Vertical spin transport in semiconductor heterostructures
The Landauer—B ttiker formalism combined with the tight-binding transfer matrix method is employed to model vertical coherent spin transport within magnetization modulated semiconductor heterostructures based on GaAs. This formalism provides excellent physical description of recent experiments co...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Цитувати: | Vertical spin transport in semiconductor heterostructures / P. Sankowski, P. Kacman, J.A. Majewski, T. Dietl // Физика низких температур. — 2007. — Т. 33, № 2-3. — С. 256-262. — Бібліогр.: 34 назв. — англ. |
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Sankowski, P. Kacman, P. Majewski, J.A. Dietl, T. 2017-12-27T10:47:28Z 2017-12-27T10:47:28Z 2007 Vertical spin transport in semiconductor heterostructures / P. Sankowski, P. Kacman, J.A. Majewski, T. Dietl // Физика низких температур. — 2007. — Т. 33, № 2-3. — С. 256-262. — Бібліогр.: 34 назв. — англ. 0132-6414 PACS: 75.50.Pp, 72.25.Hg, 73.40.Gk https://nasplib.isofts.kiev.ua/handle/123456789/127727 The Landauer—B ttiker formalism combined with the tight-binding transfer matrix method is employed to model vertical coherent spin transport within magnetization modulated semiconductor heterostructures based on GaAs. This formalism provides excellent physical description of recent experiments concerning the high tunneling magnetoresistance (TMR) in (Ga,Mn)As-based trilayers and highly polarized spin injection in p-(Ga,Mn)As/n-GaAs Zener diode. For both the TMR and the Zener spin current polarization, the calculated values compare well with those observed in the experiments and the formalism reproduces the strong decrease of the observed effects with external bias. We ascribe this decrease to the band structure effects. The role played in the spin dependent tunneling by carrier concentration and magnetic ion content is also studied. This work was partly supported by the EC project NANOSPIN (FP6-2002-IST-015728). Calculations were carried out using the resources and software at Interdisciplinary Center of Mathematical and Computer Modelling (ICM) in Warsaw. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Структура и свойства полупроводников с переходными элементами Vertical spin transport in semiconductor heterostructures Article published earlier |
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Vertical spin transport in semiconductor heterostructures |
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Vertical spin transport in semiconductor heterostructures Sankowski, P. Kacman, P. Majewski, J.A. Dietl, T. Структура и свойства полупроводников с переходными элементами |
| title_short |
Vertical spin transport in semiconductor heterostructures |
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Vertical spin transport in semiconductor heterostructures |
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Vertical spin transport in semiconductor heterostructures |
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Vertical spin transport in semiconductor heterostructures |
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vertical spin transport in semiconductor heterostructures |
| author |
Sankowski, P. Kacman, P. Majewski, J.A. Dietl, T. |
| author_facet |
Sankowski, P. Kacman, P. Majewski, J.A. Dietl, T. |
| topic |
Структура и свойства полупроводников с переходными элементами |
| topic_facet |
Структура и свойства полупроводников с переходными элементами |
| publishDate |
2007 |
| language |
English |
| container_title |
Физика низких температур |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
The Landauer—B ttiker formalism combined with the tight-binding transfer matrix method is
employed to model vertical coherent spin transport within magnetization modulated semiconductor
heterostructures based on GaAs. This formalism provides excellent physical description of recent
experiments concerning the high tunneling magnetoresistance (TMR) in (Ga,Mn)As-based
trilayers and highly polarized spin injection in p-(Ga,Mn)As/n-GaAs Zener diode. For both the
TMR and the Zener spin current polarization, the calculated values compare well with those observed
in the experiments and the formalism reproduces the strong decrease of the observed effects
with external bias. We ascribe this decrease to the band structure effects. The role played in the
spin dependent tunneling by carrier concentration and magnetic ion content is also studied.
