Interval identification of FMR parameters for spin reorientation transition in (Ga,Mn)As

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by  M. W. Gutowski

Interval identification of FMR parameters for spin reorientation transition in
(Ga,Mn)As
M.W. Gutowski,1,∗ W. Stefanowicz,1 O. Proselkov,1 J. Sadowski,1,2 M. Sawicki,1 and R. Z˙uberek1
1Institute of Physics, Polish Academy of Sciences, Warszawa, Poland
2MAX-lab, Lund University, Lund, Sweden
In this work we reports results of ferromagnetic resonance studies of a 6% 15nm (Ga,Mn)As
layer,deposited on(001)-orientedGaAs. Themeasurementswereperformedwith in-planeoriented
2 magneticfield,inthetemperaturerangebetween5Kand120K.Weobserveatemperatureinduced
1 reorientation of the effective in-plane easy axis from [110] to [110] direction close to the Curie
0 temperature.Thebehaviorofmagnetization isdescribedbyanisotropyfields,Heff(=4πM−H2⊥),
2 H2k,andH4k.Inordertopreciselyinvestigatethisreorientation,numericalvaluesofanisotropyfields
n havebeendeterminedusingpowerful–butstilllargelyunknown–intervalcalculations.Insimulation
a mode this approach makes possible to find all the resonance fields for arbitrarily oriented sample,
J which is generally intractable analytically. In ‘fitting’ mode we effectively utilize full experimental
3 information,notonlythosemeasurementsperformedinspecial,distinguisheddirections,toreliably
1 estimatethevaluesofimportantphysicalparametersaswellastheiruncertaintiesandcorrelations.

] PACSnumbers: 07.05.Kf,68.47.Fg,75.30.Gw,75.50.Pp,75.70.-i,75.70.Ak,75.70.Cn,75.70.Rf,76.50.+g
l
l
a
h I. MOTIVATION Here polar angles angles θ and ϕ refer to the orienta-
s- tion of the magnetization vector, M~, not to the orienta-
e Despite numerous and intensive studies, an origin of tion of an external field H~.
m
the in-plane uniaxial magnetic anisotropy in (Ga,Mn)As Original experimental data, taken at fixed frequency
. remains unknown.However,both its strengthand orien- 9.378GHz, are shown in Fig. 1, together with simulated
t
a tationcanbedescribedonthegroundofp-d Zenermodel spectra.
m
assuming the existence of a fictitious epitaxial strain
- [1–3]. On the other hand, as this is the leading mag-
d 2800
netic anisotropy at elevated temperatures and that the
n
120 K
o means of its controlhave alreadybeen demonstrated[4], 2600
115 K
c furtherstudiesonthisintriguingpropertyaretimelyand 110 K
[ important. In particular, a presence of the temperature- Oe] 2400 105 K
induced 90◦ rotation of the direction of the easy axis d [ 100 K
1 el 2200 95 K
6v [i1n]fmutauyreeadseevitchees.veInctotrhims caonmipmuluantiiocnatioofnmwaegnreeptiozrattioonn ance fi 2000 8900 KK
3 thetechnicalanalysisandresultsofFMRstudiesofsuch on
8 a (Ga,Mn)As layerthat exhibit the easy axis rotation at es 1800
R
2 temperatures close to its Curie temperature. 30 K
. 1600
1 The free energy density for our system, expressed by 5 K
0 anisotropy fields (H’s) rather than by more customary [11-0] [100] [110] [010] [1-10] [1-00]
1400
2 anisotropyconstants(K’s), hasthe form(gµBHM (Zee- -45 0 45 90 135 180
1
man),andinactiveout-of-planefourfoldanisotropyterm External field orientation [degrees]
:
v have been omitted):
i FIG. 1. Experimental values of FMR fields vs. external field
X F =2πM2cos2θ (1) orientation and temperature (points). Lines show computed
r 1 spectra.
a − MH cos2θ (2)
2⊥
2
1 π The numerical values of three parameters: H =
− MH sin2θ sin2 ϕ− (3) eff
2 2k 4 4πM −H2⊥, (where H2⊥ ≡ 2K2⊥/M), H2k and H4k
1 (cid:16) (cid:17) were determined for each temperature separately.
