arXiv:0704.1628v1  [cond-mat.mes-hall]  12 Apr 2007
Detection of single electron spin resonance in a double quantum dot
F. H. L. Koppens,∗ C. Buizert, I. T. Vink, K.C. Nowack, T. Meunier, L. P. Kouwenhoven, and L. M. K. Vandersypen
Spin-dependent transport measurements through a double quantum dot are a valuable tool for
detecting both the coherent evolution of the spin state of a single electron as well as the hybridization
of two-electron spin states. In this paper, we discuss a model that describes the transport cycle in
this regime, including the effects of an oscillating magnetic field (causing electron spin resonance)
and the effective nuclear fields on the spin states in the two dots. We numerically calculate the
current flow due to the induced spin flips via electron spin resonance and we study the detector
efficiency for a range of parameters. The experimental data are compared with the model and we
find a reasonable agreement.
A. Introduction
Recently, coherent spin rotations of a single electron
were demonstrated in a double quantum dot device [1].
In this system, spin-flips of an electron in the dot were in-
duced via an oscillating magnetic field (electron spin res-
onance or ESR) and detected through a spin-dependent
transition of the electron to another dot, which already
contained one additional electron. This detection scheme
is an extension of the proposal for ESR detection in a sin-
gle quantum dot by Engel and Loss [2]. Briefly, the device
can be operated (in a spin blockade regime [3]) such that
the electron in the left dot can only move to the right dot
if a spin flip in one of the two dots is induced via ESR.
From the right dot, the electron exits to the right reser-
voir and another electron enters the left dot from the left
reservoir. A continuous repetition of this transition will
result in a net current flow.
Compared to the single dot detection scheme [2], us-
ing the double-dot as the detector has two major advan-
tages. First, the experiment can be performed at a lower
static magnetic field and consequently with lower, tech-
nically less demanding, excitation frequencies. Second,
the spin detection is rather insensitive to unwanted oscil-
lating electric fields, because the relevant dot levels can
be positioned far from the Fermi energies of the leads.
These electric fields are unavoidably generated together
with the oscillating magnetic field as well.
The drawback of the double-dot detector is that spin
detection is based on the projection in the two-electron
singlet-triplet basis, while the aim is to detect single spin
rotations. However, this detection is still possible be-
cause the electrons in the two dots experience different
effective nuclear fields. This is due to the hyperfine inter-
action of the electron spins with the (roughly 106) nuclear
spins in the host semiconductor material of each quan-
tum dot [4–11]. In order to provide more insight in this
double-dot ESR detection scheme for single spin rota-
tions, it is necessary to analyze the coherent evolution of
the two-electron spin states together with the transitions
∗Electronic address: f.h.l.koppens@tudelft.nl; Kavli Institute of
NanoScience Delft, P.O. Box 5046, 2600 GA Delft, The Netherlands
in the transport cycle.
In this paper, we discuss a model that describes the
transport cycle in the spin blockade regime while includ-
ing the coherent coupling between the two dots, and the
influence of the static and oscillating magnetic field to-
gether with the effective nuclear fields on the electron
spin states. The aim is to understand how effectively
single spin resonance will affect the measured quantity
in the experiment, namely the current flow in the spin
blockade regime. The organization of this paper is as
follows. First, we will explain the transport cycle and
the mechanism that causes spin blockade. Next, we will
briefly discuss the static system Hamiltonian and the
mixing of the two-electron spin states by the effective
nuclear field. Then we add an oscillating magnetic field
to this Hamiltonian, that forms -together with the double
dot tunnelling processes- the basis of the rate equations
that describe how the density matrix of the two-electron
spin states evolves in time. The current flow, calculated
from the steady state solution of the density operator,
is then analyzed for different coherent coupling values,
magnitudes of the oscillating magnetic field, in combina-
tion with different effective nuclear fields in the two dots.
This provides further insight in the optimal conditions
for spin-flip detection with a double quantum dot.
B. Spin blockade
In the spin-blockade regime, the double-dot is tuned
such that one electron always resides in the right dot,
and a second electron can tunnel from the left reservoir
through the left and right dots, to the right reservoir [3].
This current-carrying cycle can be described with the
occupations (m, n) of the left and right dots: (1, 1) →
(0, 2) → (0, 1) → (1, 1). When an electron enters the
left dot and forms a double-dot singlet state S11 with the
electron in the right dot (S = |↑↓〉 − |↓↑〉, normalization
omitted for brevity), it is possible for the left electron
to move to the right dot, because the right dot singlet
state S02 is energetically accessible. Next, one electron
tunnels from the right dot to the right lead and another
electron can again tunnel into the left dot. If, however,
the two electrons form a double-dot triplet state T11, the
left electron cannot move to the right dot, as the right
http://arxiv.org/abs/0704.1628v1
dot triplet state T02 is much higher in energy (due to
the relatively large exchange splitting in a single dot).
The electron can also not move back to the lead and
therefore further current flow is blocked as soon as any
of the (double-dot) triplet states is formed (Fig. 1a,b).
Spin blockade only occurs if at least one of the eigen-
states of the system Hamiltonian is a pure triplet state.
If processes are present that induce transitions from all
the three (1,1) triplet states to the (1,1) singlet state,
spin blockade is lifted and a current will flow. As we will
see below, the presence of the nuclear spins in the host
semiconductor can give rise to these kind of transitions.
This can be seen most easily by adding the effect of the
hyperfine interaction to the system Hamiltonian.
C. System Hamiltonian
The system Hamiltonian is most conveniently writ-
ten in the two-electron singlet-triplet basis with the
quantization-axis in the z-direction. The basis states are
S11, T
11, T
11, T
11 and S02. The subscript m,n denotes
the dot occupancy. We exclude the T02 state from the
model, because this state is energetically inaccessible and
therefore does not play an important role in the trans-
DLR>>t DLR=0
S - S11 02
S - S11 02
-10 -5 0 5 10
DLR/t
b) a)
FIG. 1: a) A schematic of the double dot and the electro-
chemical potentials (energy relative to the (0,1) state) of the
relevant two-electron spin states. For ∆LR > t, transitions
from the S11 state to the S02 state are possible via inelastic
relaxation with rate Γin. Spin blockade occurs when one of
the T i11 states is occupied. b) Similar schematic for ∆LR = 0,
where the singlet states are hybridized. Also in this case,
spin blockade occurs when one of T i11 states is occupied. c)
Energy levels as a function of detuning. At ∆LR = 0, the
singlet states hybridize into bonding and anti-bonding states.
The splitting between the triplets states corresponds to the
Zeeman energy gµBBext.
port cycle. Furthermore, we neglect the thermal energy
kT in the description, which is justified when the bias
over the two dots is much larger than kT . The system
Hamiltonian is given by
H0 = − ∆LR|S02〉〈S02|+ t
|S11〉〈S02|+ |S02〉〈S11|
− gµBBext
|T+11〉〈T
11| − |T
11〉〈T
, (1)
where ∆LR is the energy difference between the |S11〉
and |S02〉 state (level detuning, see Fig.1a), t is the tun-
nel coupling between the |S11〉 and |S02〉 states, Bext the
external magnetic field in the z-direction and Sz
L(R) the
spin operator along z for the left (right) electron. The
eigenstates of the Hamiltonian (1) for finite external field
are shown in figure 1c. For ∆LR < t, the tunnel coupling
t causes an anti-crossing of the |S11〉 and |S02〉 states. For
∆LR < 0, transport is blocked by Coulomb blockade (i.e.
the final state |S02〉 is at a higher energy than the initial
state S11). For ∆LR ≥ 0, transport will be blocked when
one of the three triplet states becomes occupied (spin
blockade). In Fig.1a and b, we distinguish two regimes:
∆LR > t where the (exchange) energy splitting between
T 011 and S11 is negligibly small and transitions from S11
to S02 occur via inelastic relaxation with rate Γin and the
energy. A different regime holds for ∆LR < t, where S11
is coherently coupled with S02 giving rise to a finite (ex-
change) splitting between T 011 and the hybridized singlet
states. We will return to this distinction in the discussion
below.
