Selective Dynamic Nuclear Spin Polarization in Spin-Blocked Double-Dot
Changxue Deng
We study the mechanism of dynamical nuclear spin polarization by hyperfine interaction in spin-blocked double quantum dot system. We calculate the hyperfine transition rates and solve the master equations for the nuclear spins. Specifically, we incorporate the effects of the nuclear quadrupole coupling due to the doping-induced local lattice distortion and strain. Our results show that nuclear quadrupole coupling induced by the 5\% indium substitution can be used to explain the recent experimental observation of missing arsenic NMR signal in the spin-blocked double dots.
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\begin{document}
\title{Selective Dynamic Nuclear Spin Polarization in Spin-Blocked Double-Dot}
\author{Changxue Deng}
\author{Xuedong Hu}
\affiliation{Department of Physics, University at Buffalo, SUNY,
Buffalo, NY 14260-1500}
\date{\today}

\begin{abstract}
We study the mechanism of dynamical nuclear spin polarization by hyperfine
interaction in spin-blocked double quantum dot system.  We calculate the
hyperfine transition rates and solve the master equations for the nuclear
spins.  Specifically, we incorporate the effects of the nuclear quadrupole
coupling due to the doping-induced local lattice distortion and strain.  Our
results show that nuclear quadrupole coupling induced by the 5\% indium
substitution can be used to explain the recent experimental observation of
missing arsenic NMR signal in the spin-blocked double dots.  
\end{abstract}
\pacs{85.35.Be, 76.60.-k, 03.67.Lx, 
}
\maketitle
%\vskip2pc]
%\narrowtext

In many metals and semiconductors, nuclei and conduction electrons form a
coupled nonlinear spin system through the hyperfine interaction.  Such a
coupling is well studied in bulk experimental phenomena such as dynamical nuclear
spin polarization \cite{OO}.  
%
Both electron and nuclear spins in semiconductor heterostructures have been
proposed as candidates for qubits in a scalable quantum information
processor.  There are also a variety of spintronic devices where electron
spins provide novel functionalities.  For both quantum computation and
spintronics, understanding the electron-nuclear spin interaction and the
resulting coupled dynamics is crucial because of its important roles in spin
manipulation 
%(particularly for the nuclear spins, which are difficult to
%access otherwise) 
and decoherence.  
%
%Thus the study of coupled dynamics of electron and nuclear spins is a key to
%both quantum computing and spintronics.  
%
Indeed, in the past decade there have been extensive experimental and
theoretical studies of electron-nuclei hyperfine interaction in semiconductor
nanostructures \cite{Gammon-science, Kawa, decoherence, Zoller, SB-NS}.  In
this Letter, we show that nuclear quadrupole coupling can be a critical
factor in this coupled dynamics, thus needs to be carefully accounted for in
the study of spin-based quantum computers and spintronic devices.

%It has been known for a long time that optically oriented electrons 
%can be utilized to polarize lattice nuclei in bulk semiconductors 
%\cite{OO} due to the very long spin-lattice relaxation time $T_1$. 
%The nuclei interacts with electrons in the conduction band through 
%the Fermi contact hyperfine interaction leading to a nonlinear 
%electron-nuclear spin system with strong coupling. Recently there 
%have been intensive studies of understanding electron-nuclei 
%hyperfine interaction in semiconductor nanostructures 
%\cite{SB-NS, Gammon-science, Gammon, Zoller} which hold some 
%promises for future spintronic and quantum computation devices.

Dynamical nuclear spin polarization has recently been demonstrated in a spin
blocked semiconductor double quantum dot made of Ga$_{0.95}$In$_{0.05}$As
\cite{SB-NS,SB,QD-rev}.  Coulomb interaction and Pauli principle dictates that
in a double quantum dot two-electron singlet and triplet states are split, so
that proper voltage offset and bias between the double dot leads to
significant suppression in the tunnel current due to occupied triplet state
(thus the so-called spin blockade regime, which has been suggested for single
spin detection, a crucial component of quantum information processing)
\cite{SB}.  One way to lift this blockade is to apply an appropriate magnetic
field,
% 
%when singlet-triplet spin-flip transition can be facilitated by the
%electron-nuclear spin hyperfine coupling as one of the polarized triplet
%state is energetically degenerate with the singlet state \cite{SB-NS}.  This
%selective transition can then dynamically polarize the nuclear spins in the
%double quantum dot.  
%
so that electron-nuclear spin hyperfine coupling can facilitate electron
spin-flip transitions, which in turn leads to dynamical nuclear spin
polarization as was shown in Ref.~\cite{SB-NS}.  The reported experimental
observations reveal several quite unexpected phenomena \cite{SB-NS}, one of
which is that 
%
%the oscillation period/amplitude is significantly reduced when nuclear
%magnetic resonance (NMR) is applied at the Ga$^{69}$ and Ga$^{71}$
%frequencies, 
%
while nuclear magnetic resonance (NMR) signals were observed at the
$^{69}\!$Ga and $^{71}\!$Ga frequencies, no comparable response has been
found from the $^{75}\!$As nuclei.  It seems that nuclear spin polarization
cannot be built up in $^{75}\!$As nuclei even though they have the largest
concentration and the largest hyperfine coupling with the electrons among the
three nuclear species.  

