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arXiv:1109.0733v2 [cond-mat.mes-hall] 18 Oct 2011
Isotope sensitive measurement of the hole-nuclear spin interaction in quantum dots
E. A. Chekhovich1, A. B. Krysa2, M. Hopkinson2, P. Senellart3, A. Lemaˆıtre3, M. S. Skolnick1, A. I. Tartakovskii1
1Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK
2Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, UK
3Laboratoire de Photonique et de Nanostructures, Route de Nozay, 91460 Marcoussis, France
(Dated: October 19, 2011)
Decoherence caused by the hyperfine interaction with
nuclear spins is known to be the main obstacle on the
way to realization of quantum computation using sin-
gle electron spins [1–3], which led to proposals to use
valence band holes having a weaker hyperfine coupling
[4–9]. Although the hole hyperfine interaction has been
addressed recently both theoretically [10, 11] and ex-
perimentally [12–14], full understanding of the the un-
derlying physics is still lacking. Here we report on ex-
perimental measurements of the hole hyperfine inter-
action strength in three different material systems: un-
strained GaAs/AlGaAs quantum dots (QDs), and self-
assembled InGaAs/GaAs and InP/GaInP QDs. In con-
trast to previous studies we use resonant radio fre-
quency (rf) excitation to achieve selective measurement
of the hole hyperfine constant for individual isotopes.
This allows to avoid the ambiguity of previous mea-
surements relying on detection of the total Overhauser
shifts including contributions of all isotopes. We find
that the hole hyperfine constant (normalized by the
electron hyperfine constant) changes sign for different
isotopes and ranges from -15% for indium to +15%
for arsenic, revealing strong anisotropy of the dipole-
dipole hyperfine interaction. Moreover, the hole hy-
perfine constant varies for the same isotope in differ-
ent materials opening the way for better understanding
and possible optimization of the hyperfine interaction
for applications using single hole spins.
Due to the s-type of the wavefunction, the hyperfine
interaction of the conduction band electrons has a sim-
ple isotropic form (the Fermi contact interaction) and is
described by a single constant A, depending on isotope
and material [15]. Since all nuclei in III-V semiconduc-
tors have positive nuclear moments, their electron hyper-
fine constants are also positive: A > 0. By contrast, for the
p-type valence band holes, the contact interaction vanishes,
and the non-local dipole-dipole interaction dominates. As
a result, the hole hyperfine interaction is anisotropic and
its strength depends strongly on the actual form of the
Bloch wavefunction, which is usually difficult to estimate
with sufficient precision. Indeed, for a long time, it was
wrongly assumed that the hole hyperfine interaction is neg-
ligibly small. Only recently it was shown first theoretically
[10, 11] and then experimentally [13, 14] that it can be as
large as 10% of that for the electron. However, despite
its simple fundamental nature, understanding of the hole-
nuclear interaction is far from being complete.
In this work we address this problem experimentally. We
perform direct measurements of the hole hyperfine con-
stants by independently detecting electron and hole hy-
perfine shifts. This is achieved by using high resolution
photoluminescence (PL) spectroscopy of optically forbid-
den (”dark”) exciton states in single neutral quantum dots
[14]. In contrast to previous work, where similar tech-
niques were used [14], we now also apply radio-frequency
(rf) excitation, which allows selective saturation of nuclear
polarization of different isotopes. This opens the way to
isotope-sensitive probing of the valence band hole hyper-
fine interaction, revealing that in all studied materials group
III isotopes (gallium, indium) have negative hole hyperfine
constant, while it is positive for group V arsenic.
Our experiments were performed on undoped
GaAs/AlGaAs [16] and InGaAs/GaAs QD samples
without electric gates. PL of neutral QDs was measured
at T= 4.2K, in external magnetic field Bznormal to the
sample surface. QD PL was analyzed with a 1 m double
spectrometer and a CCD.
In a neutral dot electrons ↑(↓) with spin se
z=±1/2
and heavy holes ⇑(⇓) with momentum jh
z=±3/2par-
allel (antiparallel) to the growth axis Oz can form either
optically-forbidden (”dark”) excitons |⇑↑i (|⇓↓i) with the
spin projection Jz= +2(−2), or ”bright” excitons |⇑↓i
(|⇓↑i) with Jz= +1(−1) optically allowed in σ+(σ−)
polarization. QD axis misorientation or symmetry reduc-
tion leads to weak mixing of ”bright” and ”dark” states: as
a result the latter are observed in PL [17, 18].
