cl va d a ), (3)
kr\a
where t krja couples an electronic state in lead -q to one in the dot. We consider a spin conserving tunnelling; the spin-flip process
is assumed to be confined in the dot. In the nonequilibrium Green function technique Ht is the nonequilibrium part of the
Hamiltonian because it coupleacontacts with different chemical potential (if eV ^ 0), thus allowing for charge flow. Next we
apply the Keldysh technique p7| ] to determine the average current and the noise.
III. CURRENT AND NOISE
Current. The average current from the left contact into the dot is defined as II = —e(Ni), where Nl — J2ka c \ha CkLa * s
the number operator for lead L. To find the time evolution of the occupation-number operator, we use the Heisenberg equation
Nl = i[H, Nl]. The only term of the Hamiltonian which does not commute with Nl is Ht- Using Eq.(^) we obtain
lL = ^£W<4ia(*K(*)>- W
ka
To avoid further complications in the analysis due to the spin-flip term, we perform a canonical transformation []15|
in terms of which Eq.(0) becomes
I, - — — Re VTr-ff * fciT ^\ ( G tm G tkL[ \\
(6)
where Gf kr](T (t, t) = i(c k (i)di(i)) . Applying the Keldysh technique as described in [ |l7| we find
I n = ^Re J dt 2 Ti{G r (t,t 2 )V<(t 2 ,t) + G<(t,t 2 )^ a (t 2 ,t)}, (7)
where G r and G< are the nonequilibrium dot Green functions, with elements Gfj{t,t 2 ) — i(d^(t 2 )di(t)) and Glj(t,t 2 ) =
—i0(t — t 2 )({di(t), d^(t 2 )}). Here the averages are taken over the initial (t = — oo) equilibrium density matrix [p^|. The lesser
3
l'2' l i- oZjI'til <M,, ,n „<(r,a) u ,n „<(r,a)/. ,n , „<(r,a)/. '
(retarded, advanced) self-energy is given by
2 ? |tLl feT ,OJ (*9.*)-»Sr B; (*a.*) g^f'M+gklT'M,
where g^£j a ^ is the lesser (retarded, advanced) uncoupled Green function for lead L. These are defined as <7jS<r(*2,i) =
••'4 iff (*fe,(t2)),3L ff fe><) = -M(t2-t)({c kL <r(t 2 ),tf kLa (t)}) and g% La (t 2 , t) =iO(t-t 2 )({cki~(t 2 ),cL < ,{t)})' where
(8)
the tilde denotes that the operator is in the interaction picture; its time evolution is governed entirely by Eq.(|l|). In Eq.l
assume a spin-independent amplitude t k L for simplicity.
For a time-independent Hamiltonian the Fourier transform of Eq.(Q) yields
lL = 1 / ^Tr{S L< H[G» - G»] - G<(c)[S^(c) - (9)
where T, t,< (lu) and the difference S ??r '(tj) — S ?,a (w) are calculated using the expressions g r ^ l<T {^) — cJ _ £fc 1 ±i and g^ ria {^) =
2mn r] {ui)8(uj — e^o-), n n (u>) is the Fermi distribution function of the lead 77. We find S ,;< = in^T 71 and S ,,r — S ??a = — ir 7? ,
with
i fr v 4- r ?? r 17 - \
T J. T 4. /
where V'l = 2nJ2 k \t kri \ 2 8{uj - e k , ia ).
Accounting for Coulomb interaction in the Hartree-Fock approximation, we can write down a matrix Dyson equation for
the retarded Green function, G r = G 0r + G 0r £ r G r , and a Keldysh equation for the lesser Green function G< = G r S<G a ,
where G 0r is the uncoupled dot Green function. In these equations the self energies are the sum of the left and right self energies,
i.e., £( r ><) = £- L ( r ><) + £ R ( r ><). A self consistent calculation is required to calculate (nj) and (dldi), which are given by the
lesser Green function, (djdi) — J |^ImG<(w).