In this Letter we study the mechanism of dynamical nuclear spin polarization
by hyperfine interaction in a spin-blocked double quantum dot, taking into
account electron tunneling and nuclear quadrupole coupling.  We show that the
absence of $^{75}\!$As nuclear spin signal is due to the local strain created
by the presence of Indium atoms.  Specifically, the substitution of the 5\%
In for Ga breaks the crystal symmetry and induces a local lattice distortion
\cite{Quad0,Quad}.  The charge redistribution generates {\it static} electric
field gradients (EFG) at the surrounding lattice sites, which lead to finite
quadrupole coupling for the nuclei \cite{GaAs_no_quad}.  The randomly
distributed In atom only substitute one of the nearest neighbors of
As atoms, while the nearest neighbors of Ga atoms are always four As atoms. 
Therefore the local electric field gradients at the Ga nuclei sites are on
average much smaller than at the As sites.  We show that this difference of
EFG at the locations of As and Ga nuclei is the major origin of the
unexpected absence of As nuclear spin signal.  
%
%Specifically, we first calculate the nuclear spin levels accounting for
%Zeeman splitting and quadrupole coupling.  We then identify and calculate
%the nuclear spin transition rates due to hyperfine coupling, incorporating
%electron tunneling to satisfy the energy conservation requirements.

%{\it Spin blockade} (SB) has been observed in a vertical double 
%quantum dot system with single-electron tunneling characteristics 
%\cite{SB}. SB appears when the electrons in both dots have the 
%same spin polarization. A large  current suppression occurs as 
%a result of Pauli exclusion principle. However a leakage current 
%of order 1~pA through the cotunnleing channel in the spin-blockaded 
%region has been recently reported by the same group \cite{SB-NS}. 
%Two more unexpected results have been found at the singlet-triplet 
%crossing when a DC magnetic field (0.4-0.9 Tesla) is applied in 
%the 2D plane of the well in these experiments. The first of them 
%is the coherent current oscillation with a period of order 100 
%seconds in the same region due to back reaction of nuclear spins 
%\cite{SB-NS, nutation}; The other one is that the oscillation 
%period is significantly reduced when nuclear magnetic resonance 
%(NMR) is applied at the Ga$^{69}$ and Ga$^{71}$ frequencies, while 
%no comparable response has been found from the As nuclei albeit 
%it has the largest concentration. These results suggest that 
%the gallium nuclei has been dynamically polarized via the 
%hyperfine interaction. 

%A key point in these experiments is that the singlet state $|0,0\rangle =
%\frac{1}{\sqrt{2}}(| \uparrow \rangle | \downarrow\rangle - |\downarrow
%\rangle | \uparrow\rangle )$ and one of the triplet state, i.e., $|1,-1
%\rangle = |\downarrow \rangle |\downarrow \rangle$ \cite{note_ST} have the
%same energy at some intermediate magnetic field.  Hyperfine induced flip-flop
%scattering between these two-particle spin states could be driven on
%resonance because energy conservation is satisfied in the process.  This
%transition always offers nuclei with the same spin orientation so that the
%nuclei can reach to a non-vanishing polarization in equilibrium condition. 
%The process is irreversible since once the electronic system is in the
%singlet state electron will tunnel through the dots to leads.  It should be
%noted that either electron in the dots could be flipped to make a
%singlet-triplet transition. In this letter we will concentrate on nuclear
%spin dynamics, specifically we will study the steady-state nuclear spin
%polarization under various experimental conditions and propose the possible 
%relaxation mechanism that can be used to explain the origin of the absence of
%As NMR signal.

%In the samples (InGaAs/AlGaAs) that are used to conduct 
%the quantum dot transport experiments 5\% indium is introduced in the well to 
%lower the bottom of the conduction band to 32 meV below the Fermi level of
%the GaAs contacts \cite{QD-rev,SB-NS}. The substitution of 5\% In for Ga
%breaks the crystal symmetry and induce a local lattice distortion
%\cite{Quad0,Quad}.  The charge redistribution generates static electric
%field gradients (EFG) at the lattice sites.  Since all the nuclei in GaAs
%have spin $\frac{3}{2}$ with finite quadrupole moments, we expect that there
%should be strong quadrupole interactions which have been confirmed with NMR
%technique \cite{Quad0, Quad}. Quadrupole effects in Al$_x$Ga$_{1-x}$As have
%been studied extensively in optical pumping experiments \cite{chap5}.
%However this distortion has different influence on Ga and As nuclei since
%the two atoms are coupled by the valence bonds, the randomly distributed In
%atom only substitute one of the nearest neighbors of As atoms; on the other
%hand the nearest neighbors of Ga atoms are four As atoms, therefore the
%local electric field gradients at Ga nuclei are less sensitive to
%replacement of Ga atoms by In atoms.  We show that this difference of EFG at
%the locations of As and Ga nuclei is the major origin of the unexpected
%observation.

