Input: What is the effect of the proximity of superconductivity on the Kondo effect?

Context:
\section{Introduction}
\label{sec:Intro}

The exchange interactions control the magnetic order and properties of a vast number of materials
\cite{White2006Dec}
and lead to many fascinating phenomena, such as various types of the Kondo effect
\cite{Kondo,NozieresBlandin,Pustilnik_Glazman}.
Double quantum dots (DQDs), and in general multi-impurity systems, constitute
a convenient and controllable playground,
where nearly as much different exchange mechanisms compete with each other to
shape the ground state of the system.
\emph{Local exchange} between the spin of a quantum dot (QD)
and the spin of conduction band electrons gives rise to the
Kondo effect \cite{Kondo,Hewson_book}.
\emph{Direct exchange} arriving with an additional side-coupled QD may destroy it or lead to the
two-stage Kondo screening \cite{Pustilnik_Glazman,Cornaglia,Granger,ZitkoBonca,ZitkoPRB2010,Ferreira}.
In a geometry where the two QDs contact the same lead, conduction band electrons
mediate the \emph{RKKY exchange} \cite{RK,K,Y}. The RKKY interaction competes
with the Kondo effect and leads to the quantum phase transition of a still debated nature
\cite{Doniach,Jones,Affleck,Bork,Neel,KondoRKKYexp,Hans,Hans2,Fabian}.
Moreover, in DQDs coupled in series also \emph{superexchange} can alter the Kondo physics significantly
\cite{Zitko_2QDEx,Sela}.

Recently, hybrid quantum devices, in which the interplay between various magnetic correlations
with superconductivity (SC) plays an important role, have become an important direction of research
\cite{hybridQDs,SCspintronics}. In particular, chains of magnetic atoms on SC surface have proven
to contain self-organized Majorana quasi-particles and exotic spin textures
\cite{Braunecker,Klinovaja,Vazifeh,Yazdani},
while hybrid DQD structures have been used to split the Cooper pairs coherently into two entangled
electrons propagating to separated normal leads \cite{CPS1,CPS2,CPS4,CPS5,CPS9}.
The latter is possible due to non-local (\emph{crossed}) Andreev reflections (CARs),
in which each electron of a Cooper pair tunnels into different QD, and
subsequently to attached lead. Such processes give rise to an exchange mechanism \cite{Yao},
that we henceforth refer to as \emph{the CAR exchange}, which can greatly modify
the low-temperature transport behavior of correlated hybrid nanostructures.

The CAR exchange may be seen as RKKY-like interaction between
two nearby impurities on SC surface \cite{Yao}.
The effect can be understood as a consequence
of spin-dependent hybridization of the Yu-Shiba-Rusinov (YSR)
states \cite{Yu,Shiba,Rusinov} in SC contact,
caused both by the overlap of their wave functions
and their coupling to Cooper-pair condensate.
This process is the most effective when the YSR states
are close to the middle of the SC gap, {\it e.g.} in the YSR-screened phase \cite{YSRscreening}.
The mechanism presented here is essentially the same,
yet in the considered regime can be understood
perturbatively without referring to YSR states,
as a consequence of the non-local pairing induced by SC electrode.
In particular, the presence of YSR bound states close to the Fermi level
is not necessary for significant consequences for the Kondo physics,
as long as some inter-dot pairing is present.

The proximity of SC induces pairing in QDs \cite{RozhkovArovas,Buitelaar}
and tends to suppress the Kondo effect if the superconducting energy gap $2\Delta$
becomes larger than the relevant Kondo temperature $T_K$
\cite{Buitelaar2002Dec,adatomsSC,Kondo_vs_SC1,Kondo_vs_SC2,Zitko_Kondo-Andreev,Zitko_S-QD-N,IW_Sau,YSRscreening}.
Moreover, the strength of SC pairing can greatly affect the Kondo physics in the sub-gap transport regime:
For QDs attached to SC and normal contacts, it can enhance the Kondo effect
\cite{DomanskiIW,KWIW,part1}, while
for DQD-based Cooper pair splitters, it tends to suppress both the $\mathrm{SU}(2)$ and $\mathrm{SU}(4)$ Kondo effects \cite{IW_Kacper}.
Our main result is that the non-local pairing induced by superconducting
proximity effect, which gives rise to CAR exchange, can be the sole cause of the Kondo screening.
Moreover, relatively small values of coupling to SC, $\GS{}\ll U$, are sufficient for the effect to occur.
This is in contrast to the DQD system considered in Ref.~\cite{part1},
where only one of the quantum dots is proximized, such that
CAR exchange cannot arise,
and the Kondo physics becomes qualitatively
affected only for $\GS{}\sim U/2$.%