D. Singlet-triplet mixing by the nuclear spins
The effect of the hyperfine interaction with the nuclear
spins can be studied [12] by adding a static (frozen) ef-
fective nuclear field BLN (B
N ) at the left (right) dot to
the system Hamiltonian:
Hnucl = −gµB
N · SL +BRN · SR
= −gµB(BLN −BRN ) · (SL − SR)/2
−gµB(BLN +BRN ) · (SL + SR)/2. (2)
For the sake of convenience, we separate the inhomoge-
neous and homogeneous contribution, for reasons which
we will discuss later. Considering the nuclear field as
static is justified since the tunnel rates and electron spin
dynamics are expected to be much faster than the dy-
namics of the nuclear system [10, 13, 14]. Therefore,
we will treat the Hamiltonian as time-independent. The
effect of nuclear reorientation will be included later by
ensemble averaging.
We will show now that triplet states mix with the S11
state if the nuclear field is different in the two dots (in
all three directions). This mixing will lift spin blockade,
detectable as a finite current running through the dots for
∆LR ≥ 0. The effective nuclear field can be decomposed
-30 -20 -10 0 10 20 30
Magnetic field (mT)
Bext=0 B » Bext ND
-10 -5 0 5 10
10 S02
a +b +g +kS T T T11 11 11 11
-10 -5 0 5 10
a +kS T11 11
DLR/t DLR/t
FIG. 2: a) Observed current flow in the inelastic transport
regime (gµB∆LR ≫ t) due to singlet-triplet mixing by the
nuclei. b) Electrochemical potentials in the presence of Hnucl
(t ∼ ∆BN ). Singlet and triplet eigenstates are denoted by
red and blue lines respectively. Hybridized states (of sin-
glet and triplet) are denoted by dotted purple lines. For
gµBBext ≫ t, gµB∆BN , the split-off triplets (T
and T−
) are
hardly perturbed and current flow is blocked when they be-
come occupied. Parameters: t = 0.2µeV, gµBBN,L=(0.1,0,-
0.1)µeV, gµBBN,R=(-0.1,-0.2,-0.2)µeV and gµBBext=2µeV.
in a homogeneous and an inhomogeneous part (see right-
hand side of (2)). The homogeneous part simply adds
vectorially to the external field Bext, changing slightly
the Zeeman splitting and preferred spin orientation of the
triplet states. The inhomogeneous part ∆BN ≡ BLN −
N on the other hand couples the triplet states to the
singlet state, as can be seen readily by combining the
spin operators in the following way
SxL − SxR =
|S11〉〈T−11| − |S11〉〈T
11| + h.c.
i|S11〉〈T−11| − i|S11〉〈T
11| + h.c.
SzL − SzR =
|S11〉〈T 011|+ |T 011〉〈S11|
. (3)
The first two expressions reveal that the inhomogeneous
field in the transverse plane ∆BxN , ∆B
mixes the |T+11〉
and |T−11〉 states with the |S11〉. The longitudinal com-
ponent ∆BzN mixes |T 011〉 with |S11〉 (third expression).
The degree of mixing between two states will depend
strongly on the energy difference between them [5]. In
the case of gµBBext, t < gµB
〉, the three triplet
states are close in energy to the |S11〉 state. Their
intermixing will be strong, lifting spin blockade. For
gµBBext ≫ t, gµB
〉 the |T+11〉 and |T
11〉 states
are split off in energy by an amount of gµBBext. Con-
sequently the perturbation of these states caused by the
nuclei will be small. Although the |T 011〉 remains mixed
with the |S11〉 state, the occupation of one of the two
split-off triplet states can block the flow through the sys-
The effect of nuclear mixing is shown in Fig. 2 [5].
The observed current flow through the system is typi-
cally in the order of a few hundreds of fA (Fig. 2a).
At zero field, where the mixing is strongest, the current
flow is largest. Increasing the field gradually restores
spin blockade. Fig. 2b shows the energy levels for zero
and finite external field. The theoretical calculations of
the nuclear-spin mediated current flow (obtained from
a master equation approach) are discussed in references
[12, 15].
E. Oscillating magnetic field and rate equations
So far, we have seen that the occurrence of transitions
between singlet and triplet spin states are detectable as
a small current in the spin blockade regime. We will
now discuss how this lifting of spin blockade can also be
used to detect single spin rotations, induced via electron
spin resonance. The basic idea is the following. The
basic idea is the following. If the system is blocked in
e.g. | ↑〉| ↑〉, and the driving field rotates e.g. the left
spin, then transitions are induced to the state | ↓〉| ↑〉.
This state contains a singlet component and therefore a
probability for the electron to move to the right dot and
right lead. Inducing single spin rotations can therefore
lift spin blockade.
However, together with the driving field, the spin tran-
sitions are much more complicated due to the interplay of
different processes: spin resonance of the two spins, inter-
action with the nuclear fields, spin state hybridization by
coherent dot coupling and inelastic transitions from the
S(1,1) state to the S(0,2) state. In order to understand
the interplay of these processes, we will first model the
system with a time-dependent Hamiltonian and a den-
sity matrix approach. Next, we will discuss the physical
interpretation of the simulation results.
The Hamiltonian now also contains a term with an os-
cillating magnetic field in the x-direction with amplitude
Hac(t) = gµBBac sin(ωτ) · (SxL + SxR). (4)
We assume that Bac is equal in both dots, which is a
reasonable approximation in the experiment (from simu-
lations we find that the difference of Bac is 20% at most
[1]). We assume Bext ≫ BN , Bac, which allows applica-
tion of the rotating wave approximation [16]. Therefore,
we will define B1 ≡ 12Bac, which is in the rotating frame
the relevant driving field for the ESR process.
In order to study the effect of ESR and the nuclear
fields that are involved in the transport cycle, we will
construct rate equations that include the unitary evo-
lution of the spins in the dots governed by the time-
dependent Hamiltonian. This approach is based on the
model of reference [12], where the Hamiltonian contained
only time-independent terms. Seven states are involved
in the transport cycle, namely the three (1,1) triplets
|T i11〉, the double and single dot singlet states |S11〉 and
|S02〉 and the two (0,1) states | ↑01〉 and | ↓01〉, making
the density operator a 7× 7 matrix. The rate equations
based on the time-independent Hamiltonian are given in
[12]. These are constructed from the term that gives the
unitary evolution of the system governed by the Hamilto-
nian (H = H0 +Hac) dρ̂k/dτ = − i~ 〈k|[H, ρ̂]|k〉, together
with terms that account for incoherent tunnelling pro-
cesses between the states. The rate equations for the
diagonal elements are given by
= − i
〈T+11|[H, ρ̂]|T
ρ̂↑01
= − i
〈T−11|[H, ρ̂]|T
ρ̂↓01
dρ̂T 0
〈T 011|[H, ρ̂]|T 011〉+
ρ̂↑01 + ρ̂↓01
dρ̂S11
〈S11|[H, ρ̂]|S11〉+
ρ̂↑01 + ρ̂↓01
− Γinρ̂S11
dρ̂S02
〈S02|[H, ρ̂]|S02〉+ Γinρ̂S11 − ΓRρ̂S02
dρ̂↑01
ρ̂S02 − ΓLρ̂↑01
dρ̂↓01
ρ̂S02 − ΓLρ̂↓01 (5)
The rate equations for the off-diagonal elements are
0 2 4 6 8 10
Time ( s)m
transport via S02
RF on
Spin blockade
(0,  /  )
FIG. 3: Time evolution of the diagonal elements of the density
matrix for one particular nuclear configuration. Parameters:
~ω = gµB100mT, Bext =100 mT, B
N,x,y,z =(0,0,2.2) mT,
BRN,x,y,z =(0,0,0), B1 = 1.3 mT, ΓL = 73 MHz, ΓR = 73
MHz, ~Γin = gµBB
N,z and ∆LR=200µeV, t=0.3 µeV.