We first note that the static quadrupole interaction alone cannot generate any
transition among the spin eigenstates because there is no energy relaxation
mechanism.  This is different from the spin-lattice effect, where the EFG
is generated by the lattice oscillations.  There quadrupole spin relaxation
can be realized by absorbing and emitting a phonon simultaneously (i.e., a
Raman process) \cite{spin-lattice}.  In the system we consider, the nuclei
couple with the electrons, which in turn couple to the leads (reservoir)
by tunneling, so that energy conservation can be satisfied through charge
exchange between the dot and the leads.  
%
%We could associate with the electron a density of state (DOS), i.e. energy
%broadening, then calculate the transition rates of second-order processes
%induced by quadrupole and hyperfine interaction.  
%
In the following we first find the nuclear spin eigenstates including the
static quadrupole interaction, then evaluate the first-order transition rates
among these eigenstates due to hyperfine coupling.  
%
%In this Letter we apply the second approach, where quadrupole
%interaction is not treated as a perturbation. 

The static Zeeman and quadrupole Hamiltonian for a single nuclear spin is
%
\begin{equation}
H_{n} = -\omega \hat{I}_z + \nu ( \hat{I}_Z^2 - \frac{1}{3} \hat{I^2} ),
\label{eq-H_n}
\end{equation}
%
where $\omega$ is the nuclear spin Lamour frequency and $\nu$ is quadrupole
coupling strength that is proportional to the EFG at the locations of the
nuclei, $z$ is the direction of external magnetic field and $Z$ is the
principle axis of the largest EFG \cite{NMR}.  We have
neglected the asymmetrical effect of quadrupole interaction.  It was
determined experimentally that $\nu_{\text{As}} \approx \text{2.5 MHz}$ and
$\nu_{\text{Ga}}$ is much less than $\nu_{\text{As}}$ in In$_x$Ga$_{1-x}$As
for $x$ up to 0.01 \cite{Quad}.  For a typical external field $B_{\text{ext}}$ (0.8 Tesla)
applied in the experiment \cite{SB-NS}, we can assume $\nu_{\text{Ga}} \ll
\omega_{\text{Ga}} \approx 10.4 ~\text{and} ~8.2 ~\text{MHz}$ for $^{69}$Ga
and $^{71}$Ga nuclei.  Since our intention is to study the relation of As and
Ga nuclear polarization, we will not differentiate $^{69}$Ga and $^{71}$Ga
further in this Letter.  The small $\nu_{\text{Ga}}$ allows eigenstates $|\phi_i \rangle$
($i$=-3/2,-1/2,1/2 or 3/2) of $H_n$ for Ga nuclei to be expressed
perturbatively in terms of the Zeeman spin eigenstates $|m \rangle$.
%
\begin{eqnarray}
| \phi_{\pm \frac{3}{2} } \rangle &=& |\pm \frac{3}{2} \rangle - b ~|\pm
\frac{1}{2} \rangle \mp c ~|\mp \frac{1}{2}\rangle , \nonumber \\
| \phi_{\pm \frac{1}{2} } \rangle &=& |\pm \frac{1}{2} \rangle + b ~|\pm
\frac{3}{2} \rangle \mp c ~|\mp \frac{3}{2} \rangle ,
\label{eq:eigen1}
\end{eqnarray}
%
where $b=\sqrt{3}\nu_{\text{Ga}}\text{sin}\theta \text{cos}\theta
/\omega_{\text{Ga}}$ and $c = \sqrt{3}\nu_{\text{Ga}}\text{sin}^2\theta
/4\omega_{\text{Ga}}$.  $\theta$ is the angle between $z$ and $Z$ axes.  In
these calculations $x$ axis is chosen to be in the plane of $z$ and $Z$. 
These new states $|\phi_i \rangle $ are not eigenvectors of $\hat{I}_z$
anymore, though their average spin polarizations $\langle \phi_i | \hat{I}_z |
\phi_i \rangle = i + {\it O}(\nu_{\text{Ga}}^2/\omega_{\text{Ga}}^2)$.

%which is very close to $i$.

Transitions can be induced by the hyperfine interaction
among the new energy eigenstates of the Ga nuclei.  The hyperfine
Hamiltonian is
%
\begin{equation}
H_h = \sum_i^{N} A_i |\psi({\bf R}_i)|^2 \left[ \hat{I}_z^{i} \hat{S}_z +
\frac{1}{2}(\hat{I}_+^{i} \hat{S}_- + \hat{I}_-^{i} \hat{S}_+ ) \right].
\label{eq:hyper}
\end{equation}
%
Here $N \sim 10^5$ is the number of nuclei in a quantum dot; $A$ is the
hyperfine coupling constant; $\psi({\bf R}_i)$ is the electron envelope
wavefunction at the $i$th nucleus.  Using the electron density at nuclear
sites \cite{Paget} we estimate $A \approx 46~\mu$eV  and 40 $\mu$eV for 
As and Ga nuclei respectively.  Hamiltonian~(\ref{eq:hyper}) describes the interaction of an
electron with an ensemble of $N$ nuclei in the dot.  

