Imaging spin properties using spatially varying magnetic fields

0.97 MB·English

by  V. P. Bhallamudi

Imaging spin properties using spatially varying magnetic fields
V. P. Bhallamudi,1 A. J. Berger,2 D. E. Labanowski,1 D. Stroud,2 and P. C. Hammel2
1Department of Electrical and Computer Engineering,
The Ohio State University, Columbus, Ohio 43210, USA
2Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA
(Dated: January 18, 2011)
We present a novel method to image spin properties of spintronic systems using the spatially
confined field of a scanned micromagnetic probe, in conjunction with existing electrical or optical
global spin detection schemes. It is thus applicable to all materials systems susceptible to either of
thoseapproaches. Theproposedtechniquereliesonnumericalsolutionstothespindiffusionequation
1
inthepresenceofspatiallyvaryingfieldstoobtainthelocalspinresponsetothemicromagneticprobe
1
field. ThesesolutionsalsoprovideinsightintotheeffectsofinhomogeneitiesonHanlemeasurements.
0
2
PACSnumbers: 75.76.+j,75.40.Gb,85.75.-d,75.78.-n

n
a
J Spinelectronicsisarichandrapidlygrowingfieldthat Spin dynamics in nonmagnetic materials are given by

5 integratesmagneticandnonmagneticmaterialstoenable
1 a variety of spin polarized transport phenomena [1, 2]. ∂S =D ∇2S+γB×S− S +G, (1)
Muchprogresshasbeenmadeinunderstandinginjection ∂t s τs
]
l of spins into non-magnetic semiconductors, the manipu-
al lation of those spins within the semiconductor, and both where S is the spin density, Ds is the spin diffusion con-
stant, γ = gµ /¯h is the gyromagnetic ratio, B is the
h B
local and global detection of the spin states [3–6]. Sev-
- total magnetic field experienced by the spins, τs is the
s eral recent results have brought us closer to electrically-
spin relaxation time, and G represents the spin genera-
e
controlled room-temperature spintronic devices [7–15].
m tionterm,e.g. irradiationwithcircularlypolarizedlight.
However, much is still not well understood, particularly
S is a function of time t and spatial position r = (x,y)
t. regardingthemechanismsthatdegradespinpolarization, for the 2D sample that we consider.
a
intheoftencomplexmagneticenvironmentthatprevails
m The spatially dependent response of the spin polariza-
in spintronic devices. Spin lifetimes inferred from mea-
tion to the magnetic field of the probe tip can be calcu-
-
surementsareoftenmuchshorterthantheoreticalexpec-
d lated by means of a numerical technique we have devel-
n tations and the influence of impurities, inhomogeneities
oped for solving the spin diffusion equation in the pres-
o andotherdeparturesfromidealstructuresremainpoorly
ence of spatially varying vector magnetic fields. Vector
c understood.
[ precessionisanaddedcomplexitywhichisnotpresentin

the case of (scalar) charge deflection by a scanned gate.
3
Atechniquewhichcouldspatiallyresolvethelocalvari- Our technique gives steady state spatial maps of spin
v
7 ations of spin density and other key parameters, such as density as a function of the location of the micromag-
4 spinlifetime,inmulticomponentspintronicdeviceswould netic probe over the sample. The numerical solution
7 have enormous impact. Here we show that local infor- is obtained from the time-dependent form of the diffu-
3
mation about the spin properties can be encoded in the sion equation using Euler’s iterative method (please see
.
0 variation of the global spin polarization in response to supplementary material for further details). Though we
1 a spatially scanned magnetic field from a micromagnetic do not consider electric fields nor take B, G and τ to
s
0
probe. In analogy with electrostatic tip-induced scatter- betime-dependentthesesituationsarestraightforwardly
1
ing in a two-dimensional electron gas [16] the local mag- handled by our approach.