given by
dρ̂jk
= − i
〈j|[H, ρ̂]|k〉 − 1
Γj + Γk
ρ̂jk (6)
where the indices j, k ∈
T i11, S11, S02, ↑01, ↓01
label the
states available to the system. The tunneling/projection
rates Γj equal Γin and ΓR for the |S11〉 and |S02〉 states
respectively, and equal zero for the other 5 states. The
first term on the right-hand side describes the unitary
evolution of the system, while the second term describes
a loss of coherence due to the finite lifetime of the sin-
glet states. This is the first source of decoherence in our
model. The second one is the inhomogeneous broaden-
ing due to the interaction with the nuclear system. We
do not consider other sources of decoherence, as they are
expected to occur on much larger timescales.
Because we added a time-dependent term to the
Hamiltonian (the oscillating field), we numerically calcu-
late the time evolution of ρ̂(t), treating the Hamiltonian
as stationary on the timescale ∆τ ≪ 2π/ω. To reduce the
simulation time, we use the steady state solution ρ̂τ→∞
in the absence of the oscillating magnetic field as the ini-
tial state ρ̂(τ = 0) for the time evolution. At τ = 0 the
oscillating field is turned on and the system evolves to-
wards a dynamic equilibrium on a timescale set by the
inverse of the slowest tunnelling rate Γ. This new equi-
librium distribution of populations is used to calculate
the current flow, which is proportional to the occupation
of the |S02〉 state (I = eΓRρ̂S02). An example of the time
evolution of the density matrix elements is shown in Fig.
3. The figure clearly reveals that the blockade is lifted
when the oscillating field is applied. This is visible as an
increase of the occupation of the |S02〉 state.
In order to simulate the measured current flow we have
to consider the fact that the measurements are taken
with a sampling rate of 1 Hz. As the timescale of the
nuclear dynamics is believed to be much faster than 1
Hz [10, 13, 14], we expect each datapoint to be an in-
tegration of the response over many configurations of
the nuclei. The effect of the evolving nuclear system is
included in the calculations by averaging the different
values of the (calculated) current flow obtained for each
frozen configuration. These configurations are randomly
sampled from a gaussian distribution of nuclear fields in
the left and right dot (similar as in [12]). Because the
electron in the two dots interact with different nuclear
spins, the isotropic gaussian distributions in the two dots
are uncorrelated, such that
〉 and
〈B2N,x〉 = 〈B2N,y〉 = 〈B2N,z〉. For the sake of convenience
we define
〉 and σN,z =
-100 -50 0 50 100
B (mT)ext
-g B =hm wB ext g B =hm wB ext
Simulation: Inelastic transport ( )DLR>>t
h )G m sin B N/(g
B (mT)ext
-g B =hm wB ext g B =hm wB ext
-100 -50 0 50 100
Simulation: Resonant transport
t/(gm sB N)
( )DLR=0
0 50-50 100-100
RF at 460 MHz
P~-16dBm
RF off
-g B =hm wB ext g B =hm wB ext
B (mT)ext
Experimental data
FIG. 4: a). Calculated average current flow in the inelastic
transport regime. Parameters: ~ω = gµB100mT, Bext =100
mT, σN =2.2mT, B1 = 1.3mT, ΓL,R = 73 MHz, t=0.3 µeV
and ∆LR = 200 µeV. Results are similar for any value for t,
provided that ∆LR ≫ t. b) Calculated average current flow
in the resonant transport regime at zero detuning for differ-
ent values of t. Parameters: ~ω = gµB100mT, σN =2.2mT,
B1 = 1.3mT, ΓL,R = 73 MHz, Γin = 0 and ∆LR = 0. Av-
eraged over 400 nuclear configurations for t/(gµBσN ) > 0.5
and 60 configurations for t/(gµBσN) = 0.5. Simulation car-
ried out for positive magnetic fields only; values shown for
negative fields are equal to results obtained for positive field.
c) Experimental data from Ref [1] with (curve offset by 100
fA for clarity) and without oscillating magnetic field. The fre-
quency of the oscillating magnetic field is 460 MHz and the
applied power is -16dBm. Simulation carried out for posi-
tive magnetic fields only; values shown for negative fields are
equal to results obtained for positive field.
F. Simulation results and physical picture
An example of the calculated (average) current flow
as a function of Bext (Fig. 4a,b) shows a (split) peak
around zero magnetic field and two satellite peaks for
Bext = ±~ω/(gµB), where the spin resonance condition
is satisfied. This (split) peak atBext = 0 is due to singlet-
triplet mixing by the inhomogeneous nuclear field, and
the splitting depends on the tunnel coupling, similar as
the observations in [5]. The response from the induced
spin flips via the driving field is visible for the both in-
elastic and resonant transport regime, and the current
flow has comparable magnitude to the peak at Bext = 0.
The satellite peaks are also visible in the experimental
data from [1] (also shown here in Fig. 4), although the
shape and width of the satellite peaks are different, as
we will discuss later.
We want to stress that the ESR satellite peaks only
appear when an inhomogeneous nuclear field is present
in the simulations. In other words, for ∆BN = 0 and
B1 equal in both dots, spin rotations are induced in both
dots at the same time and at the same rate. Starting,
for example, from the state |T+11〉 = | ↑↑〉 transitions are
induced to the state | ↓↓〉 via the intermediate state | ↑
+ ↓〉|↑ + ↓〉/
2 = (|T+11〉+|T
11〉+2|T 011〉)/
2. No mixing
with the singlet state takes place (the evolution is in the
triplet-subspace) and no current will therefore flow.
The ESR sattelite peaks are visible for both resonant
and inelastic transport regime (Figs. 4a,b). For the res-
onant transport regime, we see that for t/σN < 5 the
sattelite peak increases in height when increasing t, sim-
ply because the coupling between the two singlet states
increases. However, further increasing t reduces the sig-
nal, and this is because the exchange splitting then plays
a more important role. Namely, increasing the exchange
splitting reduces the mixing between the T 011 state with
the hybridized singlet state by the nuclear field gradi-
ent. This mixing is a crucial element for detecting the
induced rotations of one of the two electron spins. In
the inelastic transport regime, this exchange splitting is
negligibly small and therefore the height of the sattelite
peak depends only on Γin and the driving field B1.
A study of the height of the satellite peak as a function
of B1 reveals a non-monotonous behaviour, which can
be seen in Fig. 5a. The physical picture behind this
behavior is most easily sketched by distinguishing three
regimes:
1. For B1 < σN,z, for most of the nuclear configu-
rations the spin in at most one of the two dots
is on resonance, so spins are flipped in either the
left or right dot. In that case transitions are in-
duced from e.g. | ↑↑〉 to | ↑↓〉 = |S11〉 + |T 011〉 or
|↓↑〉 = |S11〉 − |T 011〉. The resulting current flow ini-
tially increases quadratically with B1, as one would
normally expect (Fig. 5a).