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/127727 |
| citation_txt |
Vertical spin transport in semiconductor heterostructures / P. Sankowski, P. Kacman, J.A. Majewski, T. Dietl // Физика низких температур. — 2007. — Т. 33, № 2-3. — С. 256-262. — Бібліогр.: 34 назв. — англ. |
| work_keys_str_mv |
AT sankowskip verticalspintransportinsemiconductorheterostructures AT kacmanp verticalspintransportinsemiconductorheterostructures AT majewskija verticalspintransportinsemiconductorheterostructures AT dietlt verticalspintransportinsemiconductorheterostructures |
| first_indexed |
2025-11-27T07:12:25Z |
| last_indexed |
2025-11-27T07:12:25Z |
| _version_ |
1850803074290941952 |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3, p. 256–262
Vertical spin transport in semiconductor heterostructures
P. Sankowski1, P. Kacman1, J.A. Majewski2, and T. Dietl1,2,3
1 Institute of Physics, Polish Academy of Sciences, 32/46 al. Lotnikow, Warszawa 02668, Poland
2 Institute of Theoretical Physics, Warsaw University, 69 ul. Hoza, Warszawa 00681, Poland
E-mail: jacek.majewski@fuw.edu.pl
3 ERATO Semiconductor Spintronics Project
Received September 22, 2006
The Landauer—B�ttiker formalism combined with the tight-binding transfer matrix method is
employed to model vertical coherent spin transport within magnetization modulated semiconduc-
tor heterostructures based on GaAs. This formalism provides excellent physical description of re-
cent experiments concerning the high tunneling magnetoresistance (TMR) in (Ga,Mn)As-based
trilayers and highly polarized spin injection in p-(Ga,Mn)As/n-GaAs Zener diode. For both the
TMR and the Zener spin current polarization, the calculated values compare well with those ob-
served in the experiments and the formalism reproduces the strong decrease of the observed effects
with external bias. We ascribe this decrease to the band structure effects. The role played in the
spin dependent tunneling by carrier concentration and magnetic ion content is also studied.
PACS: 75.50.Pp Magnetic semiconductors;
72.25.Hg Electrical injection of spin polarized carriers;
73.40.Gk Tunneling.
Keywords: ferromagnetic semiconductors, spin transport, tunneling magnetoresistance.
1. Introduction
With the recent discovery of ferromagnetic semi-
conductors [1] compatible with III–V epitaxy [2] the
field of spintronics has expanded from all-metal [3,4]
and hybrid metal—semiconductor [5] structures to in-
clude all-semiconductor ferromagnetic devices [6].
Such devices have revealed intriguing properties, such
as control of their Curie temperature with an applied
voltage [7]. This is a consequence of the carrier-
mediated nature of the ferromagnetism in these mate-
rials, which allows manipulation of the magnetic
properties through control of the electronic subsystem
(i.e., through doping, gating, etc.) [8].
(Ga,Mn)As is believed to be one of the most promi-
sing materials for semiconductor spintronics. First of
all, it has almost the same lattice constant as GaAs
and AlAs, and with the (Ga,Mn)As/III–V hetero-
structures one can form high-quality all-semiconduc-
tor spintronic devices, such as magnetic tunnel junc-
tions. In addition, (Ga,Mn)As, based devices might
be easily integrated within other III–V-based devices.
Moreover, in recent years the optimization of the con-
ditions of MBE growth and post-growth annealing of
(Ga,Mn)As has resulted in a still rapid increase of the
Curie temperature in this compound, which now
reaches 173 K [9].
Spin-dependent phenomena in modulated magnetic
semiconductors attract a lot of interest due to the pos-
sible applications in spintronic devices. One of the
fundamental prerequisites for construction of func-
tional spintronic devices, such as spin transistors [10],
is efficient spin injection from ferromagnetic regions
into the paramagnetic ones. On the other hand, the
tunneling magnetoresistance (TMR) effect, discov-
ered by J�lliere in a Co/Ge/Fe structure [11], has al-
ready found many applications in magnetic field
sensors and magnetic random access memories, for
example. Both these phenomena have been also
recently observed in (Ga,Mn)As-based structures. It
has been demonstrated that highly spin-polarized
electron current (about 80%) can indeed be obtained
from p-(Ga,Mn)As/n-GaAs Zener diode [12].
