Electric-field dependent spin diffusion and spin injection into semiconductors

5 Pages·0.18 MB·English

by  Z. G. Yu

Electric-field dependent spin diffusion and spin injection into semiconductors
Z. G. Yu and M. E. Flatt´e
Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242
(Dated: February 1, 2008)
2
0 Wederiveadrift-diffusionequationforspinpolarizationinsemiconductorsbyconsistentlytaking
0 into account electric-field effects and nondegenerate electron statistics. We identify a high-field
2 diffusiveregimewhichhasnoanalogueinmetals. Inthisregimetherearetwodistinctspindiffusion
n lengths. Furthermore, spin injection from a ferromagnetic metal into a semiconductor is enhanced
a by several orders of magnitude and spins can be transported over distances much greater than the
J low-field spin diffusion length.

3
PACSnumbers: 72.25.Dc,72.20.Ht,72.25.Hg,72.25.Mk.
2

]
i Semiconductor devices based on the control and ma- Here we clarify the central role of the electric field on
c
s nipulation of electron spin (semiconductor spintronics) spin transport in semiconductors. We obtain the drift-
- have recently attracted considerable attention [1]. Spin diffusionequation(2)forthe spinpolarizationina semi-
l
r transportandinjectionpropertiesofsemiconductorsand conductor. Equation (2) consistently takes into account
t
m heterostructures strongly constrain the design of new electric-field effects and nondegenerate electron statis-
spintronic devices. In theoretical studies of spin trans- tics. We identify a high-field diffusive regime which has
.
t port and injection in semiconductors [2, 3, 4] the spin no analogue in metals. This regime occurs for field as
a
m polarization is usually assumed to obey the same diffu- small as 1 V/cm at low temperatures. Two distinct spin
sion equation as in metals [5], diffusion lengths now characterize spin motion, i.e., up-
-
d 2(µ µ ) (µ µ )/L2 =0, (1) stream(Lu)anddown-stream(Ld)spindiffusionlengths,
n ∇ ↑− ↓ − ↑− ↓ which can differ in orders of magnitude with realistic
o where µ is the electrochemical potential of up-spin fields: E 2.5 V/cm at T = 3 K and E 250 V/cm
c ↑(↓) ≥ ≥
(down-spin) electrons. In this diffusion equation, the at T = 300 K. These two length scales play distinctive
[
electric field does not play any role, and spin polariza- but both favorable roles in spin injection from a ferro-

1 tion decays away on a length scale of L from an injec- magnetic metal to a semiconductor. We find that the
v tion point. This is reasonable for metals because the effective semiconductor resistance determining the injec-
25 electric field E is essentially screened. For semiconduc- tion efficiency is Lu/σs rather than L(s)/σs, which may
4 tor spintronic devices, however,the semiconductor often be comparable to L(f)/σf given that Lu can be shorter
1 is lightly doped and nondegenerate, and moderate elec- than L(s) by several orders of magnitude in the high-
0 tric field can dominate the carrier motion. Equation (1) field regime. Moreover, the decay length scale for the
2 correspondsto neglecting drift in the more generaldrift- spin polarization injected into the semiconductor is L ,
d
0 diffusion equation for the spin polarization, which would be much longer than L(s) in the presence
/
t of a strong field. Our results suggest a simple and prac-
ma ∇2(n↑−n↓)+ keET ·∇(n↑−n↓)− (n(L↑(−s)n)2↓) =0, tical approach to increase spin injection and spin coher-
B ence in semiconductors, namely, increasing the electric
- (2) field, or equivalently, increasing the total injection cur-
d
n wheren n isthedifferencebetweenup-spinanddown- rent in semiconductors. Our results are consistent with
↑ ↓
o spin elect−ron densities and L(s) is the intrinsic spin dif- the significant current dependence observed for spin in-
c jectionfromFe toGaAs [9]. We further notethatstrong
fusion length.