:
v netization is modified to an extent determined by both We consider spins injected uniformly into the sample
i
X thevectorfieldofthemicromagneticprobeandthelocal bysomemeans(uniformityisnotanecessarycondition).
spin properties. This local perturbation is detected by The injected spins and the scanned dipole are assumed
r
a established global spin polarization measurement tech- tobeorientedalongˆz. Thissamplehasasmalllocalized
niques(suchasspinphotoluminescenceornon-localvolt- regioninwhichthespinlifetimeisfivetimeslongerthan
age) which, in themselves, need not provide spatially re- the rest of the sample. Panels (b) and (c) of Fig. 1 show
solvedinformation,hencethemethodisapplicabletoany spatial maps of S , the component of the spin density
z
materials system (whether optically active or inactive). parallel to the injected spin direction, for two positions
A schematic of our proposed imaging technique is shown oftheproberelativetothelifetimeinhomogeneity. Panel
in Fig. 1(a). As a magnetic dipole is scanned above the (d)presentstheimageofS ,theintegralofS overthe
z,g z
sample the dependence of a globally detected spin signal entiresample,asafunctionofthepositionofdipoleover
on position of the probe provides a map of the sample’s thesample. Theregionofslowerrelaxationrateisclearly
spin properties. distinguished from the rest of the sample.
2
uniform systems, the steady-state value of S is propor-
(cid:107)
tional to (1+θ2)−1).
B
We are interested in spatially varying fields arising
from a micromagnetic tip. The dipole field arising from
a magnetic moment m = m zˆ located at (0, 0, z ) over
z p
a sample in the xy-plane results in a spatial variation of
spin density S shown in Fig. 2 (left axis) along with
z
θ (right axis). Here we assume a sample with uni-
B
form τ and uniform injection of spins oriented parallel
s
to zˆ and approximate the probe field by B (x,y) =
dip
µ0[(3R(m·R)−mR2]/R5, where R = xxˆ +yyˆ +z zˆ
4π p
and µ is the permeability of free space. Fig. 2 shows S
0 z
bothwithandwithoutdiffusion. Theanticorrelationbe-
tweenS andθ isparticularlyevidentwhendiffusionis
z B
neglected because a given spin experiences a single mag-
netic field throughout its lifetime. Diffusion smears out
the sharp features, and if the minima in S (for D =0)
(cid:107) s
occur closer than a spin-diffusion length L away from
s
the central peak, then the peak will collapse leaving a
localized volume of suppressed spin density.
This local suppression of the spin polarization by the
probetipfieldprovidesthebasisforscannedprobeimag-
ing of a spintronic sample: the impact of this tip field is
determined by local sample properties. The image con-
trast in Fig. 1(d) results from the fact that long-lived
spins are more strongly dephased because they experi-
ence the tip field for a longer time before being replaced
by fresh, well-oriented spins. Thus, the globally aver-
aged spin density of the sample will be more strongly
FIG. 1: [Color online] Calculated vector magnetization
suppressed when the dipole is over the region of longer
demonstrating imaging of local properties using global mea-
τ . The size of this effect depends on the “global” size
surements: (a) Measurement geometry of our proposed ex- s
over which the spin polarization is integrated.
periment. Embedded in a uniform plane is a small region
(dotted line) in which τ is five times larger than throughout Toestimatethespatialresolutionofourimagingtech-
s
theremainingarea. The2Dsampleisuniformlypumpedwith nique, we consider the effect of a linear magnetic field
spins. A probe is positioned above the sample at a distance gradient: B = α(yyˆ −zzˆ) (see Fig. 3). The injected
z and its xy-coordinates are scanned to obtain the image
p spins are uniformly oriented along zˆ. The resulting vari-
presented in panel (d). (b,c) Spatial map of S (x,y) when
z ation of S along the y-axis, for α = (γτ L )−1, where
the enhanced τ region is found at (2L ,2L ) and (6L ,6L ) (cid:107) s s
s s s s s
relative to the tip. (d) S , the globally integrated S over
z,g z
theentiresampleasafunctionofprobeco-ordinates(x ,y ).