Also a high (of about 75%) TMR effect in
(Ga,Mn)As/AlAs/(Ga,Mn)As trilayers was presen-
© P. Sankowski, P. Kacman, J.A. Majewski, and T. Dietl, 2007
ted first by Tanaka and Higo [13]. Recent experi-
ments exhibit TMR even as high as 250% [14–16].
The further developement of spintronic devices re-
quires deep understanding of vertical spin transport
and a modeling tool that facilitates the design of
new devices. Since the ferromagnetic coupling in
(Ga,Mn)As is mediated by the holes [1,8], a meaning-
ful theory has to take into account the entire complex-
ity of the valence band, including the spin—orbit in-
teraction. Furthermore, the intermixing of valence
bands caused by spin—orbit coupling shortens the
spin diffusion length and makes it comparable to the
phase coherence length. This renders the models based
on the classical spin-diffusion equation, which de-
scribe satisfactorily the spin transport phenomena in
metallic MTJs, nonapplicable directly to the struc-
tures containing layers of hole-controlled diluted fer-
romagnetic semiconductors.
Recently, we have developed a computational
scheme for vertical coherent spin transport. This mo-
del combines the two-terminal Landauer—B�ttiker
formalism with the empirical multi-orbital tight-bind-
ing description of the semiconductor band structure.
In this way, the quantum character of spin transport
over the length scale relevant for the devices in ques-
tion is taken into account.
Furthermore, the tight-binding approach, in con-
trast to kp models employed so-far [17,18], allows for
a proper description of effects crucial for spin trans-
port in heterostructures such as atomic structure of in-
terfaces, effects of Rashba and Dresselhaus terms, as
well as tunnel involving k states away from the center
of the Brillouin zone. Our model has recently been ap-
plied to describe selected features of Zener—Esaki di-
odes [19,20] and TMR devices [20] and has also been
as well as it was adopted to examine an intrinsic do-
main-wall resistance in (Ga,Mn)As [21].
The present paper is organized as follows. In Sec. 2,
we present the basics of the formalism employed. In
Secs. 3 and 4, we discuss two basic phenomena men-
tioned above, namely, the injection of spin polarized
valence electrons into the conduction band of GaAs re-
alized in the p-(Ga,Mn)As/n-GaAs Zener diode, and
TMR effect in (Ga,Mn)As/GaAs/(Ga,Mn)As struc-
tures.
2. Theoretical model
We consider a prototype heterostructure which is
uniform and infinite in the x and y directions, has
modulated magnetization along the z ([001]) growth
direction, and is connected to two semi-infinite bulk
contacts denoted by L and R. In all cases considered,
we assume that the bias is applied in such a way that
spin-polarized carriers are injected from the ferromag-
netic left lead. Our goal is to calculate the electric
current in the structure and the degree of current spin
polarization outside the left lead. The typical length
of the structures studied is comparable to the phase
coherence length. Therefore, we restrict ourselves to
the vertical coherent transport regime that we treat
within Landauer—B�ttiker formalism, where the cur-
rent is determined by the transmission probability of
the ingoing Bloch state at the left contact into the out-
going Bloch state at the right contact. In the presence
of spin—orbit coupling, the spin is not a good quan-
tum number. The only conserved quantities in tunnel-
ing are the energy E and, due to the spatial in-plane
symmetry of our structures, the in-plane wave vector
k | | . We use semi-empirical tight-binding formalism to
calculate electronic states of the system for given k | |
and E and further compute transmission coefficients.
2.1. Tight-binding model
First, we describe the construction of the
tight-binding Hamiltonian matrix for «normal» GaAs
and AlAs as well as ferromagnetic (Ga,Mn)As layers
of the heterostructure. To describe the band structure
of the bulk GaAs and bulk AlAs, we use the nearest
neighbor (NN) sp d s3 5 * tight—binding Hamiltonian
(resulting in 20 spin-orbitals for each anion or cation),
with the spin-orbit coupling included [22]. This model
reproduces correctly the effective masses and the band
structure of GaAs and AlAs in the whole Brillouin
zone. With the tight-binding Hamiltonian introduced
above, each double layer (cation + anion) is repre-
sented by 40�40 matrix. It should be pointed out that
the d orbitals used in our sp d s3 5 * parametrization are
not related to the 3d semi-core states and are of no use
for description of Mn ions incorporated into GaAs.