Non-zero average nuclear spin polarization hIzialong
the Oz axis acts as an additional magnetic field on the
electron and hole spins. It is convenient to introduce the
hole pseudospin Sh
z=±1/2corresponding to the ⇑(⇓)
heavy hole state. Coupling of the electron to the nuclear
spin of isotope iis described by the hyperfine constant
Ai, whereas for the heavy hole the dipole-dipole interac-
tion with hIzi[10, 11] is described using a constant Ci
expressed in terms of the normalized heavy-hole hyperfine
constant γias Ci=γiAi. The expression for the exciton
energy taking into account the shift due to non-zero aver-
age nuclear spin polarization can be written as:
E[Sh
z, se
z] = EQD +E0[Sh
z, se
z] +
+se
zX
i
ρiAihIi
zi+Sh
zX
i
ρiγiAihIi
zi,(1)
where the quantum dot band-gap EQD and shift
E0[Sh
z, se
z]determined by the Zeeman and exchange en-
2
60 80 100
probe & PL detection
rf pulse
(e)
Experiment cycle
pump
75As 71Ga
69Ga
115In
rf power spectral
density (arb.units)
Band Z
Band Y
Band X
(d)
Frequency (MHz)
1.7130 1.7135
BZ=3.2 T
GaAs/AlGaAs
(a)
|›`Ò
|flcÒ
|›cÒ
|fl`Ò
PL energy (eV)
PL Intensity (arb. units)
1.3575 1.3580 1.3585
PL energy (eV)
〈IZ〉 < 0
〈IZ〉 > 0
BZ=8 T
InGaAs/GaAs
(b)
|›`Ò
|flcÒ
|›cÒ
|fl`Ò
645
650
655 57.75 58.00
Spectral splitting (µeV)
(c)
75As
81.00 81.25
Frequency (MHz)
69Ga
103.00 103.25
71Ga
FIG. 1. (a,b) Photoluminescence spectra of a single GaAs/AlGaAs QD at Bz≈3.2 T (a) and an InGaAs/GaAs QD at Bz≈8.0 T
(b) measured at negative (open symbols) and positive (solid symbols) nuclear spin polarization hIziinduced on the dot. Both bright
excitons |⇑↓i,|⇓↑i and both dark excitons |⇑↑i,|⇓↓i are observed. (c) Optically detected NMR spectrum of a single GaAs QD
at Bz≈8.0 T. Gallium resonances with widths of ∼30 kHz are observed at ≈81.21 MHz and ≈103.17 MHz for 69 Ga and 71Ga
respectively. Arsenic resonance observed at ≈57.9 MHz has a linewidth of ∼100 kHz, which is determined by residual elastic strain.
(d) Schematic diagram of the radio-frequency excitation spectrum used to erase nuclear polarization of different isotopes in InGaAs
QDs at Bz≈8.0 T. The solid vertical bars show resonance frequencies of gallium and arsenic derived from (c), while the dashed line
shows the calculated central frequency ≈74.14 MHz of 115In. Bands X and Z are used to destroy nuclear polarization of 75 As and
71Ga respectively. Band Y is used to erase polarization of both 115In and 69Ga simultaneously. (e) Timing diagram of the pump-probe
experiment used in the measurements of the hole hyperfine constants.
ergy [17] do not depend on nuclear polarization. The sum-
mation goes over all isotopes icontributing to the Over-
hauser shift. The relative concentration of each isotope is
given by ρi. We note that in Eq. 1 we neglect any pos-
sible variation of average nuclear spin polarization within
the volume of the QD. Also we neglect any difference of
isotope concentrations within the electron and hole local-
ization volumes.
Since mixing of the ”dark” and ”bright” excitonic states
is weak, the oscillator strength of the ”dark” states is small,
leading to their saturation at relatively low laser powers. As
a result, all four exciton states can be observed in PL only
at low excitation power. However, at this low power, opti-
cally induced nuclear spin polarization is small and weakly
depends on polarization of photoexcitation [18], and thus
the shifts of the hole spin states due to the interaction with
the nuclei cannot be measured accurately. To avoid this
problem, we use pump-probe techniques [19]. The experi-
3
ment cycle shown in Fig. 1 (e) is similar to that used in our
previous work [14]: nuclear spin polarization is prepared
with a long (6 s) high power pump pulse. Following this,
the sample is excited with a low power probe pulse, during
which the PL spectrum of both bright and dark excitons is
measured. However, in contrast to previous work, we now
add an rf pulse between the pump and probe pulses. This
pulse is formed by an oscillating magnetic field perpendic-
ular to Oz at a frequency tuned to erase transitions between
nuclear spin states of a chosen isotope (isotopes).