\begin{figure}[bt]
\centering
\includegraphics[width=1\linewidth]{Fig1.png}
\caption{
		 (a) Schematic of the considered system. Left/right (L/R) lead
		 is coupled to the first quantum dot (QD1), while superconductor
		 is attached to both QD1 and QD2.
		 (b)-(d) illustrate an example of direct spin exchange:
		 spin-up electron from the initial state (b) hops to the other QD (c) and spin-down electron
		 hops back (d). Note, that the final state is in fact the same singlet state,
		 only with opposite sign.
		 (e)-(g) show an example of process contributing to crossed Andreev reflection (CAR) exchange.
		 A Cooper pair from SC approaches DQD (e) and two singlets of the same charge
		 are formed (f), before the Cooper pair is re-emitted (g).
		 (h)-(j) present an example of RKKY process: an electron scattered off
		 one QD (h) mediates the spin exchange towards the other (i), before it is finally scattered
		 off there, too (j).
		 }
\label{fig:system}
\end{figure}

In this paper we discuss the CAR-induced Kondo screening in a setup comprising T-shaped DQD
with normal and superconducting contacts, see \fig{system}(a).
We note that despite quite generic character of CAR exchange,
and its presence in systems containing at least two localized electrons
coupled close to each other to the same SC bath,
to best of our knowledge CAR-induced screening
has hardly been identified in previous studies
\cite{CPS1,CPS2,CPS4,CPS5,CPS9,IW_Kacper,IW_Sau,Zitko_Josephson,Zitko_S2QD,Martinek2017}.
In the system proposed here [\fig{system}(a)], its presence is evident.
Moreover, CAR exchange magnitude can be directly related to the relevant energy scales, such as the Kondo
temperature, which provides a fingerprint for quantitative experimental verification of our predictions.

The paper is organized as follows. In \Sec{model} we describe the considered system
and present the model we use to study it. In \Sec{scales} the relevant energy scales are estimated
to make the discussion of main results concerning CAR-induced Kondo effect in \Sec{main} more clear.
Finally, the influence of effects neglected in \Sec{main} are presented in the following sections,
including CAR exchange interplay with RKKY interaction (\Sec{RKKY}), particle-hole asymmetry (\Sec{asym}),
couplings asymmetry (\Sec{x}) and reduced efficiency of CAR coupling (\Sec{coef}). In summary,
the effects discussed in \Sec{main} remain qualitatively valid in all these cases.
The paper is concluded in \Sec{conclusions}.