− MH (3+cos4ϕ) sin4θ (4)
16 4k
where(1)isshapeanisotropy,(2)–ordinaryuniaxialout- II. OUTLINE OF NUMERICAL PROCEDURE
of-plane anisotropy, (3) – uniaxial in-plane anisotropy,
and (4) – fourfold, in-plane anisotropy.
Weareusingintervalcalculationsnotonlyforaccurate
simulations of FMR spectra when the values of all rele-
vant parameters are known, but also to estimate (‘fit’)
∗ Correspondingauthor: [e-mail] such parameters, together with their uncertainties and
2
correlations, utilizing full information hidden in experi- negative, then the currently considered box of searched
mental data. An excellent introduction to interval arith- parameters is infeasible as it cannot describe any reso-
metics can be found in [5]. Here we limit ourselves to nance at all. If all experimental data pass this test, then
the very brief description: i) an interval x is a bounded the ranges of their resonance frequencies can be found
set of real numbers: x ≡ [a,b] = {R∋x:a6x6b}, from(5) andthe interval Q can be evaluated. Again, for
ii) it is possible to perform arithmetic operations, eval- eachdataelementitheabsolutedifferencebetweenboth
uate functions, etc., using intervals instead of numbers, sidesof(5)iscalculated,producingthe interval∆ωi.Fi-
iii) the so obtained results are intervals as well, iv) the nally, Q = maxi∆ωi (other choices are possible but we
multidimensional intervals are called boxes for obvious prefer this one). The box is considered infeasible when
reasons,and v) ordinary real numbers may be identified it’s Q exceeds the reference value, which is equal to the
with (’thin’) intervals, for example 7 = [7,7]. Similarly, lowest Q ever seen during calculations.
f(x) is an interval containing all the possible results of
It remains to comment on resonance frequency cal-
evaluation of f(x) when x∈x. Unfortunately, f(x) usu-
culation. The angles (θ,ϕ) are unknown and should be
ally overestimates the range of true values of f(x) – but
determined prior to resonance frequency evaluation, for
always contains them all. eachdataelementseparately.Startingfromfullignorance
The unconventionality of our approach to fixed fre- (θ ∈[0,π], ϕ∈[0,2π]), we divide this initial 2D box un-
quency FMR data fitting is that we try to adjust un- til its edges are shorter than, say, 0.5◦. Discarded are all
known parameters in such a way that the classical for- boxes satisfying either of two conditions: 0 ∈/ ∂F/∂θ or
mula 0 ∈/ ∂F/∂ϕ – they certainly cannot contain the equilib-
ω 2 1 ∂2F ∂2F ∂2F 2 rium position of the vector M~. Needles to say that fail-
= · − (5) ure to find such a position immediately invalidates the
gµ (Msinθ)2 ∂θ2 ∂ϕ2 ∂θ ∂ϕ
(cid:18) B(cid:19) " (cid:18) (cid:19) # searched parameter box.
is satisfied for each experimental datum.
The main difficulty lies in the fact that the partial
III. RESULTS AND DISCUSSION
derivatives of free energy density have to be evaluated
at (stable) equilibrium position of magnetization vector,
Following the procedure described in previous section,
characterized by two (initially unknown) polar angles ϕ
we have computed the values of parameters determin-
andθ,andbeingthe solution(s)oftheequation’ssystem
ing the free energy density (1–4). During computation
∇F =(∂F/∂θ,∂F/∂ϕ)=0. (6) g was kept fixed at 2.00. The results are presented in
Fig. 2. The numerical values are consistent with those
This makes the inversion of formula (5), to obtain res- reportedbyothers[7–9],obtainedbythesametechnique
onance field(s) as a function of microwave frequency ω, but at lower temperatures. Using those results, with-
essentially impossible.
Our algorithm operates on a list L of boxes. At the 60 2000
beginning,thelistcontainsonlyonemember–theinitial Oe] 40 H4||
search domain. Further we perform following steps: H [4|| 20 1000 Oe]
1. select the biggest box from the list L and 0 0 H [eff
2. bisect its longest edge obtaining two offspring H 2|| -20 -1000 eld
boxes,then remove the parent box from the list L, elds -40 Heff opy fi
3. i–nvdeissctaigradteit,eaifchinofenaesiobflet,hoertwo offspring and: otropy fi --8600 H2|| --32000000 Anisotr
– put it back on the list L, otherwise. s
Ani -100 TC
The above procedure is repeated until the list contains -120 -4000
50 60 70 80 90 100 110 120 130
onlysmallboxes.Therestiseasy,providedthefinalclus-
Temperature [K]
ter of boxes consists of only one connected component.