The direct and simultaneous measurement of the hole
and electron energy shifts due to the hyperfine interaction
is carried out by detecting the probe spectra recorded at
different magnitudes of hIi
ziprepared by the pump. Typ-
ical probe spectra of GaAs and InGaAs QDs are shown
in Figs. 1 (a) and (b) respectively. If the rf pulse con-
tains frequency components resonant only with isotope k,
then the exciton energies Ek[Sh
z, se
z]detected by the probe
pulse will be given by Eq. 1 with the sum going only
over the isotopes for which i6=k. We also perform an
experiment with no rf pulse which gives exciton energies
E[Sh
z, se
z]corresponding to initial polarization. Calculat-
ing the difference of the exciton splittings for experiments
with and without rf pulse allows to extract the electron and
hole hyperfine shifts for the k-th isotope. For example,
according to Eq. 1, the experimentally measured quan-
tity ∆Ek
hf,e = (E[⇑,↑]−E[⇑,↓]) −(Ek[⇑,↑]−Ek[⇑
,↓]) gives the magnitude of the electron hyperfine shift
ρkAkhIk
ziinduced only by nuclei depolarized by the rf
pulse (and thus corresponding to isotope kpolarization),
whereas the hole hyperfine shift ρkγkAkhIk
ziis given by
∆Ek
hf,h = (E[⇑,↑]−E[⇓,↑]) −(Ek[⇑,↑]−Ek[⇓,↑]).
From Eq. 1 we find that when nuclear spin polarization is
varied, ∆Ek
hf,h and ∆Ek
hf,e (expressed via the experimen-
tally measured exciton energies) depend linearly on each
other with the proportionality constant given by the hole
hyperfine constant γk.
We start with the analysis of the results for
GaAs/AlGaAs interface fluctuation quantum dots.
Optically detected nuclear magnetic resonance (ODNMR)
is well studied for GaAs/AlGaAs QDs [16, 20]. A typical
NMR spectrum at Bz≈8T is shown in Fig. 1 (c).
Resonances corresponding to all three isotopes (75As,
69Ga, 71 Ga) are clearly observed. We find no contribution
from 27Al isotope of the quantum well barrier and estimate
that contribution of this isotope into the total Overhauser
shift is less then 3% and can be neglected.
In order to measure the hole hyperfine interaction with
75As, we perform experiments with the rf pulse of a rect-
angular shaped spectral band 600 kHz wide with the central
frequency corresponding to the NMR resonance frequency
of 75As. Inside the band, the rf signal has a constant spec-
tral power density (a white noise type), with the power den-
sity outside the bands ∼1000 smaller than inside the band.
Application of such a pulse results in complete depolariza-
tion of arsenic spins, whereas the gallium polarization re-
mains unaffected. The dependence of ∆Ek
hf,h on ∆Ek
hf,e
for k=75As is shown in Fig. 2 (a) with squares for QD
A1.
Since both gallium isotopes have equal chemical prop-
erties (i. e. equal electron wavefunctions), we can assume
that they have the same values of the relative hole hyper-
fine interaction constants γ69 Ga =γ71 Ga . Thus measure-
ment of γGa can be accomplished by performing experi-
ment with rf pulse erasing both 69Ga and 71 Ga polarization,
achieved by applying the rf pulse consisting of two equal
spectral bands centered at corresponding resonant frequen-
cies. The result of this experiment for the same QD is
shown in Fig. 2 (a) with circles. It can be seen that de-
pendencies for both Ga and As follow linear pattern pre-
dicted by Eq. 1. Fitting gives the following values for the
hole hyperfine constants γGa =−7.0±4.0% and γAs =
+15.0±4.5%. Similar measurements were performed on
3 other GaAs QDs. The resulting values are given in Table
I. Since variation between different dots is within the ex-
perimental error, we take average values for all dots yield-
ing γGa =−7.5±2.0% and γAs = +16.0±2.5%. We
thus conclude that different isotopes have opposite signs of
the hole hyperfine constants: it is positive for arsenic and
negative for gallium. This is in contrast to previous reports
[13, 14] where negative values of γfor InP and InGaAs
QDs have been derived.
We have also performed isotope-sensitive measurements
of the hole nuclear interaction in InGaAs/GaAs QDs. How-
ever, these QDs have a more complicated nuclear spin sys-
tem. This is due to significant lattice mismatch result-
ing in strain-induced quadrupolar shifts [21]. Quadrupolar
effects shift NMR frequencies causing significant broad-
ening of the resonance [15]. The magnitudes of these
shifts for typical values of strain ∼0.02 % can be esti-
mated [22] using the known values of the tensor relating
electric field gradient and elastic strain [23]: ∆f(75As)∼
3MHz, ∆f(115In)∼5MHz, ∆f(69 Ga)∼1.5MHz,
∆f(71Ga)∼1MHz. This frequency shift can vary
strongly within the QD volume, therefore in order to erase
nuclear polarization, broadband rf excitation must be used.