The presence of Mn ions in (Ga,Mn)As is taken into
account by including the (sp—d)-exchange interac-
tions within the virtual crystal and mean-field approx-
imations. In the spirit of the tight-binding method,
the effects of an external interaction are included in
the on-site diagonal matrix elements of the tight-bind-
ing Hamiltonian. Here, the shifts of on-site energies
caused by the (sp—d)-exchange interaction are
parameterized in such a way that they reproduce ex-
perimentally obtained spin splitting: N0 0 2� � . eV for
the conduction band, and N0 12� � � . eV for the va-
lence band [23]. The other parameters of the model for
the (Ga,Mn)As material and for the NN interactions
between GaAs and (Ga,Mn)As are taken to be the
same as for GaAs. This is well motivated because the
valence-band structure of (Ga,Mn)As with small frac-
tion of Mn has been shown to be quite similar to that
of GaAs [23]. Consequently, the valence band offset
between (Ga,Mn)As and GaAs originates only from
Vertical spin transport in semiconductor heterostructures
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 257
the spin splitting of the bands in (Ga,Mn)As. The
Fermi energy in the constituent materials is deter-
mined by the assumed carrier concentration and is
calculated from the density of states obtained for
tight-binding Hamiltonian. Our calculations of the
Fermi energy for various hole concentrations are con-
sistent with the corresponding results presented in
Ref. 8.
Having determined the Hamiltonian of the system,
we are now in the position to define the current
and current spin polarization in the presence of
spin—orbit coupling.
2.2. Current and current spin polarization
For a given energy E and in-plane wave-vector k | | ,
the Bloch states in the left L and right R leads (i and j,
respectively) are characterized by the wave vector
component k� perpendicular to the layers and are
denoted by | , , ,L kL i� � and | , , ,R kR j� �, respectively. The
indices i and j indicate all possible pairs for 40 bands
described by the tight-binding Hamiltonian. The
transmission probability TL k R kL i R j, ,, , , ,� �� is a func-
tion of the transmission amplitude
t EL k R kL i R j, , | |, , , ,
( , )
� �� k and group velocities in the
left and right lead, vL i, ,� and vR j, ,� :
T E
t E
L k R k
L k R k
L i R j
L i R j
, , | |
, , |
, , , ,
, , , ,
( , )
( ,
� �
� �
�
�
�
�
k
k� �|
, ,
, ,
) .
2 v
v
R j
L i
�
�
(1)
The current flowing in the rightward direction can
now be written as [24]
j
e
d k dEf E
T
L R
BZ
L
L k R k
k
L i R j
L
�
�
�
�
�
�
�
� �
( )
( )| |
, ,, , , ,
2 3
2
� �
, , , ,
, , , ,
,
,
| |( , ),
� �
� � �
�
i R j
L i R j
k
v v
E
0
k (2)
where fL or respectively fR are the electron Fermi
distributions in the left and right interface and i j,
number the corresponding Bloch states. Plugging in
the expresion given in Eq. (1) and using the time re-
versal symmetry:
T E T EL k R k L k R kL i R j R j L i, , | | , ,, , , , , , , ,
( , ) ( ,
� � � �� � � ��k k | | ) ,
(3)
we get
� �j
e
d k dE f E f E
BZ
L R
k k
v
L i R j
L
�
�
�
�
�
� �
�
( )
( )— ( )| |
,, , , ,
,
2 3
2
� �
� �
, , ,
, , , ,
,
, , | |
, ,
, ,
( , )
i R j
L i R j
v
L k R k
R j
L i
t E
v
v
�
� �
�
�
�
�
�
0
2
k . (4)
To calculate the current one has to determine the
transmission probability, and thus the transmission
amplitude t EL k kL i R R j, | |, , , , , ( , )
� �� k and the group ve-
locities vL j, ,� of the ingoing and vR j, ,� of the outgo-
ing states. These can be obtained by solving the
Schr�dinger equation for the structure with appropri-
ate scattering boundary conditions. In our studies, we
follow closely the procedure detailed in Refs. 25 and
26, which we have generalized to the case with
spin—orbit coupling.
