Magneto-Optical Effects Enhancement in DMS Layers Utilizing 1-D Photonic Crystal

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by  M. Koba

Machine Copy for Proofreading, Vol. x, y–z, 2012
MAGNETO-OPTICAL EFFECTS ENHANCEMENT IN
DMS LAYERS UTILIZING 1-D PHOTONIC CRYSTAL
3
1
0
M. Koba
2

n Institute of Experimental Physics, Faculty of Physics,
a
University of Warsaw
J
ul. Hoz˙a 69, PL-00-681 Warszawa, Poland
4
1 National Institute of Telecommunication

ul. Szachowa 1, PL-04-894 Warszawa, Poland
]
i
c
s J. Suffczyn´ski
-
l
r
t Institute of Experimental Physics, Faculty of Physics,
m
University of Warsaw
. ul. Hoz˙a 69, PL-00-681 Warszawa, Poland
t
a
m
Abstract—We propose a theoretical design of a structure combining
-
d a diluted magnetic semiconductor (DMS) layer with one-dimensional
n photonic crystal and show that it allows for a sizable (a few-fold)
o enhancement of the magneto-optical Kerr effect (MOKE) magnitude.
c
We base our calculations on the transfer matrix method assuming
[
realistic, and optimal with respect to a sample growth, parameters.

1 We investigate representative cases of (Ga,Mn)N or (Cd,Mn)Te DMS
v
layers deposited on the top of a respectively GaN- or CdTe-based
4
distributed Bragg reflector, and highlight the applicability of the
8
0 proposed design for both: III-V wurtzite and II-VI zinc-blende DMS.
3
.
1 1. INTRODUCTION
0
3
DilutedMagneticSemiconductors(DMSs)aresemiconductorsinwhich
1
a fraction of the cations is substituted with transition metal ions,
:
v e.g., Mn or Fe. Different types of DMSs comprising III-V, II-VI, and
i
X IV-VI materials exhibit a diversity of unique magnetic, optical and
conductive properties attractive from the point of view of fundamental
r
a research as well as possible applications. The s,p − d interaction
between band carriers and localized magnetic moments in DMS leads
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to such effects as giant Zeeman splitting of bands or giant magneto-
resistance, both beneficial for the domain of information processing
and storage. GaN based DMSs are particularly appealing in the view
of possible practical implementations, as they are expected to exhibit
a ferromagnetic ordering of spins with a critical Curie temperature
exceeding room temperature [1].
As it was shown [2, 3] a correlation of magneto-optical and
magnetic properties is an essential condition for the unequivocal
assessment of the intrinsic nature of magnetization of DMS. The
magneto-optical Kerr effect (MOKE) [4] is a commonly used optical
tool for a determination of the DMS magnetization. Its magnitude
scales typically linearly with the Zeeman splitting of excitonic levels
whichisinturnlinearlyproportionaltothesamplemagnetization[5,6].
However, themagneto-optical signalfromDMSdecreases ataverylow
andavery highmagnetic ions dopinglevel. Inthefirstcaseit isdueto
a low magnitude of the Zeeman splitting, while in the latter one, due
tothedecreaseoftheopticalsignalresultingfromanenhancementofa
non-radiative carrier recombination related to ions. [7] This limits the
efficacy of MOKE, and other optical methods in DMS magnetization
determination. To the contrary, optical effects in semiconductor may
beenhancedbyimplementationofphotoniccrystals(PC).ThePCthat
are periodic dielectric nanostructures containing regions of alternate
high and low refractive index. They affect the propagation of the
electromagnetic wave similarly as the periodic potential in a crystal
affects the motion of electrons and exhibit, in particular, a forbidden
gapinfrequencydomain.[8]Althoughitwashighlightedinanumberof
works [9–14] that combining the advantages offered by both domains:
DMS and photonics, may enforce a new capabilities in current and
future devices, until recently DMS and photonics paved their way in
the modern physics mostly separately.
