1
3 074
A. B. HENRIQUES et al.
55
Equation ͑5͒ was obtained assuming the quantum mobility to
be constant within a miniband. Note also that when the spac-
ing between the doped layers is made very large (d→ϱ), the
The Shubnikov–de Haas experiment was carried out in a
superconducting magnet. The sample was placed in liquid
He at 4.2 K. The magnetoresistance measurements were
made in the constant-current mode, employing currents of
ϳ10 A, and using a four-contact geometry; the samples
were approximately square, with contacts in the corners. The
magnetoresistance oscillations were measured in magnetic
fields of intensity up to 14 T.
width of the minibands decreases (⌬ →0), and Eq. ͑5͒ takes
n
the familiar form associated with an electron gas confined in
2
0
two dimensions:
⌬
xx
X
Ϫ␣ u
n
ϳϪ2
e
cos2 u .
͑7͒
n
0
sinhX
IV. RESULTS
In the most general case, Eq. ͑5͒ will be characterized by two
oscillatory components, respective to the ‘‘belly’’ and
We obtained an experimental estimate of the quantum
mobility in each of the populated minibands by using a se-
quential process, which consisted of the following steps: ͑1͒
determination of the characteristic width of the doped layer
and of the doping period; ͑2͒ determination of the sheet car-
‘‘neck’’ extremal orbits of the mini-Fermi surface associated
with the miniband; however, as shown in Ref. 19, in periodi-
cally ␦-doped systems the neck orbit is not detectable. This
conclusion is based upon the observation that in order for the
neck orbit to be detectable, the energy-level broadening must
be smaller than the miniband width and also smaller than the
minigap to the higher-energy miniband. However, in periodi-
cally ␦-doped systems these two conditions are never met
simultaneously, hence each miniband will manifest itself
with a single oscillatory component due to the belly orbit.
This assertion leads us to conclude that in the case of peri-
odically ␦-doped semiconductors the quantum mobility ap-
pearing in Eq. ͑5͒ pertains to the belly orbit state. Therefore
if we wish to test the quantum mobility theory developed
above, the quantum mobility obtained from a fit of Eq. ͑5͒ to
the experimental data should be compared to the output of
Eq. ͑4͒ at the wave vector associated with the belly orbit
rier concentration n ͑this process outputs simultaneously the
S
width, ⌬ , Fermi energy, , and dispersion factor, ⑀ , for
n
n
n
all minibands taken into consideration͒; ͑3͒ isolation of the
magnetoresistance oscillations associated with individual
minibands; and ͑4͒ determination of the quantum mobility of
a populated miniband by fitting the appropriate equation ͓ei-
ther Eq. ͑5͒ or ͑7͔͒ to the isolated magneto-oscillatory com-
ponent, with the quantum mobility being the single fitting
parameter. In what follows, each of these steps is described
in more detail.
A. Determination of the characteristic doping layer width
and doping period
͑
i.e., kϭ0 for minibands E1, E3, . . . , and kϭ/d for mini-
It has been shown by Ulrich et al.22 that the C-V spec-
trum of a ␦-doped semiconductor is very sensitive to the
spreading of the impurity atoms. In order to estimate the
characteristic width of the doped layer in our samples, we
proceeded in the following way. Sample No. 187 with a
single Si ␦ layer was grown under the same growth condi-
tions used for the periodically ␦-doped samples. In order to
take advantage of the fact that the C-V technique presents a
higher resolution for a higher sheet concentration of confined
carriers, sample No. 187 was more highly doped than the
other samples used in this work. The areal density of con-
fined carriers in this sample was obtained from the
Shubnikov–de Haas spectrum by following the prescription
described in Ref. 5. Using the density of carriers obtained
from the SdH spectrum, theoretical C-V profiles were gen-
erated by resolving self-consistently Schr o¨ dinger and Pois-
bands E2, E4, . . . ).
For those minibands whose bandwidth is larger than the
Fermi energy, i.e. when ⌬ Ͼ , the magnetoresistance os-
cillations can be approximated by the usual expression asso-
ciated with an unrestrained free electron gas in three
dimensions,
n
n
2
1
eϪ␣u
⌬
xx
X
ϳϪ2
cos
ͩ
2 uϪ
ͪ
.
͑8͒
n
0
sinhX ͱu
4
The quantum mobility in this case can similarly be obtained
by fitting Eq. ͑8͒ to the isolated magnetoresistance oscilla-
tory component.
III. EXPERIMENT
The ␦-doped structures were grown at 640 °C by low
pressure-MOVPE in an AIX 200 reactor at 20 mbar with
a growth rate of 4.5 Å/s. The source materials were
PH ͑100%͒, TMIn ͑trimethyl indium͒, and 1% SiH diluted
son equations for the structure under bias; in the calculation
of the C-V spectrum, the doped layer was assumed to be of
Gaussian profile, and the width of the Gaussian was varied
until the theoretical C-V spectrum achieved best agreement
with the experimental one. A more detailed description of
the C-V technique and theory is presented in Ref. 2. Figure 1
shows the C-V spectrum for sample No. 187. The theoretical
curve shown in Fig. 1 was calculated assuming the donor
layer to be of width 8 Å, and it reproduces very well the
experimental C-V spectrum. This demonstrates that Si atoms
in the ␦-doping layers in our samples are spread over not
more than 2 ML of InP.
3
4
in H . On ͑100͒ Fe-doped InP substrates, first a 0.3-
2
m-thick undoped buffer was grown, followed by a periodi-
cally ␦-doped InP layer, and finally a 500-Å cap layer was
deposited. The periodical structures are composed of either
five or ten periods, with a spacing varying from 90 to 300 Å.
The dopant layers were deposited during a 25-s growth in-
terruption. The silane flux was triggered 2 s after growth
interruption, and halted 2 s before the growth was resumed.
In order to determine the width of the doped layer and the
doping period, the capacitance-voltage profile (C-V) was
measured for each sample, using an electrochemical profiler
PN4300.
To determine the doping periods of a sample, its C-V
spectrum was measured. As an example, the C-V spectrum
for sample No. 200 is shown in Fig. 2. The C-V concentra-
tion, NC-V , presents oscillations as a function of the C-V
Henriques
