The Kondo Effect and Controlled Spin Entanglement in
Coupled Double-Quantum-Dots
Albert M. Chang
Department of Physics, Duke University
Durham, NC 27708
Abstract. Semiconductor double-quantum dots represent an ideal system for studying the novel spin physics of localized
spins. On each quantum dot when the number of electrons is odd and the net spin is 1/2, a strong coupling of this
localized spin to conducting electrons in the leads gives rise to Kondo correlation. On the other hand, in the coupled
double-quantum-dot if the inter-dot antiferromagnetic interaction is strong, the two spins can form a correlated spinsinglet state, quenching the Kondo effect. This competition between Kondo and antiferromagnetic correlation is studied
in a controlled manner by tuning the inter-dot tunnel coupling. Increasing the inter-dot tunneling, we observe a
continuous transition from a single-peaked to a double-peaked Kondo resonance in the differential conductance. On the
double-peaked side, the differential conductance becomes suppressed at zero source-drain bias. The observed strong
suppression of the differential conductance at zero bias provides direct evidence signaling the formation of an entangled
spin-singlet state. This evidence for entanglement and the tunability of our devices bode well for quantum computation
applications.
Keywords: Kondo Effect, Spin-Entanglement, Double Quantum Dots.
PACS: 73.23.–b, 73.63.Kv
INTRODUCTION
Manipulation of the electron spin degree of freedom for future device applications is coming to the forefront [1-7].
In particular, spintronics is an area of research, which seeks to develop future generation spin-based devices. On a
different front, the electron spin holds great promise to form the basis of the qubit in quantum computation
applications. On a more fundamental level, the spin degree of freedom plays a major role in diverse strongly
correlated systems, such as high T c superconductors [8-10], 2-d metal insulator transition at large rs (ratio of
Coulomb to kinetic energy) [11], and Kondo systems [12-14 and references therein].
In this paper we describe our recent studies of the Kondo effect in the fully controllable semiconducting doublequantum-dot (DQD) system [15,16]. These studies are interesting from two perspective: (1) Our double-quantumdots represent the first experimental realization of the two-impurity Kondo system, which has been studied
theoretically for nearly two decades, and (2) we were able to obtain direct evidence for the controlled entanglement
of quantum dot spins, one on each dot, into a spin-singlet state. This finding has important implication for quantum
computation applications. Note that recently, Craig et al. have reported similar results with a different coupled
quantum dot scheme [17]. (Please see Marcus et al. in this volume.)
In the past decade, the semiconducting quantum dot system has proven to be a versatile and ideal system for
studying correlated effects involving the spin degree of freedom. Under appropriate conditions, an isolated spin on a
quantum dot closely models an isolated spin impurity, and through its coupling to conduction electrons in the leads
connected to the dot, can give rise to a rich variety of correlated behaviors. As a result of the tunability of the
semiconducting quantum dot system, it is possible to continuously vary a number of physical parameters to produce
qualitatively distinct behaviors, or to systematically map out the behavior of a particular phenomenon of interest.
Many interesting results have been obtained in individual dots including the even-odd effect in the filling of a dot
with electrons [18-21], Kondo physics under various conditions [18,22-28], and mixed valence behavior [29]. In
double quantum dots, many authors have investigated the splitting between symmetric and anti-symmetric
combinations of the dot wave functions [30-34] and molecular bonding anti-bonding behavior [35]. These works
did not specifically address the issue of Kondo correlations.
THE DOUBLE-QUANTUM-DOT AS A MODEL TWO-IMPURITY SPIN SYSTEM
In an idealized scenario neglecting intra-dot exchange and correlation, the filling of the quantum dot levels
proceeds in an even-odd fashion, as successively higher energy orbital levels are filled first with one excess electron,
then is paired when the next electron is added. Pauli exclusion ensures that the total spin of the quantum dot thus
alternates between S=1/2 and S=0, depending on whether the total number of electrons is odd or even. Because the
filling of the odd electron involves the next higher orbital level, the energy required has an added contribution of the
level spacing, E, in addition to the Coulomb charging energy, U. This leads to an even-odd (or odd-even)
alternation of the spacing in the addition spectrum [18,21,22,28]. Within this scenario, when the number of electron
is odd and the dot spin is 1/2, the quantum dot behaves much like a spin 1/2 magnetic impurity imbedded in a metal.