Noise. The current operator can be written as its average value plus some fluctuation, i.e., I v = I v + SI V . In our system there
are two sources of noise, namely, thermal noise and shot noise. The first one is due to thermal fluctuations in the occupations
of the leads. It vanishes for zero temperature and eV 7^ 0, but can be finite for T ^ and eV = 0. On the other hand,
shot noise is due to the granularity of the electron charge and is a nonequilibrium property of the system in the sense that it is
nonzero only when there is a finite current (eV 7^ 0). To calculate the noise (thermal+shot noise) we use the standard definition
S V n> (t - t') = ({Sf n (t), ST V > (t')})> which can also be written as S nv ' (t - t') = ({I v (t), %> (t 1 )}) - 2ljL After a straightforward
calculation, which will be presented elsewhere [|16|, we find for the noise power spectrum (dc limit) [ 19 1
SW(0) = jl £Tv{S rm ,in v r'G> - 8 m ,i{l - n r) )r"G< + G<r"G>r"'
- rc„(l - ^JGT'GT"' - rv(l - n 7) )G < T"G<T"'
- G < r ?? [(l - n,, r )G r - (1 - n,)G°]r»' + (n n G r - n ir G a )r 71 G > r'' '}.
(11)
The dc noise (zero frequency) is position independent, and it is possible to show that Sll(0) — Srr(0) = —Slr(0)
—Srl(0) JlCQ. In the next section we use the component Sll-
IV. RESULTS.
Using Eqs. (£]) and ( jlll ) we calculate current and noise for the system in Fig.l. We assume TJJ to be independent of energy,
but polarization dependent with values T% = Tq[1 + (—l) Sa ip\, = Ti and Tj 2 = if the leads have parallel alignment or
= and rj 1 = if they are antiparallel aligned. The parameter p gives the spin-splitting of the ferromagnetic band. For
example, for p = the system is unpolarized while for p = 1 the system is fully polarized. The parameter r fixes the coupling
strength between leads and dot. The sign +/- in corresponds to majority/minority spins, respectively. Here we take a =|
(a =1) as majority (minority) spins in the lead L and assume To = 0.01C7 and p = 0.5 as in Ref.[H]. The majority/minority
spin population in the right lead switches from one to the other according to the lead polarization, which can be controlled via an
external magnetic field. This simple form for TJJ is reasonable when the band is wide compared to others energies of the system.
The temperature is assumed to be ksT = Tq(1 + p). Our approximation (Hartree-Fock) does not include correlations of the
Kondo type, however we do not expect these to change our results in the present range of parameters.
A relevant quantity in transport is the spectral function, for the present spin-dependent case defined as A(ui) = iTr[G r (w) —
G a (w)], whose poles give the resonant levels which work as conduction channels. Figure 2 shows A(ui) for different applied
voltages and for R = (upper panel) or R = 0.1 (lower panel). For R = we have only one peak when eV — (labelled 1)
4
and two peaks when eV — 1.5(7 or eV — 3(7 (labelled 2,2' or 3,3', respectively). When eV = the dot is empty because the
level eo = erf = 0.5(7 is above the Fermi energies /i^ and [ir (set equal to zero), so Coulomb interaction plays no role. When
eV = 1.5(7 or 3(7, €q is below the Fermi energy of the left lead (peaks 2 or 3), consequently the electrons could go inside the
dot, creating the high energy peak at eo + U (peaks 2' or 3'), due to Coulomb interaction. The levels in the dot shift linearly with
the bias, following the assumption eo = ed — As mentioned above, this linear drop does not account for charging effects.
However, it gives reasonable qualitative results here. For R = 0.1(7 we have similar Jbehaviors but each peak in the R = case
is now split due to spin-flip. The peaks are located at e\, e 2 , e\ + U and e 2 + (7 [Fig.g(b)].
<
600
500 |- a ;
0.6
400 ho.4
0.2
300 h o
200
100
1,
i
2
eV=0
eV=1.5
eV=3
i3' 1
(a)
R=0
-2 -1
1
300
250
200
l
0.8
0.6
"0.4
0.2 I—
<
150 - o
100
50
i^L (b)
R=0.1
3 3 2 2/
-2 2
3' \TT
■1
co/U
FIG. 2: Spectral function A(uj) against energy uj for R = and R = 0.1(7. The peaks correspond to the dot levels. For R = there is one
peak for eV = (peak 1) and two peaks for eV = 1.5(7 and eV = 3(7 (peaks 2,2' and 3,3'). The extra peak (2' or 3') is due to Coulomb
interaction, since the lowest level (2 or 3) is already below the Fermi energy (here at zero), thus allowing electrons to go in the dot. For
R = 0.1(7 the peaks are split and given by eo ± R and eo ± R + U. The insets show the integral I(u)) of the spectral function. Each step gives
the area under a peak. Since the total area is normalized the last step is at one.