p p
The sensitivity of spin polarization to the position of
the probe arises from the interrelationship of the two
mechanisms that alter spin polarization: spin relaxation
and dephasing of spins in an ensemble as they precess
about the spatially inhomogeneous magnetic field vec-
tors. Fields parallel and perpendicular have contrast-
ing impacts since magnetic field parallel to the injected
spin direction will tend to maintain this orientation and
maximizespinpolarization,whiletransversecomponents
FIG. 2: [Color online] (a) Steady state spin density S (x,y)
z
cause precession, thus dephasing and a reduction of spin
for y = 0 in the presence of the field of a dipole for D = 0
s
polarization. Thiscompetitionbetweenthevariouscom- (red)andD τ /z2 =1(blue). Thedipolem zˆ [hereµ m =
s s p z 0 z
ponentsofthemagneticfieldiswellcapturedbyaquan- 10πz3)] is located at r = (0,0,z ) and uniform injection is
p p
tity θ2 = γ2B2τ2/(1+γ2B2τ2), which governs the av- assumed. The green curve shows the spin dephasing factor
B ⊥ s (cid:107) s
θ (x); note that θ =1 =⇒ S =0.5 for D =0.
erage projection of the spins along the (cid:107)-axis (e.g., for B B z s
3
gions of suppressed magnetization shown in Fig. 2) can-
not be smaller than the probe-sample separation, z , so
p
for large z this will set the resolution.
p
Understanding the impact of microscopic inhomo-
geneities on spin transport is a centrally important issue
that calls for spatially resolved studies. Particularly im-
portantareinhomogeneitiesduetothestrayfieldsarising
fromimperfectferromagneticinjectorsusedforelectrical
injection [13, 15, 23, 24]. Spin lifetime is typically ob-
tained from the width of Hanle curves, but such global
measures of spin polarization are sensitive to inhomo-
geneities, so microscopic approaches are needed to dis-
cernthevariousmechanisms thatcanaffectthe shape of
a Hanle curve.
Non-uniformity of the interface and complex domain
FIG. 3: [Color online] The resolution length scale L as a structures can cause the injected spin population to
r
function of magnetic field gradient α, where length is scaled experience significantly different magnetic fields, and
bythediffusionlengthLs,andfieldisintheunitsof(γτs)−1. lose spin-information on time scales much shorter than
(inset) Variation of S along the y-axis in the presence of a
z spin lifetime due to dephasing. Our numerical method
field of the form B = α(yyˆ −zzˆ). A Gaussian fit to the
is capable of simulating the impact of magnetic fields
numerically simulated data S is shown. The width of the
z
Gaussian may be considered to be the resolution for a given with random spatial variation. Fig. 4 shows calculated
gradient. Hanle curves in the presence of random magnetic fields.
We consider a field B = N (0,B )xˆ + N (0,B )yˆ +
r x v y v
N (0,B )zˆ where we sample normal distributions of
z v
(γτ )−1 is the half-width of the conventional Lorentzian mean zero and variance B for each field component at
s v
Hanle curve [17], is presented in the inset of Fig. 3. A every spatial point. Fig. 4 shows results for three values
Hanle curve is simply a plot of the dependence of spin of B :(cid:28) (γτ )−1, = (γτ )−1, and > (γτ )−1. We see
v s s s
polarization on a transverse magnetic field. In the ab- that the maxima can be shifted and, as in the last case,
sence of diffusion, spins at different positions along the the curves can be significantly broadened.