2. For B1 ≫ σN,z, for most of the nuclear configu-
rations two spins are rotated simultaneously due
B1 (mT)
0 0.2 0.4 0.6 0.8 2 3 4 5 6 7 8
Experimental data
Detuning ~360 eVm
Detuning ~320 eVm
Model
sN,z=1.3mT
0.10.01 1 10
B1 N,z/ 2s
FIG. 5: Height and width of the ESR satellite peak. a) Cir-
cles: calculated ESR peak height as a function of driving am-
plitude B1. Parameters: ~ω = gµB100mT, Bext =100 mT,
σN =2.2mT, ΓL,R = 73 MHz, t=0.3 µeV, ~Γin = gµBσN
and ∆LR = 200 µeV. Lines are the current measurements
for 2 different values of ∆LR. The measurements show time-
dependent (telegraph type) behavior. Therefore, the curves
are obtained by repeating sweeps of B1 and then selecting
the largest current value for each value of B1. b) Calculated
width of the ESR satellite peaks as a function of B1. For
small ESR power the peak is broadened by the random nu-
clear fluctuations, at high powers it is broadened by B1.
to power broadening of the Rabi resonance. The
stronger B1, the more the transitions occur only in
the triplet subspace (the driving field B1 that ro-
tates two spins dominates the S−T0 mixing by the
nuclear spins). As a result, the current decreases
for increasing B1.
3. If B1 ∼ σN,z the situation is more complex because
both processes (rotation of 2 spins simultaneously
and transitions from T 011 to S11) are effective. We
find that if both processes occur with comparable
rates, the overall transition rate to the singlet state
is highest. This is the reason why the current has
a maximum at B1 ≈ σN,z (Fig. 5a).
The experimental data of the ESR satellite peak height
(normalized by the zero-field current flow) for two dif-
ferent values of ∆LR are shown in Fig. 5a. In order to
compare the experimental results with the model we have
estimated the rate Γin from the measured current flow at
Bext = 0 (we found similar values for both curves). The
agreement of the experimental data with the model is
reasonable, as it shows the expected quadratic increase
with B1, as well as a comparable peak height. However,
we see that variations of the level detuning ∆LR can re-
sult in considerable differences of the measured ESR peak
height. We have two possible explanations for the devi-
ations of the experimental data with the model. First,
we have found experimental signatures of dynamic nu-
clear polarization when the ESR resonance condition was
fulfilled. We expect that this is due to feedback of the
electron transport on the nuclear spins (similar to that
discussed in [11, 15, 17]), although the exact processes
are not (yet) fully understood. Second, unwanted electric
fields affect the electron tunnelling processes, but are not
taken into account in the model. We expect that these
electric fields will not change the location and width of
the ESR sattelite peaks because this field does not couple
the spin states. It is however possible that the height of
the satellite peak is altered by the electric field because if
can affect the coupling between the S(0,2) with the S(1,1)
state.
Finally, we discuss the width of the ESR satellite peak
(Fig. 5b). If the inelastic tunnelling process between
the dots (with rate Γin) and B1 are both smaller than
σN,z, the ESR peak (obtained from simulations) is broad-
ened by the statistical fluctuations of the effective nuclear
field. For high B1, the width approaches asymptotically
the line with slope 1 (see Fig. 5b). In this regime, the
peak is broadened by the RF amplitude B1. In the ex-
periment [1], the shape of the satellite peak was different
(flat on top with sharp edges) than expected from the
model. Furthermore, the FWHM was larger than ex-
pected from just σN,z. We attribute this to feedback of
the ESR-induced current flow on the nuclear spin bath.
As a result, a clear FWHM increase with B1 could not
be observed.
It should be noted that in the simulation the central
peak is broader than the satellite peaks. From study-
ing the influence of various parameters in the model, we
conclude that the greater width of the central peak is
caused by the tranverse nuclear field fluctuations (BN,x
and BN,y), which broaden the central peak but not the
ESR satellite peaks.
We conclude that the model discussed here qualita-
tively agrees with the main features that were observed
in the double dot transport measurements that aims at
detecting (continuous wave) ESR of a single electron spin.
The details of the ESR satellite peak height and width
do not agree quantitatively with the model. We believe
these deviations can be attributed to unwanted electric
fields and feedback of the electron transport on the nu-
clear spin polarization. Improving the understanding of
these feedback mechanisms remains interesting for future
investigation as it might point towards a direction to mit-
igate the decoherence of the electron spin [12, 18].
Acknowledgments
This study was supported by the Dutch Organization
for Fundamental Research on Matter (FOM), the Nether-
lands Organization for Scientific Research (NWO) and
the Defense Advanced Research Projects Agency Quan-
tum Information Science and Technology program.
[1] F. Koppens, C. Buizert, K. Tielrooij, I. Vink, K. Nowack,
T. Meunier, L. Kouwenhoven, and L. Vandersypen, Na-
ture 442, 766 (2006).
[2] H. A. Engel and D. Loss, Phys. Rev. Lett. 86, 4648
(2001).
[3] K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha, Sci-
ence 297, 1313 (2002).
[4] A. C. Johnson, J. R. Petta, J. M. Taylor, A. Yacoby,
M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C.
Gossard, Nature 435, 925 (2005).
[5] F. H. L. Koppens, J. A. Folk, J. M. Elzerman, R. Han-
son, L. H. W. van Beveren, I. T. Vink, H. P. Tranitz,
W. Wegscheider, L. P. Kouwenhoven, and L. M. K. Van-
dersypen, Science 309, 1346 (2005).
[6] P.-F. Braun,X. Marie, L. Lombez, B. Urbaszek, T.
Amand, P.Renucci, V. K. Kalevich, K. V. Kavokin, O.
Krebs, P. Voisin and Y. Masumoto, Phys. Rev. Lett. 94,
116601 (2005).
[7] D. Gammon, Al. L. Efros, T. A. Kennedy, M. Rosen, D.
S. Katzer, D. Park, S.W. Brown, V. L. Korenev, I. A.
Merkulov, Phys. Rev. Lett. 86, 5176 (2001).
[8] I.A. Merkulov, A.L. Efros, J. Rosen, Phys. Rev. B 65,
205309 (2002).
[9] A. V. Khaetskii, D. Loss, L. Glazman, Phys. Rev. Lett.
88, 186802 (2002).
[10] W. A. Coish and D. Loss, Phys. Rev. B 70, 195340
(2004).
[11] K. Ono, S. Tarucha, Phys. Rev. Lett. 92, 256803 (2004).
[12] O. N. Jouravlev and Y. V. Nazarov, Phys. Rev. Lett. 96,
176804 (2006).
[13] R. de Sousa, S. Das Sarma, Phys. Rev. B 67, 033301
(2003).
[14] D. Paget, G. Lampel, B. Sapoval, V.I. Safarov, Phys.
Rev. B 15, 5780 (1977).
[15] J. Inarrea, G. Platero, and A. H. MacDonald
(2006), URL http://www.citebase.org/abstract?id=
oai:arXiv.org:cond-mat/0609323.
[16] C. Poole, Electron Spin Resonance, 2nd ed. (Wiley, New
York, 1983).
[17] M. S. Rudner and L. S. Levitov (2006), URL
http://www.citebase.org/abstract?id=oai:
arXiv.org:cond-mat/0609409.