The inset in Fig.||(a) shows the integrals of the spectral function, namely I(u>) — -h- A{uj')duj' for the three voltages
used. Observe that I(u>) — > 1 as ui increases. This is due to the normalization of the spectral function. For eV = the whole
area is essentially under the peak at 0.5C7 (peak 1), which explains the single step in I(u>). For eV = 1.5(7 and eV = 3(7 the
total normalized area should be distributed under the two peaks (2 and 2' or 3 and 3'), in order to keep the normalization of
A(u>). It leads to a reduction of the area of the lowest peak (2 or 3) in comparison to its eV — value. This area is given by the
first step in I(u>).
Figure |] shows current (a) and noise (b) as a function of the bias with R = (solid line) and R = 0.1(7 (dotted line) for both
P and AP configurations. Because P and AP curves for R = 0.1(7 coincide, we plotted only the AP case. The first enhancement
of the current and noise at eV = U happens when eo crosses the left chemical potential, allowing electrons to tunnel from
the emitter (left lead) to the dot and then to the collector (right lead). The current and noise remain constant until the second
level cq + U reaches /if, at eV = 3(7, when another enhancement is observed. In terms of differential conductance (<7diff) eacn
enhancement corresponds to a peak in erdiff- These peaks reflect the spectral function plotted in Fig. 2.
5
When the system changes from parallel (P) to antiparallel (AP) configurations the current is reduced. This is a typical behavior
of tunnelling magnetoresistance (TMR): the resistance increases when the system switches from P to AP configuration. The noise
is also affected by this resistance variation, showing a similar reduction. Contrasting behaviors between current and noise will
be explored elsewhere [|l6| for another set of parameters.
Looking at the effects of spin-flip on current and noise we see that the AP curves with R = Q.1U (dotted lines) tend to be
on the P curves with R = 0, thus snowing that lead alignments are less important when spin-flip plays a part. This AP current
enhancement due to spin-flip gives rise to a reduction of the TMR; since TMR = (Ip — Iap)/Iap, when Iap — > Ip we have
TMR -> 0. W. Rudzihski et a/.Q found a similar behavior for TMR.
In the inset of Fig.[| we plot theTano factor S^j^el^. For the parallel case the Fano factor remains around 0.5 for voltages
between U and 5U, except at eV — iU where it has a small peak. This average value around 0.5 is a consequence of the
symmetry of the double-barrier structure in the P case. A similar behavior is observed for the AP case with its average value
above the P case. When spin-flip is included (R = 0.1U) the AP Fano factor is shifted down, becoming close to the P result for
R = 0, with the addition of a peak close to eV = U and a double peak around eV — 3U. This peculiar double structure is a
consequence of the splitting of the dot levels when R ^ as observed in the spectral function.
6
o.oi
0.008
=5 0.006
<u
(N
^ 0.004
0.002
1
- (a)
1 1 1
*r \ i
>
r
( :
—
AP J
R:
=0
R=0.1 (AP)
,1,1,1,
FIG. 3: Current and noise as a function of the bias for parallel (P) and antiparallel (AP) alignments and with R = and 0.1(7. The curves for
R = 0.1U are only for the AP alignment; observe that these are almost on top of the P curves, except within the sloping region around U and
3(7. Both current and noise are reduced when the right lead changes its polarization from P to AP, following the typical behavior of TMR. The
inset shows a suppression of the AP-Fano factor due to spin-flip.
V. CONCLUSION
Using the Keldysh nonequilibrium technique we calculated current and noise in a ferromagnetic-quantum dot-ferromagnetic
system with Coulomb interaction and spin-flip relaxation. We have shown that the lead alignments affect both current and noise.
These are reduced when the leads rotate from the P to the AP configuration, following the typical magnetoresistance behavior.
The spin-flip relaxation is crucial to drive the current and noise in the AP case close to their values in the P case. In a way,
we can say that spin-flip makes the P and the AP configurations "degenerate" thus reducing the effect of the lead-polarization
alignment on transport. We also showed that TMR is reduced due to spin-flip, corroborating previous results in the literature.
FMS acknowledges support from the funding agencies CAPES and FAPESP (Brazil). JCE acknowledges financial support
from the Swiss NSF, DARPA and ARO.
7
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