Our work demonstrates that implementation of even a simple
photonic structure enables a meaningful enhancement of magneto-
optical effects in DMSs. We present a theoretical design of a DMS
layerdepositedonthetopofasingledistributedBraggreflector(DBR)
and show that such design leads to at least a few-fold enhancement
of the MOKE magnitude. Our approach is different from the one
developed in, e.g., Refs. [10–13], where the enhancement of magneto-
optical Faraday effect was obtained by enclosing a DMS layer within a
photonic cavity made of two DBR mirrors, or e.g., Ref. [14], where in
order to enhance the MOKE, a layer with single-domain nanomagnets
was placed under a dielectric multilayer. The simplicity of the
proposed design opens a way for its wide practical implementation.
It is worth emphasizing that, with respective material parameters,
Magneto-Optical Effects Enhancement... 3
the calculations can be easily applied to any stratified structure and
essentially toany III-VandII-VIDMS material withany magnetic ion
doping different from Mn, e.g., Fe or Co.
In what follows we introduce the structure design (Sec. 2) and
a numerical model upon which we base our calculations (Sec. 3), we
present and discuss the results (Sec. 4), and finally conclude our work
(Sec. 5).
2. STRUCTURES
Our investigation is focused on, but not bound to, the representative
casesofMn-dopedwurtziteIII-Vandzinc-blendeII-VIstructures,that
is (Ga,Mn)N and (Cd,Mn)Te, respectively. We consider realistically
designed structures. We assume a presence of a buffer layer, as
it is often introduced to GaN based structures grown on sapphire
substratesinordertoreducedensityofdislocationslimitingtheoptical
signal quality. Moreover, we sustain a small difference of refractive
indices of layers constituting the mirror, thus small contrast of their
compositions, in order to limit the undesirable effects of strain and
dislocation density resulting from the lattice mismatch.
We perform our calculations for two structures shown schemati-
cally in Fig. 1: one representing the proposed design, involving DMS
and DBR layers (Fig. 1a)), and the other one serving as a reference
(Fig. 1b)) being just the DMS layer deposited on the buffer layer (no
DBR).
More specifically, in the case of GaN-based structure, the
design presented in Fig. 1a) consists of the (Ga,Mn)N layer
(x =0.06%) thick for d = 100 nm, DBR layer constituted
Mn DMS
by N = 1 to 15 periods (thickness a = 61 nm) of alternating
Al0.05Ga0.95N/Al0.2Ga0.8N layers (with thicknesses of 28 nm and 33
nm, respectively), and Al0.1Ga0.9N buffer(thickness dbuffer = 200 nm)
deposited on a sapphire substrate. The thicknesses of the DBR mirror
layersaretunedsothemirrorstop-bandspectralpositionismatchedto
near the band gap spectral region. The assumed material parameters
arebasedonvaluestakenfromexperiment[18]toassurethattheresult
of ournumericalcalculation isasclose torealstructureperformanceas
possible. In the reference structure shown in Fig. 1b) the DBR layer is
replaced by the Al0.1Ga0.9N layer of the corresponding thickness Na.
The structure parameters for CdTe based sample are given in Sec. 4.
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Figure 1. a) The proposed sample design: DMS layer deposited on
DBR mirror, grown on buffer deposited on a sapphire substrate. In
the case of (Ga,Mn)N sample where a = 61 nm, and the number of
periods N varies from 1 to 15. b) Reference sample: DBR has been
replaced by Al0.1Ga0.9N buffer layer of the thickness Na equal to the
one of DBR mirror.
3. MODEL
The Magneto Optical Kerr Effect (MOKE) is a consequence of the
magnetic circular birefringence of the medium: the linearly polarized
electromagnetic wave after reflection from the surface of the sample
magnetized parallel to the light propagation becomes elliptically
polarized. This occurs due to magnetic field induced splitting of
bands and resulting difference of refraction coefficient for two opposite
circular polarizations of thelight. Theangle between themajor axis of
the elliptically polarized reflected light and the polarization direction
of the incident light defines the Kerr rotation Θ , which is given by
K
(e.g., Ref. [16]):
−
1 r
Θ = arg , (1)
K 2 r+
where r− and r+ represent complex amplitudes of reflectivity
coefficients for σ− and σ+ circular polarizations, respectively.