To calculate the transition rates given in Fig.~1 we must study how the system
couples to the environment to satisfy energy conservation.  In the
present case electron exchange between a dot and the leads is the most
important energy relaxation mechanism \cite{NSR}.  For triplet state
$|\downarrow \downarrow \rangle$ \cite{note_ST} the electron in the 1st (2nd) dot only
couples to the left (right) lead.  The electron tunnel Hamiltonian can be
expressed as
%
\begin{equation}
H_e = \sum_{k} \epsilon_k c_{k \downarrow}^+c_{k \downarrow}
+ \epsilon_0 d_{\downarrow}^{+} d_{\downarrow}
+ \sum_{k} \left [t_k c_{k \downarrow}^{+} d_{\downarrow} + \text{H.C.}
\right],
\label{eq-He}
\end{equation}
%
where $\epsilon_0$ is the renormalized single particle energy level.  
Since the Hamiltonian
$H_e$ is quadratic, exact solution of the retarded Green's function
$G_{\downarrow \downarrow}^{Ret} = -i \theta (t) \langle \{d_{\downarrow}(t),
d_{\downarrow}^{+}(0)\} \rangle$ can be obtained by using the equation of
motion method.  The density of state (DOS) follows from $\rho_T (\epsilon) =
-\text{Im} G_{\downarrow \downarrow}^{Ret} (\epsilon + i0^+)/ \pi$.  It is
straightforward to show that the electron triplet state is broadened
%
\begin{equation}
\rho_T(\epsilon) =  \frac{1}{2\pi} \frac{\Gamma}{(\epsilon - \epsilon_0)^2 +
\Gamma^2/4},
\label{eq:dos}
\end{equation}
%
where $\Gamma = 2\pi \sum_k |t_k|^2 \delta(\epsilon-\epsilon_k)$ is the level 
broadening for the triplet state. 
%
%For singlet state the first dot can also exchange electron with 
%right lead. The DOS of the electron in singlet state $\rho_S$ has to be
%modified from Eq.~(\ref{eq:dos}). However since the inter-dot coupling is
%much weaker than the tunneling coupling between the dot and the lead,
%$\rho_S$ can be approximated by Eq.~(\ref{eq:dos}) \cite{note_DOS}.
%
For the electron in the singlet state, the DOS should be
modified by a weak inter-dot coupling.  However, the major energy relaxation
channel is still the lead-dot tunnel coupling, so we can approximate the
singlet DOS with Eq.~(\ref{eq:dos}) as well. 

\begin{figure}[t]
\begin{center}
\epsfig{file=fig1.eps, width=4.5cm,height=3cm}
\caption{Schematic of hyperfine induced transitions among the energy
eigenstates of Hamiltonian $H_n$.  The downward arrows represent simultaneous
flip-flop scattering of nuclei and the electron in a triplet state.  $F$ and
$G$ are the nuclear spin relaxation rates induced by the effective hyperfine
magnetic field of the electron ($\frac{A}{N} I_z S_z$).}
\label{fig1}
\end{center}
\end{figure}

The first term in Eq.~(\ref{eq:hyper}) does not involve electron spin flip. 
However, it can induce transitions among the energy eigenstates due to 
mixture of different spin eigenstates in Eq.~(\ref{eq:eigen1}). The
transition rate between state $|\phi_i \rangle$ and $|\phi_j \rangle$
due to this term is given by the Fermi golden rule,
%
\begin{equation}
W_{i,j}^{z} = \frac{\pi A^2}{2\hbar N^2} |\langle \phi_i | \hat{I}_z | \phi_j
\rangle |^2 P(E_j - E_i),
\label{eq:Wz}
\end{equation}
%
where $E_i$ and $E_j$ are the eigenenergies of $H_n$.  The absorption power
$P$ is
%
\begin{equation}
P(\Delta) = \sum_{\epsilon_i,\epsilon_f} \rho (\epsilon_i)
\rho (\epsilon_f) \delta(\epsilon_f - \epsilon_i + \Delta),
\label{eq:P}
\end{equation}
%
where $\rho(\epsilon)$ is the DOS of either the electron singlet or the
triplet state.  Using Eq.~(\ref{eq:Wz}), the transition rates in
Fig.~(\ref{fig1}) can be calculated: $F$ = $6b^2A^2/(N^2 \hbar \Gamma)$ and
$G$ = $6c^2A^2/(N^2 \hbar \Gamma)$.  In deriving these expressions we have
neglected the nuclear energy difference of the initial and final state since
they are much smaller than the width of electronic energy broadening.  

