Anisotropic magneto-resistance in a GaMnAs-based single impurity tunnel diode: a tight binding approach

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by  M. O. Nestoklon

Anisotropic magneto-resistance in a GaMnAs-based single impurity tunnel diode:
a tight binding approach
M.O. Nestoklon
Ioffe Physical-Technical Institute, Russian Academy of Sciences, St. Petersburg 194021, Russia and
CNRS-Laboratoire de Photonique et de Nanostructures, 91460 Marcoussis, France
O. Krebs and P. Voisin
CNRS-Laboratoire de Photonique et de Nanostructures, 91460 Marcoussis, France
2
1 H. Jaffr`es, S. Ruttala, and J.-M. George
0 Unit´e Mixte de Physique CNRS-Thal`es-Universit´e Paris-Sud, 91767 Palaiseau Cedex, France
2

n J.-M. Jancu
a FOTON-INSA Laboratory, UMR 6082 au CNRS, INSA de Rennes, 35043 Rennes Cedex, France
J

6 Usinganadvancedtight-bindingapproach,weestimatetheanisotropyofthetunneltransmission
associated with the rotation of the 5/2 spin of a single Mn atom forming an acceptor state in

] GaAs and located near an AlGaAs tunnel barrier. Significant anisotropies in both in-plane and
l
l out-of-plane geometries are found, resulting from the combination of the large spin-orbit coupling
a
associatedwiththep-d exchangeinteraction,cubicanisotropyofheavy-holedispersionandthelow
h
C symmetry of the chemical bonds.
- 2v
s
e PACSnumbers: 75.30.Gw,73.40.Gk,75.70.Ak,73.40.Kp
m
.
t In the context of semiconductor-based spintronics
a
m GaMnAs-based tunnel diodes are of utmost interest. In effect of the barrier on the impurity state itself. Indeed,
particular,itwasrecentlydemonstratedthatthetunnel- it is known from STM studies that hole wavefunction is
-
d ing current in semiconductor heterostructures integrat- strongly affected by its structural environment8–10. In
n ing the p-type ferromagnetic semiconductor GaMnAs is this Letter we consider a model system close to experi-
o strongly affected by the direction of its magnetization, mentally achievable single impurity tunnel diode and es-
c whichcanbearbitrarilychangedbyanexternalmagnetic timatetheTAMReffectintheframeworkofanadvanced
[
field1–4. Inthisregard,twomainphenomenamustbedis- sp3d5s∗ tight-binding model11 including exchange inter-