In this semiconductor context, the advantage is that the coupling to the Fermi-sea of electrons, which reside in the
connecting electrical leads, can be tuned in a controlled fashion.
A successful model, which captures much of the correlated spin physics under the situation described above is
given by the Anderson Hamiltonian [36]:
H = HL + HR + HQD + HT ,
HL(R) = k, L(R)  k c†k ck ,
HQD = m,  m d†m dm + U m>m’ nm nm’ ,
(1)
HT = k,m, L(R) ( Vkm c†k dm +h.c.) ,
 = L R / (L + R) ,
m L(R)() = 2 kL(R) | Vkm|2 ( -k) ,
where HL , HR are the Hamiltonians for the left (top) and right (bottom) leads, HQD the Hamiltonian for the quantum
dot, and HT the dot-lead interaction. Within the quantum dot,  m represents the orbital energy levels, and  is the
level broadening due to coupling to the leads.
This model exhibits the required even-odd filling. In the case of one an odd total number of electrons, the last
excess electron occupies the next higher orbital level, which is the lowest unoccupied level, yielding a net spin of
1/2 for the quantum dot as desired. In this situation of an excess odd electron with spin 1/2, the Anderson
Hamiltonian gives rise to different scenarios depending on the ratio of the energy of last occupied orbital measured,
from the chemical potential of the leads, |o|, to the level broadening due to coupling to the leads, , i.e. |o|/.
When this ratio lies between 0 and 0.5, charge fluctuates readily on and off of the quantum dot, and the system is in
the mixed valence regime [20,29], while for a ratio in excess of 0.5, charge is essentially quantized on the dot, and
Kondo physics becomes relevant. Under this condition, the Anderson Hamiltonian can be mapped into the Kondo
Hamiltonian [19,20,36] familiar in the context of a magnetic impurity imbedded in a host metal:
H = i,j,s tij c†is cjs + J i sci  sfi .
TK  exp(-1/J) .
(2)
The corresponding Kondo energy scale in the quantum dot case is given by E K = kBTK [36]:
TK = [U] 1/2 exp{-[|o| (U + o)]/ U} .
(3)
Going from the individual quantum dot to the coupled double quantum dot case involves an additional dot-dot
tunnel-coupling term:
Hdot-dot = t (d†L dR + h.c.) .
(4)
Here we are taking the simplest case of full symmetry between the two (left/right, or top/bottom) dots, in their
electronic energy level structure and level splitting, E, on site Coulomb repulsion, U, and level broadening, , for
respective couplings to the left (top) and right (bottom) leads. Only the coupling term between the excess, last odd
electron on each dot needs be included as the relevant coupling term. The inter-dot tunnel coupling, t, gives rise to
an effective antiferromagnetic coupling, J = 4t2/U, between the two excess spins. According to theoretical analysis,
the following scenarios arise as the following parameters, t/ and J/TK, are varied [37-43]:
Assuming U to be the largest energy scale as is usually the case in experiment, when the ratio t/ < 1, this
problem maps onto the two-impurity Kondo problem discuss by Jones et al. [12-14], characterized by the
antiferromagnetic coupling, J, and Kondo scale, T K, but with an additional term due to the tunnel-coupling, which
breaks the symmetry in the even and odd channels. The basic behavior is similar to the scenario in the two-impurity
Kondo problem [37-43] where a competition between Kondo and antiferromagnetic correlations leads to a
continuous phase transition (or crossover) at a critical value of the coupling ratio, J/T K 2.5. However, the nonFermi liquid quantum critical point at J/T K ~ 2.5 is not accessible due to the breakage of the symmetry between the
even and odd channels, and a cross over behavior is expected instead [44].