\section{Model}
\label{sec:model}

The schematic of the considered system is depicted in \fig{system}(a).
It contains two QDs attached to a common SC lead.
Only one of them (QD1) is directly attached to the left (L) and right (R) normal leads,
while the other dot (QD2) remains coupled only through QD1.
The SC is modeled by the BCS Hamiltonian,
$H_{\mathrm{S}}=\sum_{\mathbf{k}\sigma}\xi_{\mathbf{k}}a_{\mathbf{k}\sigma}^{\dag}a_{\mathbf{k}\sigma}-\Delta\sum_{\mathbf{k}}(a^\dag_{\mathbf{k}\uparrow}a_{-\mathbf{k}\downarrow}^{\dag}+a_{-\mathbf{k}\downarrow}a_{\mathbf{k}\uparrow})$,
with energy dispersion $\xi_{\mathbf{k}}$, energy gap $2\Delta>0$ and $a_{\mathbf{k}\sigma}$ annihilation operator
of electron possessing spin $\sigma$ and momentum $\mathbf{k}$. The coupling between
SC and QDs is described by the hopping Hamiltonian
$H_{\mathrm{TS}}=\sum_{i\mathbf{k}\sigma}v_{\mathrm{S}i}(d^\dagger_{i\sigma}a^{}_{\mathbf{k}\sigma}+h.c.)$,
with $d^\dagger_{i\sigma}$ creating a spin-$\sigma$ electron at QD$i$. The matrix element
$v_{\mathrm{S}i}$ and the normalized density of states of SC in normal state, $\rho_{\rm S}$,
contribute to the coupling of QD$i$ to SC electrode as $\GS{i} = \pi \rho_{\rm S} |v_{{\rm S}i}|^2$.
We focus on the sub-gap regime, therefore, we integrate out SC degrees of freedom lying outside the energy gap \cite{RozhkovArovas}.
This gives rise to the following effective Hamiltonian,
$H_{\mathrm{eff}}=H_{\mathrm{SDQD}}+H_{\rm L}+H_{\rm R}+H_{\rm T}$,
where
\begin{eqnarray}
H_{\rm SDQD} 	& = &
				\sum_{i\sigma} \varepsilon_{i} n_{i\sigma}
				+\sum_{i} U n_{i\uparrow} n_{i\downarrow}
				+U' (n_1-1)(n_2-1)
				\nonumber\\
				&+&\sum_\sigma t(d^\dagger_{1\sigma}d^{}_{2\sigma} + h.c.)
				+J \vec{S}_1\vec{S}_2
				\nonumber\\
				&+&\sum_{i} \!\!\left[ \Gamma_{{\rm S}i} (d^\dagger_{i\uparrow} d^\dagger_{i\downarrow} \!+\! h.c.)
				+\Gamma_{\rm SX} (d^\dagger_{i\uparrow} d^\dagger_{\bar{i}\downarrow} \!+\! h.c.) \right]
	\label{H_DQD}
\end{eqnarray}
is the Hamiltonian of the SC-proximized DQD
\cite{IW_Kacper,Walldorf2018Feb}, with QD$i$ energy level $\varepsilon_i$,
inter-site (intra-site) Coulomb interactions $U'$ ($U$),
inter-dot hopping $t$, and CAR coupling $\GS{\rm X}$.
$n_{i\sigma}=d^\dagger_{i\sigma}d^{}_{i\sigma}$ denotes the electron number operator
at QD$i$, $n_i=n_\uparrow+n_\downarrow$, and $\bar{i}\equiv 3-i$.
Our model is strictly valid in the regime where $\Delta$ is the largest
energy scale. Nevertheless, all discussed phenomena are
present in a full model for energies smaller than SC gap.
Moreover, by eliminating other consequences of the presence of SC lead,
our model pinpoints the fact that the non-local pairing is
sufficient for the occurrence of the CAR exchange.
The presence of out-gap states shall result mainly in additional broadening of DQD energy levels,
changing the relevant Kondo temperatures.
We note that the procedure of integrating out out-gap states neglects the
RKKY interaction mediated by SC lead and other possible indirect exchange mechanisms%
 \footnote{
 Note, that by RKKY interaction we mean only such an effective exchange,
 which arises due to multiple scattering of a single electron or hole, see \fig{system}(h)-(j).
 Other mechanisms leading to the total indirect exchange are considered separately.
 In particular, in the large gap limit, exchange described in Ref.~\cite{Yao} is in fact reduced to
 the CAR exchange, and additional antiferromagnetic contribution would arise for finite gap.
 }.
To compensate for this,
we explicitly include the Heisenberg term $ J \vec{S}_1\vec{S}_2$ in
$H_{\rm SDQD}$, with $\vec{S}_i$ denoting the spin operator of QD$i$
and a Heisenberg coupling $J$ substituting the genuine RKKY exchange.

The normal leads are treated as reservoirs of noninteracting electrons,
$H_{r}=\sum_{\mathbf{k}\sigma}\varepsilon_{r\mathbf{k}}c^\dagger_{r\mathbf{k}\sigma}c^{}_{r\mathbf{k}\sigma}$,
where $c^{}_{r\mathbf{k}\sigma}$ annihilates an electron of spin
$\sigma$ and momentum $\mathbf{k}$ in lead $r$ ($r={\rm L,R}$) with the corresponding energy $\varepsilon_{r\mathbf{k}\sigma}$.
The tunneling Hamiltonian reads,
$H_{\rm T} = \sum_{r\mathbf{k}\sigma} v_{r} (d^\dagger_{1\sigma}c^{}_{r\mathbf{k}\sigma} + h.c.)$,
giving rise to coupling between lead $r$ and QD$i$ of strength $\Gamma_r = \pi \rho_r |v_r|^2$,
with $\rho_r$ the normalized density of states of lead $r$ and $v_r$ the
local hopping matrix element, assumed momentum-independent.
We consider a wide-band limit, assuming constant $\Gamma_r=\Gamma/2$
within the cutoff $\pm D = \pm 2U$ around the Fermi level.

For thorough analysis of the CAR exchange mechanism and its consequences
for transport, we determine the linear conductance between the two normal leads from
\begin{equation}
G = \frac{2e^2}{h} \pi \Gamma \int \left[ -\frac{\partial f_T}{\partial\omega} \right] \mathcal{A}(\omega) {\rm d} \omega ,
\label{G}
\end{equation}
where $f_T$ is the Fermi function at temperature $T$,
while $\mathcal{A}(\omega)$ denotes the normalized local spectral density
of QD1 \cite{fn1}.
Henceforth, unless we state otherwise, we assume a maximal CAR coupling,
$\GS{\rm X} = \sqrt{\GS{1}\GS{2}}$ \cite{IW_Kacper,Walldorf2018Feb},
$\GS{1}=\GS{2}=\GS{}$ and consider DQD tuned to the particle-hole symmetry point,
$\varepsilon_1=\varepsilon_2=-U/2$. However, these assumptions are not crucial for the results presented
here, as discussed in Secs.~\ref{sec:asym}-\ref{sec:coef}.