In order to obtain both r− and r+ values for an arbitrary one-
dimensional stack of layers we perform calculations basing on the
transfer matrix method (TMM) (e.g., Ref. [17]) taking into account
contributions from excitons, their excited states, and unbound states
to the dielectric functions of respective layers.
The calculations are made for two circular polarizations of the
Magneto-Optical Effects Enhancement... 5
3
−) σ −
n) (2.9 σ +
e( B = 1 T
R
x 2.8
e
d
n
e i2.7
v
cti
a2.6
efr A B C
R
23.54 80 3490 3500 3510 3520 3530 3540 3550 3560
Photon energy (meV)
Figure 2. Calculated real part of dielectric function of
Ga0.994Mn0.006N at B = 1 T and T = 2 K for σ− (solid line) and
σ+ (dashed line) polarizations of the light. Vertical lines indicate
respective A, B, and C exciton energies.
light and the case of external magnetic field of B = 1 T. The
determinationofexcitonenergiesisbasedonamodelinvolvingeffective
excitonic Hamiltonian of the following form [18,21–24]:
H = E0+Hvb+He−h+Hdiam+HZ +Hs,p−d. (2)
Intheabove equation E0 denotes the(Ga,Mn)N band-gapenergy,
H describes the valence band at k = 0 in a wurtzite semiconductor,
vb
He−h relatestothee-hexchangeinteractions withintheexciton, Hdiam
corresponds to an excitonic diamagnetic shift, H is an ordinary
Z
Zeeman excitonic Hamiltonian, and Hs,p−d accounts for s,p − d
exchange interaction between excitons and localized Mn ion spins.
Parameters of excitonic transitions and of the electronic band
structure in magnetic field are taken from Ref. [18] (or Ref. [15] in
the case of II-VI structure, considered below). The real part of the
dielectric function representing the refractive index of (Ga,Mn)N at B
= 1 T for σ− and σ+ circular polarizations of the light is depicted in
Fig. 2. The contribution to a dielectric function from A, B and C
excitons is seen. The excitons split in magnetic field, as indicated by
vertical lines. Splitting of excitons A and B occurs toward opposite
directions. It is also seen that the real parts of dielectric function for
σ− and σ+ polarizations differ only in the excitonic region. It will be
shown in the next section that the exciton splitting has a dominant
contribution to MOKE in the near the band gap region.
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15
d) with DBR
a w/o DBR
mr 10
e ( B = 1 T
gl 5
n
a
n
atio 0
ot
Kerr r −5 ←← e exxccitiotonn A A σ σ+−
−10 3510 3520 3530 3540 3550 3560
Photon energy (meV)
Figure 3. Calculated Kerr rotation angle versus photon energy at
B = 1 T for a structure containing (Ga,Mn)N layer with (dashed line)
and without (solid line) DBR underneath, as described in the text.
The energy position of the exciton A is indicated for both circular
polarizations of the light for a reference.
4. NUMERICAL RESULTS
Following the solution of equation 1 for parameters and dielectric
functions given in Section 2 and references cited there, we plot Kerr
rotation angle as a function of photon energy, as presented in Fig. 3.
Fig. 3 shows that the magnitude of the MOKE increases thanks
to the implementation of the proposed structure design. Comparison
of the peak-to-peak as well as area under the curve gives evidence
for around two times the MOKE enhancement. Furthermore, for
some photon energy values (e.g., 3506 - 3511 meV) the observed
enhancement is as large as four.