[18] D. Klauser, W. Coish, and D. Loss, Phys. Rev. B 73,
205302 (2006).

  Spin-dependent transport measurements through a double quantum dot are a
valuable tool for detecting both the coherent evolution of the spin state of a
single electron as well as the hybridization of two-electron spin states. In
this paper, we discuss a model that describes the transport cycle in this
regime, including the effects of an oscillating magnetic field (causing
electron spin resonance) and the effective nuclear fields on the spin states in
the two dots. We numerically calculate the current flow due to the induced spin
flips via electron spin resonance and we study the detector efficiency for a
range of parameters. The experimental data are compared with the model and we
find a reasonable agreement.

Introduction
Recently, coherent spin rotations of a single electron
were demonstrated in a double quantum dot device [1].
In this system, spin-flips of an electron in the dot were in-
duced via an oscillating magnetic field (electron spin res-
onance or ESR) and detected through a spin-dependent
transition of the electron to another dot, which already
contained one additional electron. This detection scheme
is an extension of the proposal for ESR detection in a sin-
gle quantum dot by Engel and Loss [2]. Briefly, the device
can be operated (in a spin blockade regime [3]) such that
the electron in the left dot can only move to the right dot
if a spin flip in one of the two dots is induced via ESR.
From the right dot, the electron exits to the right reser-
voir and another electron enters the left dot from the left
reservoir. A continuous repetition of this transition will
result in a net current flow.
Compared to the single dot detection scheme [2], us-
ing the double-dot as the detector has two major advan-
tages. First, the experiment can be performed at a lower
static magnetic field and consequently with lower, tech-
nically less demanding, excitation frequencies. Second,
the spin detection is rather insensitive to unwanted oscil-
lating electric fields, because the relevant dot levels can
be positioned far from the Fermi energies of the leads.
These electric fields are unavoidably generated together
with the oscillating magnetic field as well.
The drawback of the double-dot detector is that spin
detection is based on the projection in the two-electron
singlet-triplet basis, while the aim is to detect single spin
rotations. However, this detection is still possible be-
cause the electrons in the two dots experience different
effective nuclear fields. This is due to the hyperfine inter-
action of the electron spins with the (roughly 106) nuclear
spins in the host semiconductor material of each quan-
tum dot [4–11]. In order to provide more insight in this
double-dot ESR detection scheme for single spin rota-
tions, it is necessary to analyze the coherent evolution of
the two-electron spin states together with the transitions
∗Electronic address: f.h.l.koppens@tudelft.nl; Kavli Institute of
NanoScience Delft, P.O. Box 5046, 2600 GA Delft, The Netherlands
in the transport cycle.
In this paper, we discuss a model that describes the
transport cycle in the spin blockade regime while includ-
ing the coherent coupling between the two dots, and the
influence of the static and oscillating magnetic field to-
gether with the effective nuclear fields on the electron
spin states. The aim is to understand how effectively
single spin resonance will affect the measured quantity
in the experiment, namely the current flow in the spin
blockade regime. The organization of this paper is as
follows. First, we will explain the transport cycle and
the mechanism that causes spin blockade. Next, we will
briefly discuss the static system Hamiltonian and the
mixing of the two-electron spin states by the effective
nuclear field. Then we add an oscillating magnetic field
to this Hamiltonian, that forms -together with the double
dot tunnelling processes- the basis of the rate equations
that describe how the density matrix of the two-electron
spin states evolves in time. The current flow, calculated
from the steady state solution of the density operator,
is then analyzed for different coherent coupling values,
magnitudes of the oscillating magnetic field, in combina-
tion with different effective nuclear fields in the two dots.
This provides further insight in the optimal conditions
for spin-flip detection with a double quantum dot.
B. Spin blockade
In the spin-blockade regime, the double-dot is tuned
such that one electron always resides in the right dot,
and a second electron can tunnel from the left reservoir
through the left and right dots, to the right reservoir [3].
This current-carrying cycle can be described with the
occupations (m, n) of the left and right dots: (1, 1) →
(0, 2) → (0, 1) → (1, 1). When an electron enters the
left dot and forms a double-dot singlet state S11 with the
electron in the right dot (S = |↑↓〉 − |↓↑〉, normalization
omitted for brevity), it is possible for the left electron
to move to the right dot, because the right dot singlet
state S02 is energetically accessible. Next, one electron
tunnels from the right dot to the right lead and another
electron can again tunnel into the left dot. If, however,
the two electrons form a double-dot triplet state T11, the
left electron cannot move to the right dot, as the right
http://arxiv.org/abs/0704.1628v1
dot triplet state T02 is much higher in energy (due to
the relatively large exchange splitting in a single dot).
The electron can also not move back to the lead and
therefore further current flow is blocked as soon as any
of the (double-dot) triplet states is formed (Fig. 1a,b).
Spin blockade only occurs if at least one of the eigen-
states of the system Hamiltonian is a pure triplet state.
If processes are present that induce transitions from all
the three (1,1) triplet states to the (1,1) singlet state,
spin blockade is lifted and a current will flow. As we will
see below, the presence of the nuclear spins in the host
semiconductor can give rise to these kind of transitions.
This can be seen most easily by adding the effect of the
hyperfine interaction to the system Hamiltonian.
C. System Hamiltonian
The system Hamiltonian is most conveniently writ-
ten in the two-electron singlet-triplet basis with the
quantization-axis in the z-direction. The basis states are
S11, T
11, T
11, T
11 and S02. The subscript m,n denotes
the dot occupancy. We exclude the T02 state from the
model, because this state is energetically inaccessible and
therefore does not play an important role in the trans-
DLR>>t DLR=0
S - S11 02
S - S11 02
-10 -5 0 5 10
DLR/t
b) a)
FIG. 1: a) A schematic of the double dot and the electro-
chemical potentials (energy relative to the (0,1) state) of the
relevant two-electron spin states. For ∆LR > t, transitions
from the S11 state to the S02 state are possible via inelastic
relaxation with rate Γin. Spin blockade occurs when one of
the T i11 states is occupied. b) Similar schematic for ∆LR = 0,
where the singlet states are hybridized. Also in this case,
spin blockade occurs when one of T i11 states is occupied. c)
Energy levels as a function of detuning. At ∆LR = 0, the
singlet states hybridize into bonding and anti-bonding states.
The splitting between the triplets states corresponds to the
Zeeman energy gµBBext.
port cycle. Furthermore, we neglect the thermal energy
kT in the description, which is justified when the bias
over the two dots is much larger than kT . The system
Hamiltonian is given by
H0 = − ∆LR|S02〉〈S02|+ t
|S11〉〈S02|+ |S02〉〈S11|
− gµBBext
|T+11〉〈T
11| − |T
11〉〈T
, (1)
where ∆LR is the energy difference between the |S11〉
and |S02〉 state (level detuning, see Fig.1a), t is the tun-
nel coupling between the |S11〉 and |S02〉 states, Bext the
external magnetic field in the z-direction and Sz
L(R) the
spin operator along z for the left (right) electron. The
eigenstates of the Hamiltonian (1) for finite external field
are shown in figure 1c. For ∆LR < t, the tunnel coupling
t causes an anti-crossing of the |S11〉 and |S02〉 states. For
∆LR < 0, transport is blocked by Coulomb blockade (i.e.
the final state |S02〉 is at a higher energy than the initial
state S11). For ∆LR ≥ 0, transport will be blocked when
one of the three triplet states becomes occupied (spin
blockade). In Fig.1a and b, we distinguish two regimes:
∆LR > t where the (exchange) energy splitting between
T 011 and S11 is negligibly small and transitions from S11
to S02 occur via inelastic relaxation with rate Γin and the
energy. A different regime holds for ∆LR < t, where S11
is coherently coupled with S02 giving rise to a finite (ex-
change) splitting between T 011 and the hybridized singlet
states. We will return to this distinction in the discussion
below.