y-axis will sample different fields, resulting in a map- Itisinstructivetoconsider, inturn, theeffectsofeach
ping of the conventional Hanle curve onto the spatial magnetic field component on the Hanle curve, as is done
domain. Thus, spins will exist primarily in the region inFig. 5. ForanappliedHanlefieldB =B yˆ,thehalf-
h h
where |B| ≤ (γτ )−1, i.e., where |y| ≤ (γτ α)−1. The widthathalf-maximumB ofaHanlecurveinthepres-
s s 1/2
location of the surviving spin magnetization can be spa- enceofanadditionaluniformfieldB =B xˆ+B yˆ+B zˆ
u x y z
(cid:112)
tially shifted by adding a uniform offset to the gradient is given by γτ B = 1+γ2τ2(B +B )2. A trans-
s 1/2 s x z
field. verse field B reduces S (B = 0) (blue curve) and
x z,g h
A measure of the resolution may be obtained from the B ∼B is required to significantly reduce S further.
h x z,g
width of a Gaussian fit, L , to this curve. Fig. 3 shows A non-zero B (red curve) on the other hand will not
r z
this width as a function of the applied gradient. The reduce S (B = 0) but will broaden the Hanle curve,
z,g h
spatial resolution L decreases in inverse proportion to since B ∼ B is required to significantly increase the
r h z
the gradient until Lr <∼ Ls below which its rate of de- spin precession cone angle. A field By (parallel to Bh)
creases slows. This situation is reminiscent of Magnetic shifts the peak of the Hanle curve, since S is maxi-
z,g
Resonance Imaging (MRI) where spatial information is mizedwhenthetotaltransversefieldiszero(blackcurve).
encoded into field or frequency shifts through magnetic The results of Fig. 4 can be considered a superposition
field gradients [18]. As in MRI, diffusion degrades res- of the shifting and broadening seen in Fig. 5. Clearly
olution, though in the current approach it can be finer these effects can confound the determination of spin life-
than the spin diffusion length if one uses strong enough times from Hanle widths. The detailed features of the
gradients,asseeninnumericalsimulationsshowninFig. stray field generated by a rough ferromagnetic injector
3. Furthermore,itshouldbenotedthatdiffusionlengths will depend on several characteristics including rough-
for many spintronic systems are of the order of a mi- ness, saturation magnetization, domain size, and thick-
cron, especially at room temperatures. Field gradients ness. Regardless of these details, stray fields of the order
exceeding 10 G/nm are achievable with rare-earth mi- kiloGauss are readily achievable a few nanometers away
cromagnetic tips [19] such as are used in Magnetic Force from a rough injector and these can substantially affect
Resonance Microscopy [20–22] indicating that high spa- the spin polarization. In fact this has been experimen-
tialresolutionispossiblewiththistechnique. Inthecase tally observed in silicon devices [24].
of imaging with a dipole tip, the gradients (and the re- In summary, we have proposed a new technique for
4
lifetime measurements, as in Hanle curves. Microscopic
imaging of spin properties of real-world devices will be
essential for improving their spin fidelity and functional-
ity.
We gratefully acknowledge enlightening discussions
with Ron Jansen. This work was supported by National
Science Foundation through the Materials Research Sci-
ence and Engineering Center at The Ohio State Univer-
sity (DMR-0820414) and Department of Energy, Office
of Science (DE-FG02-03ER46054).
FIG. 4: [Color online] Global Hanle curves for a sample with
randomlyspatiallyvaryingmagneticfield,B =N (0,B )xˆ+ Supplementary Information
r x v
N (0,B )yˆ + N (0,B )zˆ. Such fields may be expected in
y v z v
electricalinjectiondeviceswithroughinjectors. Theinjected
Spindynamicsinnonmagneticmaterialsaredescribed
spins are assumed to be along the x-axis in this simulation.
by
∂S S
=D ∇2S+γB×S− +G, (2)
∂t s τ
s
where S is the spin density, D is the spin diffusion con-
s
stant, γ = gµ /¯h is the gyromagnetic ratio, B is the
B
total magnetic field experienced by the spins, τ is the
s
spin relaxation time, and G represents the spin genera-
tion term e.g., irradiation with circularly polarized light.