At Bz= 8 T we used three different bands of rf excita-
tion shown in Fig. 1 (d). Bands X and Z are used to erase
selectively polarization of 75As and 71 Ga: the widths of
these bands are chosen to be several times the quadrupo-
lar broadening ∆fof the targeted isotope while leaving
the other isotopes unaffected. However, the frequencies of
115In and 69 Ga are too close for these isotopes to be ad-
dressed individually (Fig. 1 (d)). For that reason, we use rf
excitation with the broad band Y, which erases polarization
of both isotopes [24].
The dependencies of ∆Ek
hf,h on ∆Ek
hf,e for InGaAs QD
B1 are shown in Fig. 2 (b) for 71Ga (circles, using rf band
Z) and 75As (squares, rf band X). As in GaAs, we find that
arsenic has a positive hole hyperfine constant while gallium
has a negative one. We also measured ∆EIn+69 Ga
hf,h as a
4
-40 -20 0 20 40
-10
-5
0
5
10
-40 -20 0 20 40 60
Hole HF shift ∆E k
hf, h (µeV)
(a)
As
Ga
GaAs/AlGaAs InGaAs/GaAs
(b)
As
71Ga
In & 69Ga
Electron HF shift ∆Ek
hf, e (µeV)
FIG. 2. The dependence of the hole hyperfine shift ∆Ek
hf,h on the electron hyperfine shift ∆Ek
hf,e for different isotopes in GaAs QD A1
(a) and InGaAs QD B1 (b). the electron hyperfine shift for isotope kis found as a difference of the spectral splitting (E[⇑,↑]−E[⇑,↓])
measured without rf-excitation and the splitting (Ek[⇑,↑]−Ek[⇑,↓]) measured after erasing nuclear polarization corresponding to the k-
th isotope by the rf pulse. In the same way, the hole hyperfine shift is measured as ∆Ek
hf,h = (E[⇑,↑]−E[⇓,↑])−(Ek[⇑,↑]−Ek[⇓,↑]).
Solid lines show fitting, their slopes are given by the corresponding relative hole-nuclear hyperfine constants γk. We find γGa ≈ −7.0%,
γAs ≈+15.0% for GaAs QD A1 and γGa ≈ −6.5%,γAs ≈+10.5% for InGaAs QD B1. Since NMR resonances of 69Ga and 115 In
in InGaAs cannot be resolved, we measure the total hyperfine shifts ∆EI n+69Ga
hf,e and ∆EI n+69Ga
hf,h produced by these isotopes. Fitting
(see text) gives γIn ≈ −16.0% for QD B1. Dashed line is a guide for an eye.
function of ∆EIn+69Ga
hf,e with rf band Y depolarizing both
115In and 69Ga nuclei. This is shown with triangles in Fig.
2 (b). It can be seen that the experimental dependency for
TABLE I. Experimentally measured hole hyperfine constants γi
for different isotopes iin several GaAs and InGaAs QDs. Error
estimates give 90% confidence trust regions. Average values for
each isotope in each material are given at the bottom of the table,
the value for indium in InP is taken from Ref. [14]
Material/QD γGa,%γI n,%γAs,%
GaAs/AlGaAs:
QD A1 -7.0±4.0 - +15.0±4.5
QD A2 -8.5±3.5 - +17.0±5.0
QD A3 -5.5±4.5 - +15.0±4.0
QD A4 -7.5±4.5 - +18.5±5.5
InGaAs/GaAs:
QD B1 -6.5±5.5 -16.0±4.0 +10.5±2.0
QD B2 -3.0±6.5 -15.5±5.0 +10.0±3.0
QD B3 -5.5±5.0 -16.0±4.0 +8.0±2.0
QD B4 -4.5±7.0 -13.0±4.5 +8.5±3.0
Average:
InGaAs/GaAs -5.0±3.0 -15.0±2.0 +9.0±1.0
GaAs/AlGaAs -7.5±2.0 - +16.0±2.5
InP/GaInP - -10.5±1.0 -
115In and 69Ga has a negative slope, with the absolute value
exceeding that of 71 Ga. Consequently, we conclude that
γIn <0and γI n < γGa.
Fitting using Eq. 1 gives the following values for the hy-
perfine constants γGa =−6.5±5.5% and γAs = +10.5±
2.0%. Similar measurements were performed on 3 other
InGaAs QDs. The resulting values are given in Table I.