For details on how to calculate mean values, variances FIG. 2. Anisotropy fields vs. temperature – computed val-
andcorrelationsbetweenthesearchedparameterssee[6]. ues. Lines are eye-guides only. Apparent lack of smoothness
Here we only clarify when the box is considered infeasi- may be attributed to inaccurate temperature readings (the
ble. First of all, each box is characterized by it’s quality measurements were taken during two separate sessions). For
factor, computedasanintervalquantityQ=[Q,Q].For better visibility, thecurves H2k and H4k are shifted horizon-
each resonance field (and its corresponding orientation) tally by−0.2K and +0.2K, respectively.
we try to evaluate the interval value of r.h.s. of expres-
sion (5). If the upper bound of so calculated interval is out any prior smoothing, we were able to determine the
3
directions of spontaneous magnetization in the interest- all should go to 0 as T → T – but they don’t. In addi-
C
ing temperature range, near T . In Fig. 3 one can see tion, the simulated angular FMR dependencies (Fig. 1)
C
that, in absence of the external field, there are only two seemnotsoaccurateasonemightexpect.Thoseintrigu-
suchdirections,antiparalleltoeachother,notfourasone ing facts, together with high reliability of intervalanaly-
might expect from symmetry arguments. The presence sis,stronglysuggestthattheuniaxialin-planeanisotropy
of external magnetic field, oriented along [110] (Earth’s maybeincorrectlyaccountedforin(1–4).Sincethesam-
field is sufficient) breaks even this symmetry, and the ples exhibiting this special behaviorare verythin, in nm
spontaneousmagnetizationalignsitselfexactlyalongthe range, then it is quite possible that the source of uniax-
external field above the transition temperature. Below ial anisotropy may be related to surface effects. Indeed,
the transition temperature the spontaneous magnetiza- AFMsurfacestudiesofMBE-grownsamples[10]revealed
tion deviates considerably from its ’natural’ [110] (or, the presence of well ordered ripples, parallel to [110] di-
equivalently,[110])position. Thismeansthatpreciseex- rection. If this was true, then the free energy expression
should be appended with the surface term [11], propor-
tional to |cos(ϕ−π/4)|. This, however, is beyond the
360 [100]
scope of a current paper.
H = 0 -
315 [110]
ees] 270 [01-0] IV. CONCLUSIONS
gr
e --
e positions [d 112382505 100 Oe 25 Oe [[[111--101000]]] atmaevlTtedhhrayoetdapinoaatwbenelraervlfayutlsolisctu.oatolIicnlluizflpoeurascrdtohiimfficauspclulaeblrtte,eepinterxoidpbseleepmrmrimoosnbesoantfbtralaeylxtepitdnhefreotimoromnebnalye--
n 50 Oe
pla 90 [010] tion acquired during FMR measurements. Its unrivaled
In- reliability allows us to state a sound hypothesis con-
45 [110]
cerning the nature of the somewhat mysterious in-plane
0 [100] uniaxial magnetic anisotropy observed in thin layers of
80 90 100 110 120 (Ga,Mn)As grown on (001)-oriented GaAs substrate.
Temperature [K]
We have also shownthat FMR technique is very help-
FIG. 3. Equilibrium position of magnetization vector vs. ful in accurate tracing the temperature dependency of
temperature for various strengths of the external field. The magnetic anisotropy in close vicinity of the Curie tem-
fieldisalwaysdirectedalonghigh-temperatureeasyaxis[110], perature,thatis wherethemagnetometricdataareleast
marked also as 135◦, and its strength is as indicated. reliable.
aminationofspontaneousmagnetization,especiallyofits
components perpendicular to the external field, is diffi-
ACKNOWLEDGMENTS
cult. The temperature range, in which the reorientation
occurs,is alsovery sensitive to the presence ofeven very
weak field. This work was supported in part by EC Net-
The behavior of anisotropy fields near T , estimated work SemiSpinNet (PITN-GA-2008-215368) and Polish
C
as ∼113 K for our sample, is somewhat unusual: they MNiSW 2048/B/H03/2008/34grant.
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