If t is tuned to t >  before the antiferromagnetic transition point can be reached, the system undergoes a
continuous transition from a separate Kondo state of individual spins on each dot (atomic-like) to a coherent
bonding-antibonding superposition of the many-body Kondo states of the dots (molecular-like) [38-43]. Both the
antiferromagnetic state and coherent bonding state exhibit a double-peaked Kondo resonance in the differential
conductance versus source-drain bias and involve entanglement of the dot spins into a spin-singlet. Therefore they
are likely closely related. The parallel-coupled case has only recently been analyzed in a model without inter-dottunnel-coupling where the antiferromagnetic coupling occurs via electrostatic coupling, yielding a discontinuous
transition [42].
In our experiments, two double-quantum-dot geometries were studied: series-coupled and parallel-coupled. The
series case corresponds to that investigated in the theoretical works. Whereas the series-coupled geometry is more
likely to be relevant for quantum computation applications [45], the parallel-coupled geometry is well suited for
studying the quantum phase transition (crossover) in the two-impurity Kondo problem as will be seen below.
Even though a clear theoretical picture is available only in series-coupled case, for both geometries the picture
described above has proven successful in providing the framework for understanding the overall features observed
in experiment. On the other hand, our successful demonstration of the controlled formation of a spin-singlet state
within the coupled DQD systems relied on direct experimental evidence alone, without the need for a theory
dependent interpretation.
EXPERIMENT
In Fig. 1(a) and 1(b), we show the device patterns for the series- and parallel-coupled double quantum dots,
respectively. The devices are basically complex, multi-gate field effect transistors, which are operated in the
quantum regime. The bright lines in the electron micrograph ending in finger-like features represent metallic gates,
which are utilized to drive away electrons residing 90 nm below the top surface, at the GaAs/Al xGa1-xAs
heterojunction, with the application of negative gate voltages. The "pincher" gates, V1 and V5 in Fig. 1(a), and V1
and V3 in Fig. 1(b), control the coupling to the leads, setting the scale for the level broadening, , due to coupling to
the leads, while the pincher gates V3 (V5 for the parallel case) control the inter-dot tunneling, t. Plunger gates V2
and V4 control separately the number of electrons on each dot. The real difficulty in the operation of such complex
devices arises from the close proximity of all the gates. Each quantum dot has a lithographic size of ~ 180 nm. The
close proximity gives rise to sizable capacitive coupling between all gates, as well as to the electron gas puddles,
which make up the quantum dots. Changing one gate voltage will inevitably affect the charge on all other gates and
in the puddles at the same time. To be able to tune the device parameters over a significant range, e.g. the inter-dot
tunneling matrix, t, without affecting other characteristics, e.g. electron number, it is necessary to experimentally
"diagonalized" the capacitance matrix [34]. Practically speaking, this means that in order to tune t, in addition to
varying the main gate voltage, V5, it is necessary to map out other gates, some of which need to be adjusted in an
amount proportional to the change in V3 (V5), which in effect, means that it is necessary to tune a linear
combination of different gates, with V3 (V5) being the dominant one.
FIGURE 1. SEM Micrographs of the (a) series- and (b) parallel-coupled double-quantum-dots (DQD’s).
When each of the two dots within a DQD is operated as an individual dot alone without forming the other dot,
the relevant gate voltages can readily be tuned to reveal a single-dot Kondo resonance in the differential
conductance, dI/dV, when an odd number of electrons reside on the dot. For example in Fig. 2 we show the Kondo
resonance near zero source-drain voltage bias for the series-coupled DQD of Fig. 1(a). Similar results were
obtainable for the parallel-coupled DQD’s. Furthermore, source-drain bias charging diagram characterization of
each dot, under the condition of weak coupling to the leads ( small) also revealed the Coulomb repulsion energy, U
~ 2 - 3 meV, and the excitation energy structure within each dot, yielding a level splitting, , in the 0.2 - 0.3 meV
range. Based on the relationship U=e2/C, the deduced capacitance provided a measure of the effective area of the
electron puddle within the quantum dot, yielding a total of 30 -60 electrons per dot.