:
v If Eq. (1) holds, spin injection from a ferromagnetic fields also substantially enhance spin injection in struc-
Xi metal to a semiconductor without a spin-selective inter- tures with an interfacial barrier.
facial barrier is virtually impossible due to the “conduc- The semiconductor we considerhere is lightly ormod-
r
a tivitymismatch”,ormoreprecisely,amismatchbetween erately n-doped (p-doped semiconductors can be ana-
effective resistances in the metal (L(f)/σf) and in the lyzed similarly), which is typical in spintronic devices.
semiconductor (L(s)/σ ) [2, 3, 4]. Here L(f) and L(s) We assume that there is no space charge and the mate-
s
arethespindiffusionlengthsfortheferromagneticmetal rial is homogeneous. The current for up-spin and down-
andthe semiconductor,andσ andσ areconductivities spin can be written as j =σ E+eD n , which
f s ↑(↓) ↑(↓) ↑(↓)
∇
for the two materials. Even for spin injection from fer- consists of the drift current and the diffusion one. Here
romagnetic semiconductors, L(f)/σ L(s)/σ , and the D is the electron diffusion constant, σ the up-spin
f s ↑(↓)
≫
spinpolarizationismuchlessthan99%,sothelargespin (down-spin) conductivity, and n the up-spin (down-
↑(↓)
injection percentages achieved from ZnMnSe [6, 7] and spin) electron density. The spin-dependent conductiv-
GaMnAs [8] are difficult to understand via Eq. (1). ity is proportional to the electron density for individual
2
(a) 108
E=0
on m)106 Ld
zati (b) h (n
ari E=2.5 V/cm gt
n pol n len104
Spi (c) sio
u
E=25 V/cm Diff102 Lu
-20000 -10000 0 10000 20000 0
10
x (nm) 0 0.02 0.04 0.06 0.08 0.1
-1
|eE|/k T (nm )
B
FIG. 1: Distribution of electron spin polarization as a func-
tion ofposition for aspin imbalanceinjected at x=0. Panel FIG.2: Up-stream(dashedline)anddown-stream(solidline)
(a),(b),and(c)arefor|eE|/kBT =0,0.001, and0.01nm−1, diffusion lengths as a function of electric field. The intrinsic
which correspond to E = 0, 2.5 V/cm, and 25 V/cm, re- spin diffusion length is L(s) =104 nm.
spectively, at T = 3 K. The intrinsic spin diffusion length is
L(s) =104 nm. the rootsofthe aboveequationmustbe positive andthe
other negative. The choice of roots is constrainedby the
spins, σ↑(↓) =n↑(↓)eνe, where the mobility νe is assumed boundaryconditionsat . Tounderstandthephysical
tobeindependentoffieldanddensity. Therateatwhich consequence of the elec±tr∞ic field on the spin transport,
spin-up(spin-down)electronsscattertospin-down(spin- we suppose that a continuous spin imbalance is injected
up)electronsisdenotedby1/τ↑↓(1/τ↓↑). Insteadystate, at x = 0, and the electric field is along the x direc-
the equations of continuity for individual spins read tion. The spin polarization will gradually dec−ay in size
as the distance from the point of injection increases. In
n n
j↑ = σ↑ E+σ↑ E+eD 2n↑ = ↑ ↓ e, Fig. 1, we plot the spin polarization as a function of
∇· ∇ · ∇· ∇ τ − τ
↑↓ ↓↑ position for different fields. In the absence of the field,
(cid:16)n n (cid:17)
j = σ E+σ E+eD 2n = ↓ ↑ e. as shown in Fig. 1(a), the spin polarization decays sym-
↓ ↓ ↓ ↓
∇· ∇ · ∇· ∇ τ − τ
↓↑ ↑↓ metrically along x and +x with a single length scale,
(cid:16) (cid:17) L(s). When an el−ectric field is applied, the decay of the
In nondegenerate semiconductors, τ−1 =τ−1 τ−1/2.
↑↓ ↓↑ ≡ spinpolarizationbecomesspatiallyasymmetric. Forspin
For a homogeneous semiconductor without space-
diffusionopposite to the fielddirection(down-streamfor
charge, local variation of electron density ∆n should be
electrons), the decay length of the spin polarization is
balanced by a local change of hole concentration ∆p. In
longer than L(s). For spin diffusion along the field direc-
doped semiconductors, spin polarization can be created
tion(up-streamforelectrons),thedecaylengthisshorter
without changing electrons or hole densities ∆n=∆p=
than L(s). As we change the strength of the field, the
0 [10], and therefore,
spatial distribution of the spin polarization can change
dramatically.
