Download Influence of the Magnetic Field on the Effective Mass and the

Journal of the Korean Physical Society, Vol. 57, No. 6, December 2010, pp. 1929∼1932
Influence of the Magnetic Field on the Effective Mass and the Rashba effect
in an In0.53 Ga0.47 As Quantum-well Structure
Youn Ho Park, Hyun Cheol Koo,∗ Kyung Ho Kim, Hyung-jun Kim, Joonyeon Chang and Suk Hee Han
Nano Convergence Device Center, Korea Institute of Science and Technology, Seoul 136-791, Korea
(Received 12 January 2010, in final form 22 July 2010)
From the temperature dependence of the Shubnikov-de Hass oscillation measurement, the effective
mass of an In53 Ga47 As quantum well is observed. The effective mass increases linearly from 0.050m0
with Ba = 1.3 T to 0.143m0 with Ba = 7 T. The external magnetic field strengthens the total
magnetic field inside the quantum well and increases the effective mass. These two resulting effects
make the spin precession in the quantum well channel fast.
PACS numbers: 85.70.-w, 85.75.-d, 75.70.-i, 85.75.Hh
Keywords: Rashba effect, Spin-orbit interaction, Effective mass, Two-dimensional electron gas
DOI: 10.3938/jkps.57.1929
I. INTRODUCTION
Manipulating the spin orientation of electrons in a
semiconductor is a promising method for use in the future electronic devices. The spin field effect transistor
(spin-FET), proposed by Datta and Das in 1990 [1], is
a good candidate for spin-based active devices to potentially replace conventional electronic devices. In the
paradigmatic spin-FET, the spins injected from a ferromagnet to a two-dimensional electron gas should be
controlled by a spin-orbit interaction (α) that can be
tuned by using the gate voltages. The parameter, α, is
a result of the Rashba effect in which the Rashba effective magnetic field (BR ) is induced in the y-direction by
moving electrons (kx ) with a perpendicular electric field
(Ez ). Previous studies showed successful spin injection
into semiconductor channels [2–4] and spin precession by
external magnetic field [3,4]. However, for device applications, the electric-field control of spin orientation is
more favorable than magnetic-field control. Gate controls of the spin-orbit interaction parameter and the injected spin orientation have been also reported [5] but
the electrical spin transport efficiency was still very low,
and the spin transport with gate modulation was implemented at low temperature.
The Shubnikov-de Hass oscillation (SdH) is usually
measured to observe α. The beat patterns of the SdH oscillation imply that two different frequency terms exist.
This means that the concentrations of spin-up and -down
electrons are different in the channel. From the temperature dependence of the oscillation amplitude, one can
also estimate the effective mass of an electron in a semi∗ E-mail:
[email protected]; Fax: +82-2-958-6851
conductor channel. In this research, we measured the
SdH oscillations at various temperatures to investigate
the external magnetic field dependence (Ba ) of the effective mass (m∗ ) in an In0.53 Ga0.47 As quantum-well structure. The spin precession angle is mainly decided by α,
which is proportional to the effective Rashba field. The
external magnetic field itself modifies the direction and
the amplitude of the Rashba field. In addition, the external magnetic field determines the effective mass of the
electrons. Therefore, the external magnetic field has an
effect on the spin precession in two ways. In the discussion part, we will also show the external field effect on
the spin precession angle.
II. EXPERIMENTS AND DISCUSSION
In order to measure the SdH oscillation, we utilize
an inverted high-electron-mobility transistor (HEMT)
structure with an In0.53 Ga0.47 As channel [6–8]. Figure
1 shows cross-sectional view of the heterostructure. The
carrier supply layer is located below the In0.53 Ga0.47 As
quantum-well layer and the channel layer is confined by
In0.52 Al0.48 As cladding layers. The top InAs is a capping
layer to protect against possible damage during the fabrication processes. The calculated energy diagram and
the charge distribution of the structure used in this experiment are also shown in Fig. 1. For the band calculation, WinGreen simulator and its data file are utilized [9]. The energy band gaps of In0.53 Ga0.47 As and
In0.52 Al0.48 As are 0.914 eV and 1.641 eV, respectively;
therefore, the energy difference of the band gaps produces the confined potential well. This potential difference is divided into the offsets of the conduction and the
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Journal of the Korean Physical Society, Vol. 57, No. 6, December 2010
Fig. 1. Cross-sectional view of the heterostructure (left)
and calculated energy band diagram with the charge density
of quantum well (right). z is the distance from the top surface. Solid and dotted lines correspond to the conduction
band potential and the charge density, respectively.