FIGURE 2. Differential conductance (dI/dV) versus source-drain voltage bias, VSD, when one dot, either the top (upper) or
bottom (lower) dot, is formed in the series-coupled DQD. An odd number of electrons are present. The Kondo resonance is seen
centered at zero-bias.
A useful method for characterizing a double dot system is to map out the charge stability diagram [34]. Here the
linear conductance is plotted in color (or gray) scale in a 2-dimensional diagram versus the respective plunger gates
V2 and V4 for controlling the electron number of each dot, along the x- and y-axis, respectively. Fig. 3 panels 1-6
show the results for the parallel-coupled case at successively larger inter-dot coupling, t, for weak coupling to the
leads (small , where Kondo correlation is negligible at accessible temperatures). The series of images display the
evolution of the CB spectrum from weak to strong coupling. For inter-dot weak coupling, the electrons separately
tunnel through the two nearly independent dots forming grid like pattern, see Fig 3-1 yielding rectangular domains.
As t increases, the domain vertices separate and the rectangles deformed into rounded hexagon, destroying the
charge quantization in each dot and reducing the energy of the polarized configuration. When separate charge
quantization is fully relaxed, the two dots merge into one and the domain boundaries become straight lines. The
evolution of the conductance pattern demonstrates the tunability of our DQD from ionic to covalent bonding states.
FIGURE 3. The logarithm of double-dot conductance as a function of plunger gate voltages V2 and V4, which
control the number of electrons on the left and right dot, respectively, in the parallel-coupled DQD of Fig. 1(b), for
increasing values of the inter-dot coupling, t, from panels 1 to 6. The lead-dot coupling is weak, and
correspondingly, the level broadening, , is small compared to the level spacing, .
To study the Kondo many body effect in the DQD, it is necessary to go the regime, where  is sizable, in order to
increase the Kondo energy scale (see Eq. 3). To access the regime discussed by the theories, it is necessary to have
t/ > 1, where a novel coherence forms between the respective Kondo many body wave functions on each dot.
Several considerations determine these important parameters  and t. To obtain the desired large t, the center
pincher gate V3 (V5 in the parallel case) was set so that the zigzag pattern in the charging diagram is barely visible
(Figs. 4 and 5), ensuring that the Kondo valleys can be located. To form the Kondo states in both dots, pincher gates
V1 and V5 (V3) were tuned to give a sizable  ~ E to ensure strong Kondo correlation. An estimate for t is based
on the charging diagram, which indicates a configuration close to the limit where the two dots are nearly merged
into a single large dot (Figs. 3 panel 6), so that the level broadening  |t|2/E should be comparable to the level
spacing E, yielding t ~ 150 eV, while  is deduced to be  150 eV from the half-width of the CB peaks.
FIGURE 4. (a) Schematic of a periodic structure of the electron spin configuration as a function of two gate voltages, V2 and
V4, which control the electron number in each dot, for the series-coupled DQD. The lines represent the position of the Coulomb
blockade peak. This zig-zag pattern changes, depending on the inter-dot coupling strength, t, controlled by V3. For simplicity,
only the spins of last electronic levels are shown. Circled regions contain the odd-odd electron number, two Kondo impurity spin
configuration corresponding to the regions 1, 3, 4, and 6 in (b). (b) Color scale plot of the measured conductance of a double dot
as a function of gate voltages V2 and V4. Red (blue) color indicates higher (lower) conductance. In valleys 1, 3, 4, and 6, a zerobias maximum is observed as shown in Fig. 6.