The calculation of the transition rates due to the hyperfine flip-flop terms
in Eq.~(\ref{eq:hyper}), which correspond to the polarized triplet to singlet
electron transitions, is more complicated.  When the two electron states are
degenerate, the hyperfine flip-flop transition is on resonance, leading
to nuclear spin polarization.  However, the transition should quickly becomes off
resonance due to the effective nuclear magnetic field $B_{\text{N}}$
experienced by the electrons. 
%
%Regarding $A I_z S_z$ as an average electron spin energy, 
%
The effective nuclear magnetic fields can be estimated as
%
\begin{equation}
B_{\text{N}} = \gamma_{\text{As}} \frac{\langle \hat{I}_z \rangle
_{\text{As}}}{I} + \gamma_{\text{Ga}} \frac{\langle \hat{I}_z \rangle
_{\text{Ga}}}{I},
\label{eq:BN}
\end{equation}
%
where $\gamma_{\rm As} \approx -2.8 $ Tesla and $\gamma_{\rm Ga} \approx -1.3$
Tesla \cite{Paget}.  As and Ga nuclei have independent spin temperatures
because their mutual flip-flop process is largely suppressed due to different
magnetic moments.  Equilibrium in nuclear spin polarization will thus be
established in each of the nuclear species independently.  $B_{\rm N}$ can be
quite large for even moderate nuclear polarization.  The energy difference of
the singlet state and triplet state due to $B_{\text{N}}$ can be approximated
by $\Delta_{ST} = -g^*\mu_B B_{\text{N}}$ assuming the energy of the singlet
state is independent of magnetic field.  The flip-flop transition rates of
hyperfine interaction from state $|\phi_i \rangle $ to state $|\phi_j
\rangle$ is thus
%
\begin{equation}
W_{i,j}^{-} = \frac{f_T \pi A^2}{8\hbar N^2}
| \langle \phi_{j} | \hat{I}_{-} | \phi_i \rangle |^2 P(\Delta_{ST} + E_{j} -
E_i),
\label{eq:W-}
\end{equation}
%
where $f_T$ is a multiplication factor due to cotunneling and the
probability of the triplet state occupation (electrons can tunnel into three 
triplet states and one singlet state, but only one triplet is relevant to 
nuclear polarization). $f_T$ should be multiplied by another factor 0.5 because 
there are two dots and the probability of nuclear spin-flip process in each of the dots is
50\%. We estimate $f_T \approx 0.05$.  
%
%Again ignoring the nuclear magnetic energy in the argument of $P$ 
%
We then find that $D$ in Fig.~(\ref{fig1}) is given as $D = f_T A^2 \Gamma
/N^2\hbar (\Delta_{ST}^2 + \Gamma^2)$.

The quadrupole interaction of As nuclei is much stronger than that of Ga
[$\nu_{\text{As}} \approx 2.5$ MHz is in the same order as the Zeeman energy
$\omega_{\text{As}} \approx 5.9$ MHz (at 0.8 Tesla)] and leads to a complete
mixing of the spin eigenstates, so that the perturbative wavefunctions
[Eq.~(\ref{eq:eigen1})] for Ga nuclei have to be replaced with numerical
solutions:
%  
%We can write the new energy eigenstates as
%linear combinations of $|m \rangle$:
%
\begin{equation}
|\phi_i \rangle = \sum_{m} u_{im} |m \rangle.
\label{eq:eigen2}
\end{equation}
%
The expectation value of $\hat{I}_z$ of these state is $\langle \phi_i |
\hat{I}_z | \phi_i \rangle = \sum_{m} |u_{im}|^2 m$.  For example, we find
that when $\alpha \equiv \nu_{\text{As}} / \omega_{\text{As}} =0.3$, 
$\langle \phi_i | \hat{I}_z | \phi_i \rangle $ = 1.419, 0.558, -0.528 and
-1.450 for $i$ = 3/2, 1/2, -1/2 and -3/2 respectively.  All the matrix
elements of $\hat{I}_z$ and $\hat{I}_-$ can be easily obtained from
Eq.~(\ref{eq:eigen2}), and the transition rates can be calculated numerically
using Eq.~(\ref{eq:Wz}) and (\ref{eq:W-}).
%
%$\langle \phi_i | \hat{I}_z  | \phi_j \rangle$ and $\langle \phi_i |
%\hat{I}_- | \phi_j \rangle$ can be inserted into Eq.~(\ref{eq:Wz}) and
%Eq.~(\ref{eq:W-}) to evaluate the transition rates numerically.  
%
One should keep in mind that the illustration of the various transitions in
Fig.~(\ref{fig1}) is only applicable to the perturbative analysis for Ga
nuclei, while for As nuclei all transitions are possible.  However the
concept of polarization ($D$) and depolarization ($F$ and $G$) processes
illustrated in Fig. (\ref{fig1}) should still be valid.

After obtaining all the transition rates $W^z$ and $W^-$ we can proceed to
construct the master equations \cite{NMR}
%
\begin{equation}
\frac{d p_i} {dt} = \sum_{j \ne i} W_{j,i} p_j - \sum_{j \ne i} W_{i,j} p_i,
\label{eq:rate}
\end{equation}
%
where $W_{i,j} = W_{i,j}^z +W_{i,j}^-$, and $p_{i}$ is the probability of 
nuclear spins in state $| \phi_{i} \rangle$.  In Eq.~(\ref{eq:rate}) we have
neglected the spin-lattice relaxation, which is very weak at liquid Helium
temperature.  Here we can use the spin temperature approximation
because the nuclear spin build-up time ($\approx$ 100 s) \cite{SB-NS} is much
longer than the transverse spin relaxation $T_2~ (\approx 10^{-4} ~{\rm s})$.
We solve for the steady-state polarization: $d p_i/dt = 0$.  
%Only three of the four equations are independent, the remaining equation is 
%the normalization condition $\sum_{i} p_{i} = 1$.  
These equations are highly nonlinear, as the
equilibrium polarizations depend on the transition rates, while to calculate
the transition rates $W^-$ one must know the effective nuclear magnetic
field, which itself depends on the nuclear spin polarization.  We solve the
set of equations self-consistently.  The average nuclear polarization is $P =
\sum_i p_i \langle \phi_i | \hat{I}_z | \phi_i \rangle = \langle \hat{I}_z
\rangle /I$.  We can then calculate the effective nuclear magnetic fields
$B_{\text{N}}$ with Eq.~(\ref{eq:BN}).  
%The updated transition rates are used
%to find the new nuclear spin polarizations until self-consistency is
%achieved.