S is a function of time t and spatial position r = (x,y)
for the 2D sample that we consider. While all of the
terms can be functions of t and r, we will consider B, G
and τ which are independent of t, and D will be in-
s s
FIG.5: [Coloronline]GlobalHanlecurvesforvariouscasesof dependent of both space and time. The formalism can
spatially uniform magnetic fields, B , applied in addition to
u be readily extended to include electric fields, spin-orbit
thesweptHanlefield,B . Thesecurvesarevalidforallvalues
h effects, hyperfine coupling and a third spatial dimension
of D since globally averaged spin density is insensitive to
s
for a more comprehensive treatment. Also, we are usu-
diffusion (see Supplementary Information), when the sample
dimensions are much greater than L . allyinterestedinthesteady-statesolution∂S/∂t=0and
s
S , the parallel component of S. We will refer to “paral-
(cid:107)
lel”and“perpendicular”withrespecttotheinjectedspin
imaging spin properties of spatially inhomogeneous sam- direction. ExperimentallyS isthemostcommonlymea-
(cid:107)
ples that achieves applicability to a wide variety of ma- sured quantity. However, it should be noted that we can
terials by relying on proven spin polarization detection evaluate any component of the spin in our simulations.
techniques such as electrical and optical detection. The If B and τ are position-independent, the steady-state
s
localizedinformationisobtainedbydetectingthemodifi- differentialequationcanbesolvedanalyticallybyFourier
cation to the local spin density by the confined magnetic transform. In the steady state, the Fourier-transformed
field of a magnetic dipole scanned over the sample. To equation for the spin density S(k) takes the form
thisend,wehavedevelopedamethodforsimulatingspin
density in a medium with spatial or temporal inhomo- −k2D S(k)+γB×S(k)−S(k)/τ +G(k)=0
s s
geneities of the magnetic field, lifetime, or diffusion con- =⇒S(k)=[(k2D +τ−1)I−B]−1G(k) (3)
s s
stant, regardless of injection or detection technique. As
in MRI, this technique has a spatial resolution inversely
where S(k) and G(k) are vectors. I is the 3×3 unit
proportionaltothegradientofthemagneticfielddownto
matrix. We introduce the 3 × 3 matrix
lengthscalescomparabletothespindiffusionlength. Be-
 
lowthis,resolutionimprovesmoreslowlywithincreasing 0 B −B
z y
gradient. Our numerical analysis of spin transport em- B =γ−Bz 0 Bx ,
phasizestheimportanceofexperimentalcharacterization B −B 0
y x
of microscopic inhomogeneity, and of models that incor-
porate these phenomena. In particular, inhomogeneities The real-space spin density is then given by the inverse
can confound the interpretation of spin polarization and Fourier transform of S(k).
5
Animportantcaseexperimentallyisgivenbytheglob- where δˆ represents the unit vectors along the spatial
ally averaged spin polarization grid. We iterate until S does not change appreciably
with time. We have verified that the analytical results
S(cid:107),g = τ(cid:82)(cid:82)ASG(cid:107)(r()rd)2dr2r, (4) fdruocmedthbeyatnhaislymticeathloedx.prSeessvieornasl dexerpievreidmaenbtoavlegalorebarleparnod-
s A (cid:107)
spatially localized Hanle curves measured under varying
forG=G ˆ(cid:107)andtheareaofintegration,A,extendsover conditions[3, 6] can also be qualitatively reproduced by
(cid:107)
the entire space. Note that S ∝ S (k = 0) and thus this method.
z,g (cid:107)
from Eq. 3 is independent of D and is given by
s
1
S = (5)
(cid:107),g 1+θ2
B
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approximated as
∇2S(r,t)∼ 1 (cid:88)[S(r+∆xδˆ,t)−S(r,t)] (7)
(∆x)2
δˆ