In Figs. 4, 5, and 6, we show the assignment of stability valleys, as well as spin status of the DQD, the latter
based on the differential conductance shown in Figs. 5(b) and 6. In the series case (Figs. 4 and 6), we observed a
Kondo anomaly only in the valleys, where one excess odd electron resides on each of the two dots. In contrast, for
the parallel case (Fig. 5), a Kondo resonance is visible when either or both of the dots are occupied by an odd
number of electrons.
FIGURE 5. (a) Device characterization in the Kondo regime for the parallel-coupled DQD: Charging diagram for
the third cool-down depicted in a grayscale plot of the conductance crest as a function of gate voltages V2 and V4,
for a sizable inter-dot coupling, t < . (b) The differential conductance (dI/dV) traces in different valleys and the
spin configuration of the highest lying occupied electronic levels on each dot. Note that a single upward-pointing
arrow denotes only an unpaired electron, and is not intended to represent the actual direction of spin alignment.
FIGURE 6. Differential conductance traces for the series-coupled DQD from valleys 1 to 6 in Fig. 4(b). Trace 4
and 6 are magnified by a factor of 2. The Kondo resonance peaks were observable only in the valleys with an oddodd electron configuration for the two dots. The periodicity is consistent with the diagram in Fig. 4(a). A unique
feature of the Kondo resonance is the appearance of a double-peaked feature.
The first indications of inter-dot coherence in the Kondo effect were obtained in the series geometry [15]. These
are clearly visible in the data shown in Fig. 6, in valleys labeled 1, 3, 4, and 6, as a double-peaked resonance
structure centered near zero source-drain bias. From these odd-odd valleys, it is clear that unless the DQD’s are
tuned to a precise, symmetrical configuration, the double-peaked structure can appear quite asymmetrical.
FIGURE 7. . (a) dI/dV versus VSD in valley 4 of Fig. 6 (and Fig. 4(a)) for the series-coupled DQD, at two
values of the inter-dot coupling, t. A slight reduction in t from the value shown in Fig. 6 resulted in a substantial
reduction in the double-peak splitting, and in the magnitude of the dI/dV, as indicated by the x2 factor for the solid
curve, compared to the dotted curve. (b) In-plane magnetic field dependence of the symmetric Kondo peak 4 in Fig.
6, from 0 to 1.25 T in increments of 0.25 T. The curves are offset by 0.02 e2/h for clarity. The splitting behavior
results from the Zeeman effect. (c) Temperature dependence of dI/dV. From 40 mK (to), to 500 mK (bottom), offset
by 0.01 e2/h for successive traces. The overall conductance structure decreases as the temperature increases. (d)
Conductance at zero bias; the temperature T is shown on a logarithmic scale.
conductance to increase initially and then decrease.
Increasing the T causes the
In Fig. 7, the variation of the symmetrical Kondo double-peak (valley 4) was studied as a function of t, the interdot coupling, temperature, and in-plane magnetic field. The observed behaviors were all consistent with the
interpretation based on Kondo physics [16,37-43]. In Fig. 7(a), as the inter-dot coupling t was slightly reduced, it
was observed that the spacing between them decreased precipitously, while at the same time the magnitude of the
Kondo anomaly became significantly reduced. This sensitive of the linear and differential conductances to t in this
series geometry precluded us from accessing the predicted crossover between a single-peaked Kondo resonance, and
the observed double-peaked behavior. In these data, the focus was in establishing inter-dot coherence in the
presence of strong Kondo correlations. The behavior at larger values of t was not investigated. The symmetric
double-peaked feature only exhibited a relatively shallow dip feature at zero-bias, in between the peaks. Although
based on theoretical interpretation, this observation already signaled the formation of a spin-singlet state, the
shallowness of the feature precluded the establishment of a singlet-state based on the experimental evidence alone,
independent of theory. Note that in the presence of a spin-singlet state, the Kondo resonance is expected to be
quenched at zero-bias.