In the studies of the Zener tunneling diode, we are
mostly interested in the degradation of the spin polar-
ization of carriers passing from the source (Ga,Mn)As
to a paramagnetic n-GaAs drain. To quantify this deg-
radation, we introduce the spin polarization of the
outgoing Bloch state [27] defined as
P E R k R kR k R i R iR i, | | , , , ,, ,
( , ) , | | , ,
�
� � �� �k s٠(5)
where Ω is the direction of the spontaneous magneti-
zation in the ferromagnetic semiconductor and s is the
spin operator. Then, we can define the total spin po-
larized current:
j
e
d k dE f E f Es
BZ
L R
k k
v
L i R j
�
�
�
�
�
� �
( )
[ ( )— ( )]| |
,, , , ,
2 3
2
� �
L i R j
L i R j R i
v
L k R k R kT E P
, , , ,
, , , , , ,
,
, , | | ,( , ) .
� �
� � �
�
��
0
k (6)
The spin polarization of the coherently transmitted
current is now equal to
P
j
js
s� . (7)
This scheme follows closely some ideas introduced re-
cently in studies of the spin depolarization of holes
in the (Ga,Mn)As/GaAlAs/p-GaAs structures [27]
and will be employed to analyze the spin polarization
of current in the Zener tunnel diode.
3. Results and discussion
In this Section we employ the formalism described
in the previous Section to the structures studied exper-
imentally. We start the discussion with the Zener tun-
nel diode and further we discuss TMR structures
based on (Ga,Mn)As ferromagnetic semiconductors.
258 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3
P. Sankowski, P. Kacman, J.A. Majewski, and T. Dietl
3.1. Zener tunnel diode
A rather high
80% spin polarization of the tunnel-
ing electrons from the valence band of (Ga,Mn)As to
the GaAs conduction band has been recently obtained
in Zener diodes [12,28]. This opens new perspectives
for applications of electron spin injection. However,
the experimentally observed degree of current spin po-
larization exhibits a sharp decrease with the applied
bias [12,28]. In the present paper we analyze theoreti-
cally the transport in the Zener diode structures as a
function of the applied bias employing the developed
theoretical scheme involving transfer matrix formal-
ism described in Sec. 2. However, within this formal-
ism, one is not able to consider the whole structure in-
vestigated experimentally, because of computational
constraints. Therefore, we perform the analysis of the
intriguing phenomenon of current drop with applied
bias in two steps. First, we model the charge transport
in the entire Zener diode (corresponding to the experi-
mental structure) under bias using continuum theory
and a commercial simulation tool [19]. This allows
self-consistent calculations of semiconductor hetero-
structures taking into account band-to-band tunnel-
ing, recombination, and impact ionization. By these
simulations a detailed understanding of the charge dis-
tribution and the band lineup in the entire device can
be achieved. Next, we consider the most essential part
of the device consisting of p-Ga1�xMn xAs/n-GaAs
tunnel junction using the calculated potential profile
in the tight-binding Hamiltonian. According to the
self-consistent calculations, the scattering region con-
sists of approximately 120 monolayers of
Ga1�xMn xAs and GaAs, and corresponds roughly to
the depletion zone of the diode. In Fig. 1 we show the
calculated potential profile in the structures used in
experiments [12]. This potential profile has been ob-
tained in self-consistent calculations for external re-
verse bias ofV0 = 1.8 V, i.e., the voltage for which the
interband tunneling process starts. This potential pro-
file is the basis for the calculation of the spin-current
polarization within the transfer matrix formalism. An
additional component of the potential profile used in
the transfer matrix formalism is the bias �V that cor-
responds to the potential drop across the Zener diode
when the bias on the whole spin-LED is changed. We
assume a linear potential drop which fairly well ren-
ders the real band bending across the p–n junction.