∆n +∆n =0. (3)
↑ ↓
We define two quantities L , L ,
d u
Here ∆n = n n /2, and n is the total elec-
↑(↓) ↑(↓) − 0 0 eE eE 2 1 −1
tron density in equilibrium. From Poisson’s equation, L = | | + + , (4)
∇·E = −(∆n↑+∆n↓)e/ǫ = 0. By using the Einstein’s d (cid:20)−2kBT s(cid:16)2kBT(cid:17) (L(s))2 (cid:21)
relation, D =k Tν /e, where k is the Boltzmann con-
B e B
stant and T is temperature, we obtain the differential
eE eE 2 1 −1
eizqautaiotnionin(s2e)mfoicronn↑du−ctno↓r,s,thweitmheLa(ssu)r=e o√fDthτe.spin polar- Lu =(cid:20)2|kBT| +s(cid:16)2kBT(cid:17) + (L(s))2 (cid:21) . (5)
Equation(2),togetherwiththelocalchargeneutrality
The distribution of the spin polarization in Fig. 1 is
constraintEq. (3),dramaticallyaltersthespintransport
then described by n n exp( x/L ) for x>0, and
behaviorinsemiconductorsfromthatexpectedfromEq. ↑− ↓ ∼ − d
n n exp(x/L ) for x < 0. Thus L (L > L(s))
(1). The general form of solution to Eq. (2) (restricting ↑ − ↓ ∼ u d d
and L (L <L(s)) are the down-stream and up-stream
variation to the x-direction) is u u
spin diffusion lengths, respectively.
n↑ n↓ =Aexp( x/L1)+Bexp( x/L2), Figure 2 shows Ld and Lu as a function of the elec-
− − − tric field. In the absence of the field, the down-stream
where λ = 1/L and λ = 1/L are the roots of the and up-stream lengths are equal to the intrinsic diffu-
1 1 2 2
quadraticequation,λ2 λeE/k T 1/(L(s))2 =0.Oneof sion length L(s). With increasing field the down-stream
B
− −
3
diffusion length Ld increases,whereas the up-streamdif- 100
fusion length L decreases. A high-field regime for spin
u
wtrhaenrsepoerEt i/nkseTmi=con1d/uLc(tso).rsIcnanthbies dreegfiinmeed,bLy Ean>dELc, 10-1 σf /σs=10
c B u d
deviate fromL(s) considerablyandthe spindiffusionbe- α0
n σ /σ=102
havior is qualitatively different from that in low fields. o10-2 f s
We emphasize that since L(s) is large in semiconductors, cti
e
thisregimeisnotbeyondrealisticfieldswheremostspin- nj σ /σ=103
tronicdevicesoperate. Foratypicalspindiffusionlength, n I10-3 f s E
pi
L(s) = 104 nm [11], E = 25 V/cm at T = 300 K and S
c FM S
E =0.25 V/cm at T =3 K. 10-4
c
0 x
The physics of the field effects on the spin diffu-
-5
sion becomes clearer at the strong-field limit, where 10 0 0.02 0.04 0.06 0.08 0.1
eE /K T 1/L(s). In this limit, the electrons move |eE|/k T (nm-1)
| | B ≫ B
with velocity E ν and so does the spin polarization.
e
| |
Ld is simply the distance over which the carriers move FIG. 3: Spin injection efficiency α0 as a function of elec-
within the spin life time τ, L E ν τ = E e Dτ = tric field. Dot-dashed, solid, and dashed lines correspond to
d ≃ | | e | |kBT
(L(s))2 eE /k T. For the up-stream diffusion length L σf/σs = 10, 100, and 1000, respectively. Other parameters
at this|lim|it,BL k T/eE , which simply correspondus arepf =0.8,L(f) =100nm,L(s) =104 nm. Theinsetshows
u B
≃ | | theschematic injection structure.
to a Boltzmann distribution of electrons in a retarding
field.