FIGURE 8. The differential conductance, dI=dV, for the parallel-coupled DQD versus VSD for different interdot coupling, t, set by different values of V5 in the vicinity of V5  0.6 V. (a) From top to bottom: V5 = 0.6155V
+ V, V= 0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4.0, 4.5 mV, second cool-down. (b) From top to bottom: V5 =
0.5995 V + V, V= 3, 2.5, 2, 1.7, 1.5, 1.3, 0.8, 0.3, 0 mV, first cool-down, offset by 0.005e2/h for each trace. (c)
From top to bottom: V5 = 0.5940 V + V, V=1.5, 1.2, 0.9, 0.6, 0.3, 0.0, 0.3, 0.6, 0.9, 1.2, 1.5, 1.8, 2.1,
2.4, 2.7, 3.0, 3.3, 3.6, 4.0, 4.3, 4.6 mV, third cool-down as measured in Kondo valley 2 of Fig. 5, offset
by 0.02e2/h for each trace. The traces vary in different cool-downs due to the rearrangement in the occupation of
charges in defects and traps. (d), (e) Selected data from (c) without offset. (d) The second quantum state regime
(from top to bottom: V5 = 0.5943, 0.5940, 0.5937, 0.5934 V). (e) The first quantum state regime (from top to
bottom: V5 = 0.5961, 0.5964, 0.5970, 0.5986 V).
To fully access a sizable range of t for the purpose of establishing the sought-after crossover behavior of the
Kondo resonance from a single to double-peaked behavior, the parallel geometry was studied [16]. Fig. 8 shows
several sets of dI/dV versus Vbias in the odd-odd valley, as t was tuned. Two distinct regimes of behavior were
evident in dI/dV (see Fig. 8(a)-(e)). The main features in the first regime (see Fig. 8(a),(e)) were the clear presence
of a single peak centered at zero-bias, an increasing peak width and an increasing linear conductance dI/dV|V=0 with
increasing t. Here only the first regime is observable due to a limited tuning range in t. In the second regime where t
was increased further (see Fig. 8(b), (d)), the single peak developed into two peaks. In some Kondo valleys, both
types of behavior were observed as shown in Fig. 8(c),(d), in which the transition is seen to take place in a
continuous manner. In the transition region, the single peak broadened and its zero-bias value approached a
maximum, became flat before dropping as the peak gradually split into two. The suppression of the zero-bias
conductance on the double-peaked side can be attributed to the emergence of spin-singlet correlation between the
two dot spins [37-43]. This emergence of the spin-singlet can be viewed as strong evidence that a controlled
formation of this entangled spin-state is taking place. The maximum dI/dV|V=0 is about ~ 0.1e2/h, reduced below the
theoretical unitary limit of 4e2/h. This reduction can occur when the energy levels in the two dots differ and the dotlead coupling is asymmetric [41,42].
FIGURE 9. The linear conductance dI=dV|V=0 above background, peak splitting V, width W, and peak
amplitude(s) extracted by first subtracting a 2-Boltzmann simulated background signal followed by a two or oneGaussian fit [15]. The data are taken from Fig. 8(c) for the parallel-coupled DQD. The fit and data are virtually
indistinguishable. (a) Background subtraction. The fitting results are shown in (b) –(e). The double peak feature
visibly disappears for V5 < 0.5943 V, marked by the dashed line, where V W and a one-Gaussian fit is equally
viable as a two-Gaussian fit.
In Fig. 9 we summarize key parameters deduce from systematic analysis and curve fitting to experimental data.
The double-peak feature visibly disappeared at V5~ -0.5943V as indicated by the dash line roughly coinciding with
the g dI/dV|V=0 above background) maximum position. The peak splitting, =eV, dramatically reduces from a
maximum of ~ 120 eV to ~ 0 when V5 is slightly reduced from 0.5925 V to ~ 0.595 V and correspondingly t is
reduced by less than 4% from an initial value ~ 150 eV.