In Fig. 4 the Kerr rotation angle is plotted as a function of the
photonenergyandthenumberNofperiodsconstitutingtheDBR.The
magnitude of the MOKE increases with the increasing N, saturates for
about N = 4 and stays roughly unchanged after further increase of
N. The saturation effect seen in Fig. 4 indicates a limitation of the
proposed method for the MOKE enhancement, however Fig. 4 shows
that already for a number of the DBR periods as small as 3 or 4 the
MOKE is significantly enhanced with respect to ”no DBR” case. This
points toward ease of the sample fabrication, an important factor in a
view of possible practical implementations of the proposed design.
Finally, we apply the proposed sample design and conduct
respective calculations for the structure involving zinc-blende II-VI
DMS, (Cd,Mn)Te. In contrast to the wurtzite (Ga,Mn)N where three
excitons contribute to the dielectric function, in the case of CdTe
Magneto-Optical Effects Enhancement... 7
d)
a
mr 20
gle ( 10
n
n a 0
o
ati−10
ot
Kerr r−1250 10 3550 3560
3540
5 3520 3530
Number of periods (−) 0 3510 Photon energy (meV)
Figure 4. Calculated Kerr rotation angle for a structure involving
Ga0.994Mn0.006N combined with DBR layer as described in the text,
plotted versus photon energy and number N of periods constituting
DBR mirror.
based alloys, a zero field splitting of the valence band is typically
negligible and one deals with a single exciton contribution to the
material dielectric function. We set the (Cd,Mn)Te layer thickness
d = 76 nm, and Mn quantity amounting to 4%. The DBR layer
DMS
consistsofCd0.95Mg0.05TeandCd0.7Mg0.3Te,withthicknessesof58nm
and 70 nm, respectively. The number of periods within DBR amounts
to 3. In this design, there is no buffer layer, and the substrate is made
of GaAs. As can be seen in Fig. 5, presenting the Kerr rotation angle
as afunction of thephotonenergy, theuseof theBragg mirrorleads to
a few fold enhancement of the MOKE, quantified by the integrated of
the area under the curve. There are energy ranges where the MOKE
amplitude is even more enhanced, e.g., around 1610 meV and 1630
- 1650 meV. Thus, implementation of the DBR in the case of the
(Cd,Mn)Te provides a considerable MOKE increase. This indicates
the utility of the proposed design in the case of both, III-V and II-VI
DMS.
5. SUMMARY
In this paper, we present the theoretical design of diluted magnetic
semiconductor layer combined with the one-dimensional photonic
structure (Bragg mirror) and show that even for a relatively small
numberofperiodsandarelativelysmallcontrastofrefractiveindicesof
layers constitutingtheDBR,asignificantamplification oftheobserved
magneto-opticaleffects(MOKE)relatedtoDMSlayerisachieved. The
presented design allows for a broadening of the range of magnetic ion
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0.3 with DBR exciton A σ− →
e (rad) 0.2 B w= /0o. 1D TBR exciton A σ+ →
gl
n 0.1
a
n
o
ati 0
ot
err r−0.1
K
−0.2 1580 1600 1620 1640 1660 1680
Photon energy (meV)
Figure 5. Calculated Kerr rotation angle versus photon
energy at B = 0.1 T and T = 4.2 K for a structure containing
Cd0.96Mn0.04Te layer with (dashed line) and without (solid line)
Cd0.95Mg0.05Te/Cd0.7Mg0.3Te Bragg mirror underneath. The position
of the A exciton is indicated for both circular polarizations of the light
for a reference.
concentrations within the DMS for which MOKE could be observed.
In particular, it should enable detection of sizable MOKE even for
boundary (very low or very high) concentrations of the magnetic
dopant, where the quality of magnetooptical signal decreases. The
proposed design is valid for the case of both, wurtzite III-V and zinc-
blende II-VI DMS.
ACKNOWLEDGMENT
The work was supported by the European Research Council through
the FunDMS Advanced Grant (#227690) within the ”Ideas” 7th
Framework Programme of the EC, the Austrian Fonds zur F¨orderung
der wissenschaftlichen Forschung-FWF (P22477, P20065 and N107-
NAN), and by polish NCBiR project LIDER.
We thank Tomasz Dietl for valuable discussions.
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