The general solution can be written as
A similar field-dependent diffusion phenomenon has
bnifoenerintyobcnaserrriviseerdssubainnstdidtouspttueedddiebsdyem∆inicpocnahdnaudrgcteLo(rtssr)ai[ns1s2pr]e.ogratIrndoeffdamcatis-, e1J (cid:18)µµ↑↓ (cid:19)= σ↑f +x σ↓f (cid:18)11(cid:19)+C1eL(xf) −11//σσ↑f↓f !,
↑ ↓ (6)
−
theintrinsicchargediffusionlength,Eq. (2)becomesthe
diffusionequationfor the disturbance ofminoritycarrier whereJ is the totalelectroncurrent,whichis aconstant
in n-doped semiconductors. It is knownthat the electric throughout the structure in steady state. In the semi-
field leads to two distinct chargediffusion lengths in this conductor, according to Eqs. (2) and (3),
case as well as a modification of carrier injection [12].
∆n = ∆n =C exp( x/L ), (7)
↑ ↓ 2 d
Asanapplicationofourfield-dependentspintransport − −
theory, we study how the electric field affects spin injec- and J =σsE. In order to matchboundary conditions at
tionfrom aferromagneticmetal to a semiconductor. We the interface between the metal and the semiconductor,
consider a simple one-dimensional spin injection struc- it is desirable to know the electrochemical potentials for
ture to elucidate the underlying physics of electric field up-spin and down-spin electrons in the semiconductor,
and nondegenerate electron statistics effects. This injec- which are related to the electron density for individual
tion structure,as shownin the inset of Fig. 3, comprises spins via
a semi-infinite metal (x < 0) and a semi-infinite semi-
2∆n
conductor(x>0). Electronsareinjectedfromthemetal µ =k T ln 1+ ↑(↓) +eEx C . (8)
↑(↓) B n − 0
to the semiconductor, and therefore, the electric field is 0
(cid:16) (cid:17)
antiparalleltothex-axis. Intheferromagneticmetalthe This relation can be readily derived based on the def-
electrochemicalpotentialsforindividualspinssatisfythe inition of the electrochemical potential in nondegener-
equations [13], atesemiconductorsn exp[(µ +eψ)/k T],where
↑(↓) ↑(↓) B
∝
E dψ/dx.
≡−
The three unknown coefficients C (i = 0,1,2) in
i
d2 µ↑ = (D↑fτ↑↓)−1 −(D↑fτ↑↓)−1 µ↑ , Eqs. (6)-(8) will be determined by the boundary con-
dx2 (cid:18)µ↓ (cid:19) −(D↓fτ↓↑)−1 (D↓fτ↓↑)−1 !(cid:18)µ↓ (cid:19) denittioinnsteraftacteh,ei.ein.,tenrofacsep.in-flFiopr sacatctleearinnganadt tthreanisnptaerr--
face and no interface resistance, both the electrochem-
ical potential and the current for individual spins are
whereDf isthe up-spin(down-spin)electrondiffusion
↑(↓) continuous, giving rise to three independent equations:
constant. In metals the conductivity and the diffusion (1) µ (0−) = µ (0+), (2) µ (0−) = µ (0+), and (3)
↑ ↑ ↓ ↓
constant are related via σf /Df =e2N (E ), and j (0−) j (0−) = j (0+) j (0+). The current can be
↑(↓) ↑(↓) ↑(↓) F ↑ − ↓ ↑ − ↓
N↑(↓)(EF)istheup-spin(down-spin)densityofstatesat calculated using j↑(↓) =σ↑(↓)d(µ↑d(x↓)/e).