\begin{figure}[t]
\begin{center}
\epsfig{file=fig2.eps, width=8.5cm, height=4.8cm}
\caption{ Ratio of Ga and As nuclei polarization as a function of the relative
quadrupole interaction strength $\alpha ~(=\nu_{\text{Ga}}/\nu_{\text{As}})$
for $\Gamma=10~\mu$eV (panel a) and $\Gamma=30~\mu$eV (panel b).
$\Gamma$ is the width of level broadening due to electron tunneling between
the lead and the dot. The three curves in each graph represent three different
As quadrupole coupling strengths $\nu_{\text{As}}/\omega_{\text{As}} $ 
= 0.1, 0.3 and 0.5. $\theta$ has chosen to be 45 degrees for both As and Ga
nuclei.
}
\label{fig2}
\end{center}
\end{figure}

Figure~\ref{fig2} shows the calculated ratio of Ga and As nuclei polarization
as a function of $\alpha$, which is taken as a free parameter in our
calculations.  We consider As quadrupole effects with three different
strengths.  $\nu_{\text{As}}/\omega_{\text{As}} = 0.3$ corresponds to
$\nu_{\text{As}} \approx 1.8$ MHz.  In this case the Ga nuclear polarization
is two orders of magnitude greater than that of As nuclei when $\alpha
\approx$ 0.01.  This difference can qualitatively explain the experimental
observation that the As NMR signal is missing in measuring the current
oscillation period as a function of the AC magnetic field frequency.  We also
find the ratio is very sensitive to electron level width $\Gamma$. 
Figure~\ref{fig2} shows that the polarization ratio in the case of
$\Gamma=30~\mu$eV is roughly $\frac{1}{5}$ of that in the case of
$\Gamma=10~\mu$eV.  Since $D \propto \Gamma/(\Delta_{ST}^2 + \Gamma^2 )$, the
triplet to singlet transition rate decreases much faster when $\Gamma$ is
smaller, resulting in larger polarization ratio.  

Figure~\ref{fig3} shows how Ga nuclear spin polarization changes with the
relative quadrupole coupling strength $\alpha$.  It is obvious from the two
panels that $P_{\text{Ga}}$ increases rapidly as $\alpha$ decreases from 0.1
to 0.01.  The Ga polarization saturates to unity for smaller $\alpha$, and
larger $\Gamma$ leads to larger Ga polarization.

\begin{figure}[t]
\begin{center}
\epsfig{file=fig3.eps, width=8.5cm, height=4.8cm}
\caption{ Ga nuclear spin polarization as a function of the relative
quadrupole interaction strength $\alpha$ for $\Gamma=10~\mu$eV (panel a) and
$\Gamma=30~\mu$eV (panel b). All the parameters are the same as those used in
Fig. 2.}
\label{fig3}
\end{center}
\end{figure}

%As an illustration to our calculation, we consider the dynamical nuclear
%polarization of nuclei with spin $I=\frac{1}{2}$.  Equation~(\ref{eq:rate})
%in steady state can be solved easily and we obtain
%
%\begin{equation}
%\frac{P_{\text{As}}}{P_{\text{Ga}}} 
%               = \frac{D(B_{\text{N}}) + 2F_{\text{Ga}}}
%                      {D(B_{\text{N}}) + 2F_{\text{As}}},
%\end{equation}
%
%where $D(B_{\text{N}})$ and $F$ have the same meaning as in the above
%discussion.  We have used the fact the hyperfine constants of Ga and As are
%approximately equal ($D_{\text{As}} \approx D_{\text{Ga}}$). In the limit of 
%$F_{\text{Ga}} \rightarrow 0$  and $D(B_{\text{N}}) \ll F_{\text{As}}$ 
%(i.e., off resonance), $P_{\text{As}}/P_{\text{Ga}} \approx
%D(B_{\text{N}})/2F_{\text{As}} \ll 1$. 

The physical picture of the master equation calculation is actually quite
straightforward.  Both Ga and As have a polarization rate and a
depolarization rate.  The As nuclear spin depolarization rate is much larger
than that of Ga due to the quadrupole coupling.  Thus there exists a regime
of polarization rate where nuclear spin polarization can be built up in Ga
but not in As.

%At this point we would like to clarify the relation of the present work and 
%some of the previous studies where As polarization has been detected in spite
%of the presence of quadrupole interactions (say in AlGaAs)
%\cite{Gammon-science}. 
%
In our study As depolarization rates are greatly enhanced because the energy
levels of the dots are broadened through tunnel coupling to the leads, and
that the singlet-triplet transition becomes off resonance once the
polarization of Ga nuclei is established.  In previous experiments concerning
dynamical nuclear polarization, optical pumping creates a constant
polarization rate for the nuclei with the help of hyperfine interaction; and
the relaxation process due to carrier recombination overshadows the effect of
quadrupole interaction, so that As polarization can still be built up
\cite{OO}.
 