D. Singlet-triplet mixing by the nuclear spins
The effect of the hyperfine interaction with the nuclear
spins can be studied [12] by adding a static (frozen) ef-
fective nuclear field BLN (B
N ) at the left (right) dot to
the system Hamiltonian:
Hnucl = −gµB
N · SL +BRN · SR
= −gµB(BLN −BRN ) · (SL − SR)/2
−gµB(BLN +BRN ) · (SL + SR)/2. (2)
For the sake of convenience, we separate the inhomoge-
neous and homogeneous contribution, for reasons which
we will discuss later. Considering the nuclear field as
static is justified since the tunnel rates and electron spin
dynamics are expected to be much faster than the dy-
namics of the nuclear system [10, 13, 14]. Therefore,
we will treat the Hamiltonian as time-independent. The
effect of nuclear reorientation will be included later by
ensemble averaging.
We will show now that triplet states mix with the S11
state if the nuclear field is different in the two dots (in
all three directions). This mixing will lift spin blockade,
detectable as a finite current running through the dots for
∆LR ≥ 0. The effective nuclear field can be decomposed
-30 -20 -10 0 10 20 30
Magnetic field (mT)
Bext=0 B » Bext ND
-10 -5 0 5 10
10 S02
a +b +g +kS T T T11 11 11 11
-10 -5 0 5 10
a +kS T11 11
DLR/t DLR/t
FIG. 2: a) Observed current flow in the inelastic transport
regime (gµB∆LR ≫ t) due to singlet-triplet mixing by the
nuclei. b) Electrochemical potentials in the presence of Hnucl
(t ∼ ∆BN ). Singlet and triplet eigenstates are denoted by
red and blue lines respectively. Hybridized states (of sin-
glet and triplet) are denoted by dotted purple lines. For
gµBBext ≫ t, gµB∆BN , the split-off triplets (T
and T−
) are
hardly perturbed and current flow is blocked when they be-
come occupied. Parameters: t = 0.2µeV, gµBBN,L=(0.1,0,-
0.1)µeV, gµBBN,R=(-0.1,-0.2,-0.2)µeV and gµBBext=2µeV.
in a homogeneous and an inhomogeneous part (see right-
hand side of (2)). The homogeneous part simply adds
vectorially to the external field Bext, changing slightly
the Zeeman splitting and preferred spin orientation of the
triplet states. The inhomogeneous part ∆BN ≡ BLN −
N on the other hand couples the triplet states to the
singlet state, as can be seen readily by combining the
spin operators in the following way
SxL − SxR =
|S11〉〈T−11| − |S11〉〈T
11| + h.c.
i|S11〉〈T−11| − i|S11〉〈T
11| + h.c.
SzL − SzR =
|S11〉〈T 011|+ |T 011〉〈S11|
. (3)
The first two expressions reveal that the inhomogeneous
field in the transverse plane ∆BxN , ∆B
mixes the |T+11〉
and |T−11〉 states with the |S11〉. The longitudinal com-
ponent ∆BzN mixes |T 011〉 with |S11〉 (third expression).
The degree of mixing between two states will depend
strongly on the energy difference between them [5]. In
the case of gµBBext, t < gµB
〉, the three triplet
states are close in energy to the |S11〉 state. Their
intermixing will be strong, lifting spin blockade. For
gµBBext ≫ t, gµB
〉 the |T+11〉 and |T
11〉 states
are split off in energy by an amount of gµBBext. Con-
sequently the perturbation of these states caused by the
nuclei will be small. Although the |T 011〉 remains mixed
with the |S11〉 state, the occupation of one of the two
split-off triplet states can block the flow through the sys-
The effect of nuclear mixing is shown in Fig. 2 [5].
The observed current flow through the system is typi-
cally in the order of a few hundreds of fA (Fig. 2a).
At zero field, where the mixing is strongest, the current
flow is largest. Increasing the field gradually restores
spin blockade. Fig. 2b shows the energy levels for zero
and finite external field. The theoretical calculations of
the nuclear-spin mediated current flow (obtained from
a master equation approach) are discussed in references
[12, 15].
E. Oscillating magnetic field and rate equations
So far, we have seen that the occurrence of transitions
between singlet and triplet spin states are detectable as
a small current in the spin blockade regime. We will
now discuss how this lifting of spin blockade can also be
used to detect single spin rotations, induced via electron
spin resonance. The basic idea is the following. The
basic idea is the following. If the system is blocked in
e.g. | ↑〉| ↑〉, and the driving field rotates e.g. the left
spin, then transitions are induced to the state | ↓〉| ↑〉.
This state contains a singlet component and therefore a
probability for the electron to move to the right dot and
right lead. Inducing single spin rotations can therefore
lift spin blockade.
However, together with the driving field, the spin tran-
sitions are much more complicated due to the interplay of
different processes: spin resonance of the two spins, inter-
action with the nuclear fields, spin state hybridization by
coherent dot coupling and inelastic transitions from the
S(1,1) state to the S(0,2) state. In order to understand
the interplay of these processes, we will first model the
system with a time-dependent Hamiltonian and a den-
sity matrix approach. Next, we will discuss the physical
interpretation of the simulation results.
The Hamiltonian now also contains a term with an os-
cillating magnetic field in the x-direction with amplitude
Hac(t) = gµBBac sin(ωτ) · (SxL + SxR). (4)
We assume that Bac is equal in both dots, which is a
reasonable approximation in the experiment (from simu-
lations we find that the difference of Bac is 20% at most
[1]). We assume Bext ≫ BN , Bac, which allows applica-
tion of the rotating wave approximation [16]. Therefore,
we will define B1 ≡ 12Bac, which is in the rotating frame
the relevant driving field for the ESR process.
In order to study the effect of ESR and the nuclear
fields that are involved in the transport cycle, we will
construct rate equations that include the unitary evo-
lution of the spins in the dots governed by the time-
dependent Hamiltonian. This approach is based on the
model of reference [12], where the Hamiltonian contained
only time-independent terms. Seven states are involved
in the transport cycle, namely the three (1,1) triplets
|T i11〉, the double and single dot singlet states |S11〉 and
|S02〉 and the two (0,1) states | ↑01〉 and | ↓01〉, making
the density operator a 7× 7 matrix. The rate equations
based on the time-independent Hamiltonian are given in
[12]. These are constructed from the term that gives the
unitary evolution of the system governed by the Hamilto-
nian (H = H0 +Hac) dρ̂k/dτ = − i~ 〈k|[H, ρ̂]|k〉, together
with terms that account for incoherent tunnelling pro-
cesses between the states. The rate equations for the
diagonal elements are given by
= − i
〈T+11|[H, ρ̂]|T
ρ̂↑01
= − i
〈T−11|[H, ρ̂]|T
ρ̂↓01
dρ̂T 0
〈T 011|[H, ρ̂]|T 011〉+
ρ̂↑01 + ρ̂↓01
dρ̂S11
〈S11|[H, ρ̂]|S11〉+
ρ̂↑01 + ρ̂↓01
− Γinρ̂S11
dρ̂S02
〈S02|[H, ρ̂]|S02〉+ Γinρ̂S11 − ΓRρ̂S02
dρ̂↑01
ρ̂S02 − ΓLρ̂↑01
dρ̂↓01
ρ̂S02 − ΓLρ̂↓01 (5)
The rate equations for the off-diagonal elements are
0 2 4 6 8 10
Time ( s)m
transport via S02
RF on
Spin blockade
(0,  /  )
FIG. 3: Time evolution of the diagonal elements of the density
matrix for one particular nuclear configuration. Parameters:
~ω = gµB100mT, Bext =100 mT, B
N,x,y,z =(0,0,2.2) mT,
BRN,x,y,z =(0,0,0), B1 = 1.3 mT, ΓL = 73 MHz, ΓR = 73
MHz, ~Γin = gµBB
N,z and ∆LR=200µeV, t=0.3 µeV.