Fig. 3. Observation of the effective mass. (a) A plot of
ln(A/T ) versus T at a magnetic field Ba = 1.8 T. (b) External
magnetic field dependence of the effective mass.
Fig. 2. (Color online) Results of Shubnikov-de Hass measurements at temperatures of 1.8 K, 3 K, 4 K, 5 K, and 6
K. The curves are shifted to allow the amplitudes to be easily compared. The bias current is 15 µA. The inset is an
enlarged graph of the SdH oscillation at T = 1.8 K, which
clearly presents the node positions of the beat pattern.
valence bands. From the band simulation, we found that
the conduction band offset between In0.53 Ga0.47 As and
In0.52 Al0.48 As layers is 0.492 eV. As shown in Fig. 1, the
quantum well is asymmetric, and the charge inside the
channel is mainly concentrated on the side of the carrier supply layer, which induces a strong internal electric
field, which produces the Rashba magnetic field. In this
case, the spin-orbit interaction field arises even without
an external electric field. For the SdH oscillation experiment, a 64 µm wide Hall bar was defined by using
photolithography and ion milling. To test the channel
formation, we checked to ensure that no current flows in
the area outside the patterned channel. The channel resistance was measured in a perpendicular magnetic field
and the bias current is 15 µA.
Figure 2 shows the SdH oscillation results which yield
the spin-orbit interaction parameter and the effective
mass of the electron. The inset of Fig. 2 shows the beat
pattern with node positions at 1.8 K. The beat pattern
is due to the Rashba effect inducing a density imbalance
between spin-up and -down electrons. From the beat frequency (fbeat ) of the SdH oscillation and the relationship
of α = ~fbeat /2k0 m∗ , we can extract the spin-orbit interaction parameter, α, where k0 = (2πnS )0.5 is a mean
Fermi wave vector and ~ is the reduced Planck constant
[5,6,10]. Note that fbeat is obtained from the oscillation
curve of the channel resistance versus 1/Ba , so the unit
of fbeat is the same as that of Ba . At T = 1.8 K, we
estimate α to be 6.02 × 10−12 eVm and the carrier concentration, nS to be 2.01 × 1012 cm−2 . Note that spin-up
and -down subbands individually have various electron
valleys but the average masses of the two spin subbands
are almost same. We inserted the average effective mass
of all electrons.
We also observed the SdH oscillation at T = 1.8 K, 3 K,
4 K, 5 K, and 6 K, as shown in Fig. 2. The temperature
dependence of the amplitude of the SdH oscillation gives
us the effective mass of the electrons. The oscillatory
part, ∆ρxx , which is the magnetic field dependence of
channel resistivity, can be expressed as [11,12]
∆ρxx
χ
−π
2πε
=2
exp
cos
− π , (1)
2ρ0
sinh(χ)
ω c τq
~ωc
where χ = 2π 2 kB T /~ωc , ωc = eBa /m∗ , and =
π~2 n/m∗ . Note that τq is the quantum scattering time,
Ba is the magnetic field, kB is the Boltzmann constant,
T is the absolute temperature, and n is the electron con-
Influence of the Magnetic Field on the Effective Mass and the Rashba effect · · · – Youn Ho Park et al.
Fig. 4. External field dependence of the total magnetic
field (Btotal ), which is the vector sum of the Rashba field
(BR ) and the external field (Ba ).
centration in the channel. Approximating sinh(χ) by
exp(χ)/2, we can express the amplitude A of the SdH
oscillation as a function of the external magnetic field as
[12]
A
2π 2 kB m∗
ln
≈C−
T,
(2)
T
e~Ba
where C does not vary with temperature. Using Eq. (2),
ln(A/T ) as a function of T gives a straight line with a
slope of (-2π 2 kB m∗ /e~Ba ). From this relationship, we
can finally calculate the effective mass of electrons, m∗ .