Since the variation between different dots is within the ex-
perimental error, we take average values for all dots yield-
ing γGa =−5.0±3.0% and γAs = +9.0±1.0%. The
hyperfine constant of indium can also be estimated from the
experimental results. This requires an additional assump-
tion that both gallium isotopes have the same degrees of
spin polarization hI69 Ga
zi=hI71Ga
zias a result of nuclear
spin pumping. Such assumption is justified by the fact that
both isotopes have the same spin I= 3/2and both become
polarized due to the hyperfine interaction with the optically
polarized electrons. Since γ69 Ga =γ71 Ga , we can calcu-
late the Overhauser shifts of 69 Ga from the measured shifts
of 71Ga. For that we need to take into account the ratio of
natural abundances of this isotopes, ρ69Ga /ρ71Ga ≈1.5,
and the ratio of the absolute magnitudes of the electron hy-
perfine constants, A69 Ga /A71Ga, equal to the ratio of the
magnetic moments µ69 Ga/µ71Ga ≈0.79. Thus the elec-
tron (hole) hyperfine shifts of indium can be written as
∆EIn
hf,e(h)= ∆EI n+69Ga
hf,e(h)−ρ69 Ga
ρ71Ga
µ69Ga
µ71Ga ∆E71 Ga
hf,e(h). Using
this expression for the fitting, we find γIn =−16.0±4.0%
for QD B1 with an average for 4 QDs of γIn =−15.0±
5
2.0%.
We can now compare these results with previous reports
[12–14]. In our previous work, [14] we used InP/GaInP
QDs. Using the rf-induced depolarization techniques re-
ported here for InGaAs QDs we found that the contribution
of gallium and phosphorus isotopes into the Overhauser
shift is small compared to the contribution of indium (less
than 10%). Therefore, the value γ=−10.5±1.0% re-
ported previously corresponds to the indium hole hyperfine
constant γIn in InP. On the other hand, Fallahi et al [13] re-
ported negative γ=−9.0±2.0% for InGaAs QDs (aver-
age for all isotopes). For InGaAs QDs studied in this work
we find a much smaller average hyperfine constant (mea-
sured without isotopic sensitivity) γ≈ −2.0%. This can
be explained if we take into account that InGaAs QDs used
in this work have large abundance of gallium with smaller
absolute value of γcompared to indium. This is revealed
by the short PL wavelength (∼915 nm) in our sample com-
pared to the longer wavelengths (∼950 nm) reported by
Fallahi et al. [13] As a result, the negative average hole
hyperfine interaction found in Ref. [13] can be explained
by a significant contribution of indium, dominating due to
its large magnetic moment ∼3.5 times greater than that for
arsenic which has a positive hyperfine constant.
The hyperfine interaction of the valence band holes has
been considered in several theoretical papers [10, 11, 25].
The hyperfine interaction of the conduction band electrons
has a contact (Fermi) form and therefore depends only on
the electron wavefunction density at the nucleus site [15].
By contrast for the valence band holes, the hyperfine cou-
pling is dominated by the dipole-dipole interaction. As a
result, calculations of the hole hyperfine constants mea-
sured in this work requires averaging over the spatial co-
ordinates using an explicit expression for the Bloch wave-
functions, which are not known. One approach is to use
the spherical approximation taking p-type atomic orbitals
to approximate the real Bloch wavefunctions [11, 25]. The
hole hyperfine constants γcalculated in this way have the
same signs for all isotopes in contradiction with our experi-
mental observations. The exact reason for this discrepancy
is not yet fully understood. A possible explanation is that
the real Bloch wavefunction with the symmetry imposed
by the crystal symmetry deviates strongly from the spheri-
cal approximation resulting in variation of γ, including the
sign reversal, for the isotopes with the opposite charges.
In conclusion, we have employed optical spectroscopy
of single quantum dots to measure hole hyperfine constants
γfor individual isotopes in tree types of III-V semicon-
ductor quantum dots. Strong variation of γfor different
isotopes and for different material systems has been found.
This opens the way for improved modeling of microscopic
wavefunctions and deeper insight into fundamental proper-
ties of semiconductors on the atomic scale. Better under-
standing of these properties will allow to engineer mate-
rials with the valence band hyperfine interaction optimized
for future applications requiring highly coherent hole spins.
The authors are thankful to M. M. Glazov for fruitful
discussions, and D. Martrou for help with the GaAs sam-
ple growth. This work has been supported by EPSRC Pro-
gramme Grant No. EP/G601642/1, the Royal Society, and
ITN Spin-Optronics.
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hf,e was measured as a function of the spectral
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hf,e first increases and then
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