Fermi energy. It is readilyseen that the aboveequations The spin injection in the semiconductor is usually de-
lead to Eq. (1) if L(f) = [(D↑fτ↑↓)−1 +(D↓fτ↓↑)−1]−1/2. fined via the spin polarization of the current, α(x) =
4
[j (x) j (x)]/J, which is found to be proportional to resistance mismatch makes it virtually impossible to re-
↑ ↓
−
the spin polarization of the electron density n n , alize an appreciable spin injection from a ferromagnetic
↑ ↓
−
metal to a semiconductor. However, the more general
n↑(x) n↓(x) kBT description of the spin transportin semiconductors indi-
α(x)= − 1 . (9)
n − eEL cates that the effective semiconductor resistance to be
0 d
(cid:16) (cid:17) compared with L(f)/σ should be L /σ rather than
f u s
Thus the solution of n↑ n↓ in Eq. (7) indicates α(x)= L(s)/σ . Since L can be smaller than L(s) by orders
− s u
α0e−x/Ld, where α0 is the spin injection efficiency. We of magnitude in the high-field regime, this “conductiv-
obtain an equation for α0, noting 1 kBT/eELd = ity mismatch” obstacle may be overcome with the help
−
kBT/eELu, of strong electric fields, or equivalently, large injection
−
currents [14]. For example, if the parameters of a spin
2L(f)(α0 pf) kBT kBT/eELu+α0 injection device are as follows, p =0.8, L(f) =100 nm,
− = ln− , (10) f
(1−p2f)σf eEσs −kBT/eELu−α0 L(s) = 104 nm, and σf = 100σs, at zero field the spin
injection efficiency is 0.04%, which can be increased to
where σ =σf +σf, and p =(σf σf)/σ is the spin 4.2% at eE /k T = 0.02 nm−1, which corresponds to
f ↑ ↓ f ↑ − ↓ f | | B
polarization in the metal. We solve Eq. (10) and plot E =50V/cm,or J =50A/cm2 foratypicalsemicon-
the spin injection efficiency α0 as a function of the elec- |du|ctor conductivity| σ|s = 1 (Ω cm)−1, at T = 3 K. This
tric field in Fig. 3. We see that the electric field can may explain the large spin injection percentages from
enhance the spin injection efficiency considerably. When ZnMnSe to ZnSe [6, 7] and from Fe to GaAs [9, 15], as
∆n /n 1, i.e., small spin polarization in the semi- well as the dramatic increase in spin injection with cur-
↑(↓) 0
≪
conductor, α(x) can be expressed in a compact form, rent in Ref. [9]. Finally, we note that spin injection
enhancement from a spin-selective interfacial barrier be-
L(f) Lu −1 pfL(f) − x tween the ferromagnetic metal and the semiconductor,
α(x)= (1 p2)σ + σ (1 p2)σ e Ld. (11) which has been identified in the low-field regime [3, 4],
h − f f si − f f becomes more pronounced in the high-field regime.
This remarkable expression shows that the electric-
field effects on spin injection can be described in terms In summary, we have derived the drift-diffusion equa-
of the two field-induced diffusion lengths. Both diffusion tion for spin polarization in a semiconductor by consis-
lengths affect spin injection favorably but in a different tently taking into account electric-field effects and non-
manner. The up-stream length Lu controls the relevant degenerate electron statistics. This equation provides
resistance in the semiconductor, which determines the a framework to understand spin transport in semicon-
spin injection efficiency. With increasing field this effec- ductors. We have identified a high-field diffusive regime
tive resistance,Lu/σs, becomes smaller,andaccordingly which has no analogue in metals. In this regime, there
the spin injection efficiency is enhanced. The transport aretwodistinctspindiffusionlengths,i.e.,theup-stream
distance ofthe injected spinpolarizationin the semicon- and down-stream spin diffusion lengths. The high-field
ductor,however,iscontrolledbythedown-streamlength description of the spin transport in semiconductors pre-
Ld. As the field increases, this distance becomes longer. dicts that the electric field can effectively enhance spin
We now contrast Eq. (11) with that obtained by pre- injection from a ferromagnetic metal into a semiconduc-
vious calculations [2, 3, 4] based on Eq. (1). The spin tor and substantially increase the transport distance of
injection thespinpolarizationinsemiconductors. Ourresultssug-
gest that the “conductivity mismatch” obstacle in spin
L(f) L(s) −1 pfL(f) − x injection may be overcome with the help of high field
α(x)= + e L(s) (12)
(1 p2)σ σ (1 p2)σ injection in the diffusive regime.
h − f f s i − f f
is given by the zero-field result of Eq. (11). As L(f) WewouldliketothankN.Samarthforpointingoutthe
≪
L(s) and σf σs, the effective resistance in the metal, relevance of resistance mismatch for spin injection from
≫
L(f)/σf, is much less than its counterpart in the semi- ferromagneticsemiconductors. This workwassupported
conductor, L(s)/σ . Thus Eq. (12) suggests that this by DARPA/ARO DAAD19-01-0490.
s
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5
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