In conclusion, we have studied the dynamical nuclear polarization mechanism in
spin-blockade double-dot system.  Specifically, we calculate the hyperfine
induced polarization rates and depolarization rates among the eigenstates of
the nuclear spins.  We show that the average spin polarization in steady
state of different nuclear species (Ga and As) could differ by two orders of
magnitude due to static quadrupole interaction induced by lattice
distortions.  Our results can thus be used to explain the recent NMR
experiments conducted in such system.  Our calculation suggests both caution
and promise.  Doping is widely used in semiconductors to tailor electronic
properties.  However, as we point out in this Letter, there can be unexpected
side effects in material properties, such as increased nuclear spin
relaxation in the present case.  Conversely, controlled doping can also be
used to differentiate parts of a system (such as Ga and As nuclei here), so
that selective operations become possible.  How to utilize the additional
control provided by doping while overcoming its negative effects can be
critical to future quantum information processors and/or spintronic devices.

We acknowledge helpful conversations with K. Ono, S. Tarucha, and S. Das
Sarma.  We thank partial financial support by ARDA and ARO.

\begin{thebibliography}{99}

\bibitem{OO} F. Meier, and B.P. Zakharchenya, {\em Optical Orientation}
(North-Holland, Amsterdam, 1984).

\bibitem{Gammon-science} D. Gammon {\it et al.}, 
%S.W. Brown, E.S. Snow, T.A. Kennedy, D.S. Katzer, and D. Park, 
Science {\bf 277}, 85 (1997).

%\bibitem{Gammon} D. Gammon {\em et al.}, Phys. Rev. Lett. {\bf 86}, 5176
%(2001).

\bibitem{Kawa} R.K. Kawakami {\em et al.}, Science {\bf 294}, 131 (2001). 

\bibitem{decoherence} A.V. Khaetskii and Yu.V. Nazarov, Phys. Rev. B {\bf 64},
125316 (2001); J. Schliemann, A.V. Khaetskii, and D. Loss, {\it ibid.} {\bf
66}, 245303 (2002); R. de Sousa and S. Das Sarma, {\it ibid.} {\bf 67},
033301 (2003); {\it ibid.} {\bf 68}, 115322 (2003).

\bibitem{Zoller} A. Imamo\={g}lu, E. Knill, L. Tian, and P. Zoller,
Phy. Rev. Lett. {\bf 91}, 017402 (2003).

\bibitem{SB-NS} K. Ono and S. Tarucha, cond-mat/0309062.

\bibitem{SB} K. Ono, D.G. Austing, Y. Tokura, and S. Tarucha, Science {\bf
297}, 1313 (2002).

\bibitem{QD-rev} L.P. Kouwenhoven, D.G. Austing, and S. Tarucha, Rep. Prog.
Phys. {\bf 64}, 701 (2001).

\bibitem{Quad0} E.H. Rhoderick, J. Phys. Chem. Solids {\bf 8}, 498 (1958).

\bibitem{Quad} W.E. Carlos, S.G. Bishop, and D.J. Treacy, Phy. Rev. B {\bf
43}, 12512 (1991); 
%W.E. Carlos, S.G. Bishop, and D.J. Treacy, 
Appl. Phy. Lett. {\bf 49}, 528 (1986).

\bibitem{GaAs_no_quad} The crystal structure of {\it pure} GaAs is
zinc-blende with cubic symmetry.  No noticeable field gradient is
expected due to this symmetry.

\bibitem{spin-lattice} J.V. Kranendonk, Physica {\bf 20}, 781 (1954).

\bibitem{NMR} A. Abragam, {\em The Principles of Nuclear Magnetism}
(Oxford University Press, London, 1961); C.P. Slichter, {\em Principles of
Magnetic Resonance} (Spinger-Verlag, Berlin, 1996).

%\bibitem{nutation} S.I. Erlingsson, O.N. Jouravlev, and Y.V. Nazarov,
%cond-mat/0309069.

\bibitem{Paget} D. Paget, G. Lampel, B. Sapoval, and V.I. Safarov, Phys. Rev.
B {\bf 15}, 5780 (1977).

\bibitem{NSR} Y.B. Lyanda-Geller, I.L. Aleiner, and B.L. Altshuler, Phys. Rev.
Lett. {\bf 89}, 107602 (2002).

\bibitem{note_ST} We have assumed that the electron g-factor is negative and
the triplet is the ground state at zero magnetic field. One could equivalently
discuss the other triplet state $|1,1 \rangle = |\uparrow \rangle |\uparrow
\rangle $ if the singlet is the ground state when there is no magnetic field
\cite{Jpn}.

\bibitem{Jpn} T. Inoshita, K. Ono, and S. Tarucha, J. Phys. Soc. Jpn. {\bf 72}
183 (2003).


%\bibitem{note_DOS} We have evaluated the exact form of DOS in this case.
%$\rho_S(\epsilon) = \frac{\Gamma}{2\pi} \frac{(\epsilon - \epsilon_0)^2 +
%t_c^2 + \Gamma^2/4}
%{\left [(\epsilon-\epsilon_0)^2 - t_c^2 - \Gamma^2/4 \right ]^2 + 
%\Gamma^2(\epsilon-\epsilon_0)^2}$. $t_c$ is the inter-dot coupling
%coefficient and 
%a symmetrical double-dot is considered for simplicity. When $t_c$ (a few tens
%$\mu$eV) 
%is small $\rho_S \approx \rho_T$.