1 tinguished: (i) the tunneling magnetoresistance (TMR) actions in the effective semi-classical approach6,7. In ad-
v whichisthedependenceofthetunnelcurrentwiththere- ditiontoastrongdependencyoftunnelprobabilityonthe
9 spective parallel or antiparallel magnetic configurations angle between Mn spin orientation and the [001] growth
3
of two ferromagnetic layers separated by a tunnel bar- direction, our calculations reveal an unexpected in-plane
4
1 rier giving rise to spin-valve effects and (ii) the tunnel- anisotropy of comparable magnitude, with the [110] and
. ing anisotropic magnetoresistance (TAMR) which is the [¯110] eigenaxis characteristic of the C2v symmetry12,13.
1
changeofthetunnelcurrentasafunctionofthemagneti-
0 We model TAMR using the following scheme: we
zation direction of a (single) magnetic layer with respect
2 consider a single Mn impurity in a GaAs layer near
1 to a reference coordinate system. So far, experimental a thin AlGaAs barrier (see Fig. 1) and calculate the
v: results1–3 were obtained in the metallic regime, but it neutral acceptor bound state. Tunneling transmission
wassuggestedthattheoriginofTAMRcouldbestillim-
i from this hole bound state through the AlGaAs barrier
X putedtotheanisotropicshapeoftheholestateboundto
is estimated as an integral of the wavefunction tail
r the isolated Mn impurity in a GaAs host5. Various the- in the GaAs ”collector” layer on the right side of the
a oretical approaches indicate that the bound hole state barrier (z > 0): (ψ (r)) (cid:82) |ψ (r)|2d3r . We do not
is highly anisotropic6–8 and more extended in directions h z>0 h
consider here the ”injector” problem, i.e. the way a hole
perpendicular to the Mn spin orientation. As a result,
is injected onto the impurity bound state from the left
for in-plane magnetization, the exponential tail of hole
of the structure. This step can be ensured by p-doping
bound state is more extended and tunneling probability
the left electrode with shallow acceptors like Be, as was
higher. Ontheonehand,thevalidityofthisexplanation done in the context of STM studies9.
might be a matter of discussion for Mn densities close
to the Mott limit (few percent Mn, considering a Bohr
For the physical validity of our calculations, all what
radiusof1nm). Ontheotherhand,eventhisqualitative
matters is that the current be limited by the tunnel bar-
pictureneedstobecheckedbycalculationsincludingthe
rier in Fig. 1. The hole density of states (DOS) at the
2
∫|ψ(r)|2dr
E
v
GaAs:Mn p-type GaAs
AlGaAs
0
z║[001] _ 10] → →
[1 S S
FIG. 1. Schematic representation of a Mn impurity in the
GaAslayernearathinAl Ga Asbarrier. Wavefunctionof
0.3 0.7
the hole bound to the acceptor penetrates this barrier. Tun-
neling probability is proportional to the integral of the hole
−12 −11 −10 −9 −8 −7 −6 −5 −4 −3
wavefunction over the region to the right of the barrier.
Log|ψ|²
[001]
FIG. 2. (Color online) Cross-section of the neutral acceptor
right collector like originally investigated by Bardeen14 wavefunction in the (110) plane containing the Mn dopant.
is not explicitely included, as it just enters a prefactor Wavefunction amplitude is color-coded according to the log
that does not depend on Mn spin orientation. Still, this scale. Atomic positions are shown with circles : red and
blue for As and Ga, green for Mn, and white for the virtual
simple approach is only a zeroth order estimate of the
Al Ga cation in the tunnel barrier.
tunnel probability as it does not take into account the 0.3 0.7
spin structureof hole wave-functions. One may convince
himself that this should be valid as soon as the kinetic
energy at which the hole is injected into the collector
is high enough (a few 10 meV’s). Indeed, in this case than one Mn atom per thousand in the GaMnAs alloy.
the admixture of heavy and light characters is such that In real structures Mn atoms would be distributed ran-
the selectivity of transmission into the heavy- and ligh- domly in the alloy while in the calculations they form a
holebandsshouldessentiallyvanish. Aroughestimation periodic array on a square lattice. We have checked that
showsthatthiscorrespondstoactualexperimentalsitua- calculation results are stable with respect to Mn spatial
tion. Notethatamorecompletetreatmentwouldrequire distribution. Anotherpointworthtobementionedisthe
full description of a specific collector geometry. use of sp3d5s∗ model, compared to the sp3 model used
To calculate wavefunctions we use the model and pa- in previous papers6,7. While sp3 is a qualitatively ac-
rameterization of Ref. 9 and include p-d exchange inter- ceptable basis to discuss valence band properties in III-
action in the mean-field approach previously introduced V compounds, it is well-known16 that it fails to repro-
byTangandFlatt´e6. Thiswellestablishedmethod,valid duce accurately the Luttinger parameters that describe
in the ferromagnetic regime, actually corresponds to an valencebanddispersion. sp3d5s∗ isactuallytheminimal
Ising coupling between the hole and Mn 3d5 electron tight-binding model which allows fitting all the param-
spins. It amounts to treating the Mn 5/2 spin as a clas- eters of valence band over a large energy range, and in
sical magnetic momentum7. In the paramagnetic regime particular, the cubic anisotropy (warping) of the heavy-
a complete quantum treatment based on Heisenberg ex- hole dispersion. This is essential for an accurate account
change coupling15 would be definitely required, but it of the kinetic energy part of the impurity hamiltonian.
is much more difficult to implement self-consistently in In order to image the wavefunctions, the local density of
the tight-binding formalism. It is also worth mentioning states at atomic sites is approximated by gaussian func-
thatforagiven(classical)Mnspinorientation, themag- tionswitharadiusdependingonorbitaltype(∼1.5-2˚A).
neticacceptorgroundstateisnon-degenerate,incontrast Thecross-sectionsofthewavefunctionofaholebound