given by
dρ̂jk
= − i
〈j|[H, ρ̂]|k〉 − 1
Γj + Γk
ρ̂jk (6)
where the indices j, k ∈
T i11, S11, S02, ↑01, ↓01
label the
states available to the system. The tunneling/projection
rates Γj equal Γin and ΓR for the |S11〉 and |S02〉 states
respectively, and equal zero for the other 5 states. The
first term on the right-hand side describes the unitary
evolution of the system, while the second term describes
a loss of coherence due to the finite lifetime of the sin-
glet states. This is the first source of decoherence in our
model. The second one is the inhomogeneous broaden-
ing due to the interaction with the nuclear system. We
do not consider other sources of decoherence, as they are
expected to occur on much larger timescales.
Because we added a time-dependent term to the
Hamiltonian (the oscillating field), we numerically calcu-
late the time evolution of ρ̂(t), treating the Hamiltonian
as stationary on the timescale ∆τ ≪ 2π/ω. To reduce the
simulation time, we use the steady state solution ρ̂τ→∞
in the absence of the oscillating magnetic field as the ini-
tial state ρ̂(τ = 0) for the time evolution. At τ = 0 the
oscillating field is turned on and the system evolves to-
wards a dynamic equilibrium on a timescale set by the
inverse of the slowest tunnelling rate Γ. This new equi-
librium distribution of populations is used to calculate
the current flow, which is proportional to the occupation
of the |S02〉 state (I = eΓRρ̂S02). An example of the time
evolution of the density matrix elements is shown in Fig.
3. The figure clearly reveals that the blockade is lifted
when the oscillating field is applied. This is visible as an
increase of the occupation of the |S02〉 state.
In order to simulate the measured current flow we have
to consider the fact that the measurements are taken
with a sampling rate of 1 Hz. As the timescale of the
nuclear dynamics is believed to be much faster than 1
Hz [10, 13, 14], we expect each datapoint to be an in-
tegration of the response over many configurations of
the nuclei. The effect of the evolving nuclear system is
included in the calculations by averaging the different
values of the (calculated) current flow obtained for each
frozen configuration. These configurations are randomly
sampled from a gaussian distribution of nuclear fields in
the left and right dot (similar as in [12]). Because the
electron in the two dots interact with different nuclear
spins, the isotropic gaussian distributions in the two dots
are uncorrelated, such that
〉 and
〈B2N,x〉 = 〈B2N,y〉 = 〈B2N,z〉. For the sake of convenience
we define
〉 and σN,z =
-100 -50 0 50 100
B (mT)ext
-g B =hm wB ext g B =hm wB ext
Simulation: Inelastic transport ( )DLR>>t
h )G m sin B N/(g
B (mT)ext
-g B =hm wB ext g B =hm wB ext
-100 -50 0 50 100
Simulation: Resonant transport
t/(gm sB N)
( )DLR=0
0 50-50 100-100
RF at 460 MHz
P~-16dBm
RF off
-g B =hm wB ext g B =hm wB ext
B (mT)ext
Experimental data
FIG. 4: a). Calculated average current flow in the inelastic
transport regime. Parameters: ~ω = gµB100mT, Bext =100
mT, σN =2.2mT, B1 = 1.3mT, ΓL,R = 73 MHz, t=0.3 µeV
and ∆LR = 200 µeV. Results are similar for any value for t,
provided that ∆LR ≫ t. b) Calculated average current flow
in the resonant transport regime at zero detuning for differ-
ent values of t. Parameters: ~ω = gµB100mT, σN =2.2mT,
B1 = 1.3mT, ΓL,R = 73 MHz, Γin = 0 and ∆LR = 0. Av-
eraged over 400 nuclear configurations for t/(gµBσN ) > 0.5
and 60 configurations for t/(gµBσN) = 0.5. Simulation car-
ried out for positive magnetic fields only; values shown for
negative fields are equal to results obtained for positive field.
c) Experimental data from Ref [1] with (curve offset by 100
fA for clarity) and without oscillating magnetic field. The fre-
quency of the oscillating magnetic field is 460 MHz and the
applied power is -16dBm. Simulation carried out for posi-
tive magnetic fields only; values shown for negative fields are
equal to results obtained for positive field.
F. Simulation results and physical picture
An example of the calculated (average) current flow
as a function of Bext (Fig. 4a,b) shows a (split) peak
around zero magnetic field and two satellite peaks for
Bext = ±~ω/(gµB), where the spin resonance condition
is satisfied. This (split) peak atBext = 0 is due to singlet-
triplet mixing by the inhomogeneous nuclear field, and
the splitting depends on the tunnel coupling, similar as
the observations in [5]. The response from the induced
spin flips via the driving field is visible for the both in-
elastic and resonant transport regime, and the current
flow has comparable magnitude to the peak at Bext = 0.
The satellite peaks are also visible in the experimental
data from [1] (also shown here in Fig. 4), although the
shape and width of the satellite peaks are different, as
we will discuss later.
We want to stress that the ESR satellite peaks only
appear when an inhomogeneous nuclear field is present
in the simulations. In other words, for ∆BN = 0 and
B1 equal in both dots, spin rotations are induced in both
dots at the same time and at the same rate. Starting,
for example, from the state |T+11〉 = | ↑↑〉 transitions are
induced to the state | ↓↓〉 via the intermediate state | ↑
+ ↓〉|↑ + ↓〉/
2 = (|T+11〉+|T
11〉+2|T 011〉)/
2. No mixing
with the singlet state takes place (the evolution is in the
triplet-subspace) and no current will therefore flow.
The ESR sattelite peaks are visible for both resonant
and inelastic transport regime (Figs. 4a,b). For the res-
onant transport regime, we see that for t/σN < 5 the
sattelite peak increases in height when increasing t, sim-
ply because the coupling between the two singlet states
increases. However, further increasing t reduces the sig-
nal, and this is because the exchange splitting then plays
a more important role. Namely, increasing the exchange
splitting reduces the mixing between the T 011 state with
the hybridized singlet state by the nuclear field gradi-
ent. This mixing is a crucial element for detecting the
induced rotations of one of the two electron spins. In
the inelastic transport regime, this exchange splitting is
negligibly small and therefore the height of the sattelite
peak depends only on Γin and the driving field B1.
A study of the height of the satellite peak as a function
of B1 reveals a non-monotonous behaviour, which can
be seen in Fig. 5a. The physical picture behind this
behavior is most easily sketched by distinguishing three
regimes:
1. For B1 < σN,z, for most of the nuclear configu-
rations the spin in at most one of the two dots
is on resonance, so spins are flipped in either the
left or right dot. In that case transitions are in-
duced from e.g. | ↑↑〉 to | ↑↓〉 = |S11〉 + |T 011〉 or
|↓↑〉 = |S11〉 − |T 011〉. The resulting current flow ini-
tially increases quadratically with B1, as one would
normally expect (Fig. 5a).