In Fig. 2 the dependence of the current spin polar-
ization on the external bias �V in the Zener tunneling
diode is depicted. One can see that at low bias, the
current spin polarization is of the order of 0.6, in good
agreement with the experimental results (0.7–0.8).
Interestingly, the current spin polarization decreases
rapidly with �V. The strong dependence of current
spin polarization on bias is again in agreement with
the experimental findings, though a direct comparison
is hampered by the fact that the exact relation be-
tween �V and the total bias V applied to the device is
known only from simulations. Having these obstacles
in mind and to reduce computational burden, the cal-
culations for higher bias (�V � 0.1 V) have been per-
formed with lower accuracy. The latter is probably re-
sponsible for the spread of current spin polarizations
obtained in this region.
It is pretty obvious that the degree of spin polariza-
tion of the tunnel current may depend on these intrin-
sic features of (Ga,Mn)As layers. For example, it is
Vertical spin transport in semiconductor heterostructures
Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3 259
AlGaAs AlGaAsGaAs
(N+)
(N) (P) (P)
(P+)
vb
Cb
Zener diode
G
aA
s
(G
a,
M
n
)A
s
Fig. 1. Potential profile for the entire experimental struc-
ture at an external bias of V0 = 1.81 V. In the calculations
of the coherent current only the Zener diode region is
considered.
0.2
0 0.1 0.2 0.3 0.4 0.5
0.4
0.6
0.8
1.0
�V, V
C
u
rr
e
n
tp
o
la
ri
za
tio
n
Fig. 2. The calculated current spin polarization as a func-
tion of external bias for the Zener diode.
well known that magnetic characteristics of
(Ga,Mn)As depend strongly on both hole and manga-
nese concentrations [8]. To investigate this problem,
we have calculated the dependence of current spin po-
larization on the hole concentration p and Mn content
x. For these calculations we assume that the electron
concentration is n � 1019 cm �3 as indicated by the ex-
perimental results in Ref. 12. The dependencies of the
current spin polarization on the hole concentration p
and Mn content x � 0 08. are depicted in Fig. 3. These
results show a strong decrease of the tunneling current
spin polarization with the increase of hole concentra-
tion. The spin injection in the Zener diode depends
also crucially on the content of magnetic ions in the
Ga1�xMn x layer. For x � 0 08. , we obtain the spin
current polarization of the order of 60%, which agrees
fairly well with the observations in [12].
Another problem in studies of spin currents is the
role of anisotropy, since (Ga,Mn)As layers exhibit a
variety of anisotropic properties [8,29,30]. We have
also studied anisotropy effects in spin polarization of
the tunneling current in the Zener diode within devel-
oped the scattering formalism developed, gaining
some insight into physical mechanism leading to such
anisotropy [31].
3.2. Tunneling magnetoresistance
Now we demonstrate how the whole TMR semicon-
ductor structures can be modeled by the formalism de-
veloped. In our calculations of the TMR effect we con-
sider the structure consisting of three layers. The two
half-infinite leads are build made of ferromagnetic
p-type Ga1�xMn xAs. The middle, scattering region is
composed of the nonmagnetic GaAs, which forms
a barrier for the holes. We compare the tunneling
currents in two configurations, i.e., with parallel
(ferromagnetic — FM) and the antiparallel (anti-
ferromagnetic — AFM) alignments of the spontane-
ous magnetizations in the leads. We define the TMR
as
TMR �
I I
I
FM AFM
AFM
—
, (8)
where IFM and IAFM are the currents in the FM and
AFM configurations of the spontaneous magneti-
zations in the leads, respectively. In Fig. 4,a the
TMR values obtained, for a given (8%) Mn content
and a set of different hole concentrations in the FM
layers are plotted as a function of the applied bias.
As can easily be seen from the Fig. 4, TMR depends
strongly on the hole concentration. As TMR is deter-
mined primarily by the spin polarization of the carri-
ers at the Fermi level, this is in agreement with the
result of Ref. 8 stating that the higher the hole con-
centration the smaller is the polarization at the Fermi
level. For p � 3 5 1020. cm �3, which is the typical
hole concentration in (Ga,Mn)As samples with high
Mn content [32], a TMR of about 250% has been ob-
tained.