In addition to varying t to access the double-peaked regime for a fixed electron number on each dot as shown in
Fig. 8, it was also possible to change the Kondo correlation of the system by increasing the number of electron on
one of the two dots by one unit without changing t, as shown in Fig. 10. The double-peaked features in valleys 1
and 3 revert to the usual single peaked case as expected. The temperature dependence of dI/dV was also distinct in
the two regimes (see Fig. 11). In the single peak regime, the zero-bias dI/dV, decreased logarithmically with T (Fig.
11(a)). In contrast, when double peaks appeared, the zero-bias dI/dV exhibited a non-monotonic behavior (Fig.
11(b)), where with increasing T the zero-bias dI/dV slightly increased initially, then slowly decreased before
increasing again when T exceeded T K. This unusual non-monotonic behavior in was even more clearly observable
in the data from an earlier (first) sample cool-down, as shown in Fig. 12. Together, these evidences point to
qualitatively different phases in the two regimes and the existence of a quantum transition between them.
FIGURE 10. Top panel--Charging diagram for the parallel-coupled DQD in the first cool-down. Valleys 1 and
3 contain the odd electron-odd electron configuration, and a double-peaked Kondo resonance is observed in the
dI/dV. In contrast, increasing the electron number by one for either dot (valleys 2, 4, and 6) causes the Kondo
resonance to revert to the usual single-peaked behavior in the absence of inter-dot Kondo coherence.
FIGURE 11. Temperature dependence of dI/dV versus VSD for the parallel-coupled DQD within the Kondo
valley of Fig. 8(c) at slightly displaced voltage settings: (a) First quantum state regime. (b) Second regime near the
transition point. Insets: dI/dV|V=0 versus T (large dots are from curves shown).
To date no theoretical work has addressed the parallel-coupled DQD with inter-dot tunnel coupling.
Nevertheless, because the Kondo anomaly occurs under slightly non-equilibrium conditions it is likely that we may
identify the observed transition with the quantum critical phenomenon discussed in the two-impurity Kondo
problem for the series geometry based on the following evidence: (a) a continuous evolution from the single- to
double peaked behavior, (b) a maximum in dI/dV|V=0 near the transition point, (c) different behaviors in the
temperature dependence of the zero-bias conductance, and (d) a strong renormalization of the peak splitting, ,
compared to the estimated t or antiferromagnetic coupling, J, close to the transition. These features are all in
qualitative agreement with predictions for a series-coupled DQD [37-43]. Semi-quantitatively, it is informative to
roughly estimate key parameters and compared these to observed splitting,   120 eV. Within the theoretical
scenarios,  must be compared to 4t  600 eV or 2J = 8t2/U  180  eV. (Note that in the open dot regime U is
expected to be reduced from its closed dot value by roughly 1/3 [18], yielding U  1meV.) The reduction of 
compared to 4t and 2J are in agreement with theory and lends further credibility to our identification of the quantum
transition.
FIGURE 12. Non-monotonic temperature behavior of dI/dV|V=0 in the double-peaked Kondo resonance for the
parallel-coupled DQD in an earlier first cool-down of the sample.
In summary, our analysis clearly indicates that the Kondo resonance undergoes a transition from a single-peaked
to a double-peaked behavior, as the inter-dot coupling, t, is increased, while the zero-bias conductance above
background, g, attains a maximum and becomes suppressed on the double-peak side. These evidences strongly
indicate that the DQD systems undergo a transition between quantum states as t is increased. At the same time, on
the double-peaked side for sufficiently large t, the differential conductance at zero-bias is strongly suppressed,
indicating the vanishing of the spin and the associated Kondo correlation. This behavior constitutes clear and direct
evidence for the formation of an S=0, spin-singlet entangled state within the double-quantum-dot system, and bodes
well for the future use of coupled semiconductor quantum dots in quantum computation applications.
ACKNOWLEDGMENTS
This work was supported in part by NSF Grants No. DMR-9801760 and No. DMR-0135931.
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