2. For B1 ≫ σN,z, for most of the nuclear configu-
rations two spins are rotated simultaneously due
B1 (mT)
0 0.2 0.4 0.6 0.8 2 3 4 5 6 7 8
Experimental data
Detuning ~360 eVm
Detuning ~320 eVm
Model
sN,z=1.3mT
0.10.01 1 10
B1 N,z/ 2s
FIG. 5: Height and width of the ESR satellite peak. a) Cir-
cles: calculated ESR peak height as a function of driving am-
plitude B1. Parameters: ~ω = gµB100mT, Bext =100 mT,
σN =2.2mT, ΓL,R = 73 MHz, t=0.3 µeV, ~Γin = gµBσN
and ∆LR = 200 µeV. Lines are the current measurements
for 2 different values of ∆LR. The measurements show time-
dependent (telegraph type) behavior. Therefore, the curves
are obtained by repeating sweeps of B1 and then selecting
the largest current value for each value of B1. b) Calculated
width of the ESR satellite peaks as a function of B1. For
small ESR power the peak is broadened by the random nu-
clear fluctuations, at high powers it is broadened by B1.
to power broadening of the Rabi resonance. The
stronger B1, the more the transitions occur only in
the triplet subspace (the driving field B1 that ro-
tates two spins dominates the S−T0 mixing by the
nuclear spins). As a result, the current decreases
for increasing B1.
3. If B1 ∼ σN,z the situation is more complex because
both processes (rotation of 2 spins simultaneously
and transitions from T 011 to S11) are effective. We
find that if both processes occur with comparable
rates, the overall transition rate to the singlet state
is highest. This is the reason why the current has
a maximum at B1 ≈ σN,z (Fig. 5a).
The experimental data of the ESR satellite peak height
(normalized by the zero-field current flow) for two dif-
ferent values of ∆LR are shown in Fig. 5a. In order to
compare the experimental results with the model we have
estimated the rate Γin from the measured current flow at
Bext = 0 (we found similar values for both curves). The
agreement of the experimental data with the model is
reasonable, as it shows the expected quadratic increase
with B1, as well as a comparable peak height. However,
we see that variations of the level detuning ∆LR can re-
sult in considerable differences of the measured ESR peak
height. We have two possible explanations for the devi-
ations of the experimental data with the model. First,
we have found experimental signatures of dynamic nu-
clear polarization when the ESR resonance condition was
fulfilled. We expect that this is due to feedback of the
electron transport on the nuclear spins (similar to that
discussed in [11, 15, 17]), although the exact processes
are not (yet) fully understood. Second, unwanted electric
fields affect the electron tunnelling processes, but are not
taken into account in the model. We expect that these
electric fields will not change the location and width of
the ESR sattelite peaks because this field does not couple
the spin states. It is however possible that the height of
the satellite peak is altered by the electric field because if
can affect the coupling between the S(0,2) with the S(1,1)
state.
Finally, we discuss the width of the ESR satellite peak
(Fig. 5b). If the inelastic tunnelling process between
the dots (with rate Γin) and B1 are both smaller than
σN,z, the ESR peak (obtained from simulations) is broad-
ened by the statistical fluctuations of the effective nuclear
field. For high B1, the width approaches asymptotically
the line with slope 1 (see Fig. 5b). In this regime, the
peak is broadened by the RF amplitude B1. In the ex-
periment [1], the shape of the satellite peak was different
(flat on top with sharp edges) than expected from the
model. Furthermore, the FWHM was larger than ex-
pected from just σN,z. We attribute this to feedback of
the ESR-induced current flow on the nuclear spin bath.
As a result, a clear FWHM increase with B1 could not
be observed.
It should be noted that in the simulation the central
peak is broader than the satellite peaks. From study-
ing the influence of various parameters in the model, we
conclude that the greater width of the central peak is
caused by the tranverse nuclear field fluctuations (BN,x
and BN,y), which broaden the central peak but not the
ESR satellite peaks.
We conclude that the model discussed here qualita-
tively agrees with the main features that were observed
in the double dot transport measurements that aims at
detecting (continuous wave) ESR of a single electron spin.
The details of the ESR satellite peak height and width
do not agree quantitatively with the model. We believe
these deviations can be attributed to unwanted electric
fields and feedback of the electron transport on the nu-
clear spin polarization. Improving the understanding of
these feedback mechanisms remains interesting for future
investigation as it might point towards a direction to mit-
igate the decoherence of the electron spin [12, 18].
Acknowledgments
This study was supported by the Dutch Organization
for Fundamental Research on Matter (FOM), the Nether-
lands Organization for Scientific Research (NWO) and
the Defense Advanced Research Projects Agency Quan-
tum Information Science and Technology program.
[1] F. Koppens, C. Buizert, K. Tielrooij, I. Vink, K. Nowack,
T. Meunier, L. Kouwenhoven, and L. Vandersypen, Na-
ture 442, 766 (2006).
[2] H. A. Engel and D. Loss, Phys. Rev. Lett. 86, 4648
(2001).
[3] K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha, Sci-
ence 297, 1313 (2002).
[4] A. C. Johnson, J. R. Petta, J. M. Taylor, A. Yacoby,
M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C.
Gossard, Nature 435, 925 (2005).
[5] F. H. L. Koppens, J. A. Folk, J. M. Elzerman, R. Han-
son, L. H. W. van Beveren, I. T. Vink, H. P. Tranitz,
W. Wegscheider, L. P. Kouwenhoven, and L. M. K. Van-
dersypen, Science 309, 1346 (2005).
[6] P.-F. Braun,X. Marie, L. Lombez, B. Urbaszek, T.
Amand, P.Renucci, V. K. Kalevich, K. V. Kavokin, O.
Krebs, P. Voisin and Y. Masumoto, Phys. Rev. Lett. 94,
116601 (2005).
[7] D. Gammon, Al. L. Efros, T. A. Kennedy, M. Rosen, D.
S. Katzer, D. Park, S.W. Brown, V. L. Korenev, I. A.
Merkulov, Phys. Rev. Lett. 86, 5176 (2001).
[8] I.A. Merkulov, A.L. Efros, J. Rosen, Phys. Rev. B 65,
205309 (2002).
[9] A. V. Khaetskii, D. Loss, L. Glazman, Phys. Rev. Lett.
88, 186802 (2002).
[10] W. A. Coish and D. Loss, Phys. Rev. B 70, 195340
(2004).
[11] K. Ono, S. Tarucha, Phys. Rev. Lett. 92, 256803 (2004).
[12] O. N. Jouravlev and Y. V. Nazarov, Phys. Rev. Lett. 96,
176804 (2006).
[13] R. de Sousa, S. Das Sarma, Phys. Rev. B 67, 033301
(2003).
[14] D. Paget, G. Lampel, B. Sapoval, V.I. Safarov, Phys.
Rev. B 15, 5780 (1977).
[15] J. Inarrea, G. Platero, and A. H. MacDonald
(2006), URL http://www.citebase.org/abstract?id=
oai:arXiv.org:cond-mat/0609323.
[16] C. Poole, Electron Spin Resonance, 2nd ed. (Wiley, New
York, 1983).
[17] M. S. Rudner and L. S. Levitov (2006), URL
http://www.citebase.org/abstract?id=oai:
arXiv.org:cond-mat/0609409.
[18] D. Klauser, W. Coish, and D. Loss, Phys. Rev. B 73,
205302 (2006).