with the four-fold degeneracy of a non-magnetic neutral to the Mn acceptor are shown in Fig. 2 for two direc-
acceptor state. tions of the Mn spin, in the (110)-plane containing the
In practice, we use a supercell formalism, with a 8624 impurity. Thewavefunctionamplitudeiscolor-codedac-
atom supercell of 7 ML×7 ML×22 ML, which is approx- cording to the log scale below the panels, and atomic
imately 4 nm×4 nm in lateral directions and 12 nm in positions are indicated by colored dots (see figure cap-
growth direction. Technically, we destroy the effects of tion). Left panel displays the situation for a Mn spin
periodicity in the [001] growth direction by adding an along the [001] direction, and right panel for a Mn spin
AlAs barrier to the left of the GaMnAs layer, that fully in [1¯10] direction. While it is not obvious from these fig-
decouples adjacent cells. Periodic boundary conditions ures how tunneling probability can be affected, it may
in lateral direction produce non-negligible effects on the be concluded that the hole wavefunction changes signifi-
bound state energy and on the tails of the wavefunction, cantlyinsizeasalreadyemphasizedinRef.6. Theeffect
that must be carefully characterized. However, these ef- of barrier proximity is clearly visible from the left/right
fectsmayhardlybeconsideredasanartefactofthemodel asymmetry of wavefunction tails, and the role of lateral
aslongasdistancebetweenMnatomscorrespondstoless periodicity is evident from the non vanishing tails in the
3
1.10 tion. If the spin is oriented along the growth direction,
1.08
thebarrierliesinthedirectionwheretheexponentialde-
1.06
1.05
1.04 cay is stronger, giving a lower tunneling probability by
1.02 more than 20%, which was defined in a previous letter
1.00 1.00 as a negative TAMR18. Wavefunction deformation due
0.98
u.0.95 0.96 to the barrier proximity and large cubic anisotropy due
a. 0.94 to valence band warping (see Fig. 2) do not change this
ψ|², 0.90 −0.4−0.2φ0,. 0π 0.2 0.4 qualitative argumentation significantly. In-plane varia-
∫| tion (inset in Fig. 3) is more intriguing. It has similar
0.85
φ, 0 amplitude as the zenithal variation, and shows the char-
0.80 φ,−π/4 acteristicC2v symmetry,withpronounceddifferencesbe-
φ, π/4 tween [¯110] and [110] directions. We have checked that
0.75 thisresultisnotacalculationartifactduetolateralperi-
0.0 0.1 0.2 0.3 0.4 0.5
θ, π odicboundaryconditionsbyrotatingthesupercellorien-
tation, which is equivalent to rotating the square lattice
FIG. 3. Dependence of wavefunction integral on right side ofMnatoms: thisrotationhasnoinfluenceonthecalcu-
of barrier on Mn spin orientation. a) dependence on zenithal latedtunnelcurrentanisotropy. Thissuggeststhatthese
angle θ for different values of the azimuthal angle ϕ with results would be qualitatively unchanged for the ran-
respect to [100], b) azimuthal plot of in-plane anisotropy. domly positioned Mn atoms. In-plane anisotropy is also
nearly unaffected by the lateral supercell size, although
the variation of bound-state eigenenergy with spin ori-
entation increases with decreasing size (or, equivalently,
with increasing Mn concentration). The calculated in-
verticaldirectioninFig.2(supercellinthecalculationsis plane anisotropy of about 14% actually results from a
few ML larger in lateral direction than shown in figure). combination of three factors reducing the symmetry of
Neutral acceptor binding energy decreases when the im- bound state wavefunction : i) a native anisotropy of the
purity gets closer to the barrier17. For Mn located at 1 bulk wavefunction, which is clearly evidenced by remov-
ML from the barrier, this reduction amounts to 20 meV. ing the barriers in the supercell while calculating the in-
The eigenenergies also change with Mn spin orientation: tegral over the same part of the wavefunction ; ii) the
this is associated with the magnetization-dependence of left/right asymmetry of the wavefunction due to the in-
the wavefunction tails, and the related change in tunnel terface proximity ; and iii) the interface C symmetry
2v
coupling of acceptors in adjacent cells7 . The amplitude itself12,13. These contributions interfere and can hardly
of this change for Mn located at 1 ML from the barrier be separated in the final result.
as a function of angle between Mn spin and growth di- In summary, we have estimated the anisotropy of the
rection is almost 10 meV. Under in-plane rotation of the tunnel transmission from the fundamental A0 hole state
spindirectionthismodulationofeigenenergiesisreduced attached to a single Mn dopant in a GaAs host matrix,
but still amounts to about 4 meV. The calculated tun- coupledtoareservoirthroughanAlGaAstunnelbarrier.
neling probability is extremely sensitive to the impurity The simulated structure displays pronounced anisotropy
distance from the barrier, decreasing exponentially with of the tunnel transmission as a function of Mn classical
characteristic length ∼3.7 ˚A. However, for all distances spindirection. Significantin-planeanisotropyofthetun-
it shows similar angle dependencies. In the following, we nel current is obtained. Yet, a full treatment of Heisen-
focus on the 1 ML distance corresponding to Fig. 2. berg (instead of Ising) spin coupling and direct account
Calculation results are illustrated in Fig. 3. Mn spin ofappliedmagneticfieldwouldbehighlydesirableinor-
direction is defined with the zenithal angle θ with re- dertofullyassertthepotentialofsingleimpurityTAMR
spect to [001] growth direction and in-plane azimuthal effect.
angle ϕ with respect to [100] axis. The dependency on θ Acknowledgment: Thisworkwassupportedby“Trian-
followsthequalitativeexplanationgivenintheintroduc- gle de la Physique”, by CNRS-RAS joint laboratory IL-
tion: the wavefunction of the hole localized at acceptor NACSandbytheEuropeanNetworksforInitialTraining
has a somewhat oblate shape and follows Mn spin direc- (ITN): SemiSpinNet.
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