Figure 3(a) shows a plot of ln(A/T ) versus T at Ba =
1.8 T, and it has a linear dependence on temperature.
Using the same procedure, the values of effective mass
are determined at several values of Ba as shown in Fig.
3(b). The field dependence of the effective mass is due to
the conduction band nonparabolicity [12]. For Ba < 1.3
T, we cannot precisely determine the effective mass because the amplitude of the SdH oscillations is not clear.
However, we can observe a weak field dependence of the
effective mass below Ba = 2 T. Therefore, with zero
magnetic field, the effective mass may approach 0.05m0 ,
which is the well-known value of the effective mass of
In0.53 Ga0.47 As bulk.
The spin-orbit interaction strength is expressed in the
form of an effective magnetic field. We can allow the
spin-splitting energy (2αk 0 ) to be the field-induced energy term (gµB BR ), where µB is the Bohr magneton.
The Rashba magnetic field, BR , is given by [13]
BR =
2αk0
,
µB g
(3)
where g is the electron g-factor. In our calculation, the
value of α at low fields (0T - 2T) is utilized and α generally does not vary with the external field. The external
magnetic field also changes the total magnetic field, as
well as the effective mass. As shown in Fig. 4, the amplitude and the direction of the total magnetic field (Btotal )
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Fig. 5. Spin precession angle for Ba = 0 and 3.02 T.
are decided by the vector sum of the Rashba field (BR )
and the external field (Ba ). With increasing external
field, the total field also increases, and the direction of
the total field rotates toward the direction of the external field. The total field increases up to 8.58 T when
an external field of 7.03 T is applied. In general, the
injected spins precess around the Rashba field, but with
an external field, the spin’s precession axis changes to
the direction of the total field.
Next, we consider the influence of the external magnetic field on the Rashba precession. In general, the spin
precession angle is θSP = 2m∗ xα/~2 , where xis the traveling distance of the spin-polarized electron. Using Eq.
(3), we can rewrite θSP as a function of the Rashba field
as θSP = µB gm∗ xBR /k 0 ~2 . Considering the external
field, BR is replaced by Btotal , and the expression becomes
θSP =
µB gm∗ xBtotal
.
k0 ~2
(4)
From Eq. (4), the spin precession angle is proportional
to m∗ Btotal . Note that when estimating θSP at Ba =
0, we use m∗ = 0.05m0 . As shown in Fig. 5, with an
external magnetic field, the spin precession angle is substantially increased. When x = 1 µm, θSP is 757◦ (452◦ )
with Ba = 3.02 T (0 T). The large precession angle is
advantageous for high-density applications because the
rapid precession allows the channel to be shorter. In
this calculation, we assumed that the spin orientation
of the injected electrons was perpendicular to the direction of the total field. In real applications, this assumption is not easily satisfied because the external field
affects the spin injector and detector. Therefore, with
an external magnetic field, the signal margin in real applications should be significantly reduced while the spin
precession frequency should be still enhanced. An alternative method to avoid the effect of the external field on
the spin source would be to develop a structure in which
only the channel is locally magnetized.
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Journal of the Korean Physical Society, Vol. 57, No. 6, December 2010
III. CONCLUSION
REFERENCES
In conclusion, we investigated the influence of the magnetic field on the Rashba effect in an In0.53 Ga0.47 As
quantum well by measuring the SdH oscillation. From
the temperature dependence of the SdH oscillations, we
obtained the external magnetic field dependence of the
effective mass of the electron by comparing the amplitudes of the oscillations. This result helped us to estimate the magnetic field effect on the Rashba effect
because the external magnetic field modifies the effective mass and the total induced magnetic field inside
the channel. The effective mass increases with increasing external magnetic field, Ba . The observed effective
mass changed from 0.050m0 to 0.143m0 when Ba was
increased from 1.3 T to 7 T. The spin precession angle
was also noticeably enhanced. For Ba = 3 T, the spin
precession angle increases by 67% if we assumes the injected spin orientation iss perpendicular to the direction
of the total magnetic field.
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ACKNOWLEDGMENTS
This work was supported by the Korea Institute of
Science and Technology (KIST) Institutional Program
and the Korea Research Council of Fundamental Sciences and Technology (KRCF) Degree and Research
Center (DRC) Program.