Recent experiments show that in Ga1�xMn xAs
samples the Fermi energy of the hole liquid remains
equal about 220 meV for a wide range of Mn content
and that the hole concentration depends very little on
x [33] — thus, in the following we have calculated a
TMR for different x in the magnetic layers, while the
hole concentration in these regions is kept equal to
p � 3 5 1020. cm �3. The results of this study are pre-
sented in Fig. 4,b.
Surprisigly enough, our simple model fully repro-
duces the experimental data: for structures with 8% of
260 Fizika Nizkikh Temperatur, 2007, v. 33, Nos. 2/3
P. Sankowski, P. Kacman, J.A. Majewski, and T. Dietl
1001
0
0.2
0.4
0.6
0.8
1.0
a
0 0.04 0.08
b
C
u
rr
e
n
ts
p
in
p
o
la
ri
za
tio
n
x
10
0.2
0.4
0.6
0.8
1.0
C
u
rr
e
n
ts
p
in
p
o
la
ri
za
tio
n
p, 10 cm19 –3
Fig. 3. Spin current polarization in p-Ga1�xMnxAs/n-GaAs
as a function of hole concentration p for x � 008. (a); Mn
content x for p � �35 1020. cm�3 (b). The bias applied to
the structure is V � 005. V.
1
2
3
4
5
6
7
2.5
2.0
1.5
1.0
0.5
1 10 100 0 0.04 0.08
x
3.0
T
M
R
T
M
R
p, 10 cm19 –3
a b
Fig. 4. TMR in Ga1�xMnxAs/GaAs/Ga1�xMnxAs as a
function of hole concentration p for fixed Mn content
x � 008. (a); as a function of Mn content for fixed hole
concentration p � �35 1020. cm�3 (b). The bias applied to
the structure is V � 005. V.
Mn we obtain the TMR of the order of 250%, as ob-
served recently by Chiba et al. [15]; for 4% Mn the
calculations lead to a TMR of the order of 60%, in per-
fect agreement with the experimental observations
[13,34]. Therefore, our calculations seem to suggest
that for obtaining high TMR, large exchange split-
tings, i.e., high concentrations content of magnetic
ions, are needed. Unfortunately, the presented in
Fig. 4 dependence suggests that the attempts to in-
crease the hole concentration in (Ga,Mn)As, in order
to obtain higher Curie temperature, may result in
much smaller TMR in the structure.
4. Summary
We have analyzed the vertical coherent spin
transport in (Ga,Mn)As-based heterostructures
using a tight-binding model together with the
Landauer—B�ttiker formalism. Our studies reproduce
quantitatively the recently observed high TMR
in (Ga,Mn)As/(Al,Ga)As/(Ga,Mn)As trilayers.
Within the formalism we are able to study spin polar-
ization of the current. The theoretical calculations of
(Ga,Mn)As/(Al,Ga)As Zener diodes demonstrate
large spin polarization of the injected current in excel-
lent agreement with experimental results. The model
reproduces as well the experimentally observed strong
dependence of the spin polarization of the injected
current and TMR effect on the applied bias voltage,
both in Zener and TMR heterostructures. It should be
pointed out that our calculations do not take into ac-
count, e.g., the interfacial roughness or the scattering
on impurities or defects, magnons, and other physical
effects that were previously ascribed to this intriguing
bias dependence. Instead, the formalism employed de-
scribes carefully the electronic structure of the
heterostructure, especially at the interfaces. In con-
trast to the standard ( )k p -method, the scattering for-
malism based on the tight-binding scheme takes into
account all the effects resulting from the electric field
in the depletion zone, in particular Rashba and
Dresselhaus terms which are essential for the lost of
spin polarization. These features make the approach
particularly suited for studying phenomena related to
spin-polarized tunneling.
This work was partly supported by the EC project
NANOSPIN (FP6-2002-IST-015728). Calculations
were carried out using the resources and software at
Interdisciplinary Center of Mathematical and Com-
puter Modelling (ICM) in Warsaw.
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