%\bibitem{chap5} V.G. Fleisher and I.A. Merkulov, Chapter 5 in Ref. \cite{OO}. 

%\bibitem{spin-lattice} J.A. McNeil, and W.G. Clark, Phys. Rev. B {\bf 13}
%4705 (1976).

\end{thebibliography}

\end{document}

cond-mat0402428.extracted_bib
\begin{thebibliography}{99}

\bibitem{OO} F. Meier, and B.P. Zakharchenya, {\em Optical Orientation}
(North-Holland, Amsterdam, 1984).

\bibitem{Gammon-science} D. Gammon {\it et al.}, 
%S.W. Brown, E.S. Snow, T.A. Kennedy, D.S. Katzer, and D. Park, 
Science {\bf 277}, 85 (1997).

%\bibitem{Gammon} D. Gammon {\em et al.}, Phys. Rev. Lett. {\bf 86}, 5176
%(2001).

\bibitem{Kawa} R.K. Kawakami {\em et al.}, Science {\bf 294}, 131 (2001). 

\bibitem{decoherence} A.V. Khaetskii and Yu.V. Nazarov, Phys. Rev. B {\bf 64},
125316 (2001); J. Schliemann, A.V. Khaetskii, and D. Loss, {\it ibid.} {\bf
66}, 245303 (2002); R. de Sousa and S. Das Sarma, {\it ibid.} {\bf 67},
033301 (2003); {\it ibid.} {\bf 68}, 115322 (2003).

\bibitem{Zoller} A. Imamo\={g}lu, E. Knill, L. Tian, and P. Zoller,
Phy. Rev. Lett. {\bf 91}, 017402 (2003).

\bibitem{SB-NS} K. Ono and S. Tarucha, cond-mat/0309062.

\bibitem{SB} K. Ono, D.G. Austing, Y. Tokura, and S. Tarucha, Science {\bf
297}, 1313 (2002).

\bibitem{QD-rev} L.P. Kouwenhoven, D.G. Austing, and S. Tarucha, Rep. Prog.
Phys. {\bf 64}, 701 (2001).

\bibitem{Quad0} E.H. Rhoderick, J. Phys. Chem. Solids {\bf 8}, 498 (1958).

\bibitem{Quad} W.E. Carlos, S.G. Bishop, and D.J. Treacy, Phy. Rev. B {\bf
43}, 12512 (1991); 
%W.E. Carlos, S.G. Bishop, and D.J. Treacy, 
Appl. Phy. Lett. {\bf 49}, 528 (1986).

\bibitem{GaAs_no_quad} The crystal structure of {\it pure} GaAs is
zinc-blende with cubic symmetry.  No noticeable field gradient is
expected due to this symmetry.

\bibitem{spin-lattice} J.V. Kranendonk, Physica {\bf 20}, 781 (1954).

\bibitem{NMR} A. Abragam, {\em The Principles of Nuclear Magnetism}
(Oxford University Press, London, 1961); C.P. Slichter, {\em Principles of
Magnetic Resonance} (Spinger-Verlag, Berlin, 1996).

%\bibitem{nutation} S.I. Erlingsson, O.N. Jouravlev, and Y.V. Nazarov,
%cond-mat/0309069.

\bibitem{Paget} D. Paget, G. Lampel, B. Sapoval, and V.I. Safarov, Phys. Rev.
B {\bf 15}, 5780 (1977).

\bibitem{NSR} Y.B. Lyanda-Geller, I.L. Aleiner, and B.L. Altshuler, Phys. Rev.
Lett. {\bf 89}, 107602 (2002).

\bibitem{note_ST} We have assumed that the electron g-factor is negative and
the triplet is the ground state at zero magnetic field. One could equivalently
discuss the other triplet state $|1,1 \rangle = |\uparrow \rangle |\uparrow
\rangle $ if the singlet is the ground state when there is no magnetic field
\cite{Jpn}.

\bibitem{Jpn} T. Inoshita, K. Ono, and S. Tarucha, J. Phys. Soc. Jpn. {\bf 72}
183 (2003).


%\bibitem{note_DOS} We have evaluated the exact form of DOS in this case.
%$\rho_S(\epsilon) = \frac{\Gamma}{2\pi} \frac{(\epsilon - \epsilon_0)^2 +
%t_c^2 + \Gamma^2/4}
%{\left [(\epsilon-\epsilon_0)^2 - t_c^2 - \Gamma^2/4 \right ]^2 + 
%\Gamma^2(\epsilon-\epsilon_0)^2}$. $t_c$ is the inter-dot coupling
%coefficient and 
%a symmetrical double-dot is considered for simplicity. When $t_c$ (a few tens
%$\mu$eV) 
%is small $\rho_S \approx \rho_T$.

%\bibitem{chap5} V.G. Fleisher and I.A. Merkulov, Chapter 5 in Ref. \cite{OO}. 

%\bibitem{spin-lattice} J.A. McNeil, and W.G. Clark, Phys. Rev. B {\bf 13}
%4705 (1976).

\end{thebibliography}