\section{Estimation of relevant energy scales}
\label{sec:scales}

Since we analyze a relatively complex system, let us build up the understanding of its behavior starting
from the case of a QD between two normal-metallic leads, which can be obtained in our
model by setting $t=\GS{}=J=U'=0$. Then, the conductance as a function of temperature, $G(T)$, grows
below the Kondo temperature $T_K$ and reaches maximum for $T\to 0$, $G(T\!=\!0)=G_{\rm max}$.
At particle-hole symmetry point, the unitary transmission is achieved, $G_{\rm max}= G_0 = 2e^2/h$;
see short-dashed line in \fig{G-T}(a).
An experimentally relevant definition of $T_K$ is that at $T=T_K$
$G(T)=G_{\rm max}/2$. $T_K$ is exponentially small in
the local exchange $J_0 = 8\Gamma / (\pi \rho U)$, and is approximated by
$T_K \approx D \exp[-1/(\rho J_0)]$ \cite{Hewson_book}.

The presence of a second side-coupled QD, $t,U'>0$, significantly enriches the physics of the system
by introducing direct exchange between QDs, see \fig{system}(b-d).
In general, effective inter-dot exchange can be defined as energy difference between
the triplet and singlet states of isolated DQD,
$J^{\mathrm{eff}} = E_{S=1} - E_{\rm GS}$. Unless $U$ becomes very large, superexchange can be neglected
\cite{Zitko_2QDEx} and $J^{\mathrm{eff}}$ is determined by \emph{direct exchange}, $J^{\mathrm{eff}}\approx 4t^2/(U-U')>0$.
When the hopping $t$ is tuned small \cite{CPS1}, one can expect $J^{\mathrm{eff}}\lesssim T_K$, which
implies the two-stage Kondo screening \cite{Pustilnik_Glazman,Cornaglia}.
Then, for $T \ll T_K$, the local spectral density of QD1 serves as a band of width $\sim T_K$ for QD2.
The spin of an electron occupying QD2
experiences the Kondo screening below the associated Kondo temperature
\begin{equation}
T^* = a T_K \exp(- b T_K / J_{\rm eff})
\label{Tstar}
\end{equation}
with $a$ and $b$ constants of order of unity \cite{Pustilnik_Glazman,Cornaglia}.
This is reflected in conductance, which drops to $0$ with lowering $T$, maintaining characteristic
Fermi-liquid
$G\sim T^2$ dependence \cite{Cornaglia}; see the curves indicated with squares
in \fig{G-T}(a). Similarly to $T_K$, experimentally relevant definition of $T^*$ is that
$G(T\!=\!T^*) = G_{\rm max}/2$. Even at the particle-hole
symmetry point $G_{\rm max} < G_0$, because the single-QD strong-coupling fixed point
is unstable in the presence of QD2 and $G(T)$ does not achieve $G_0$ exactly,
before it starts to decrease.

The proximity of SC gives rise to two further exchange mechanisms that
determine the system's behavior. First of all, the (conventional)
\emph{RKKY interaction} appears, $J \sim \GS{}^2$ \cite{RK,K,Y}.
Moreover, the \emph{CAR exchange} emerges as a consequence of finite $\GS{}$ \cite{Yao}.
It can be understood on the basis
of perturbation theory as follows. DQD in the inter-dot singlet state may absorb
and re-emit a Cooper pair approaching from SC; see \fig{system}(e)-(g). As a second-order
process, it reduces the energy of the singlet, which is the ground state of isolated DQD.
A similar process is not possible in the triplet state due to spin conservation.
Therefore, the singlet-triplet energy splitting $J^{\mathrm{eff}}$ is increased (or generated for $t=J=0$).
More precisely, the leading ($2$nd-order in $t$ and $\GS{}$) terms
in the total exchange are
\begin{equation}
J^{\mathrm{eff}} 	\approx 	J + \frac{4t^2}{U-U'+\frac{3}{4}J} + \frac{4\GS{}^2}{U+U'+\frac{3}{4}J}.
\label{Jeff}
\end{equation}
Using this estimation, one can predict $T^*$ for finite $\GS{}$, $t$ and $J$ with \eq{Tstar}.
Apparently, from three contributions corresponding to:
(i) RKKY interaction, (ii) direct exchange and (iii) CAR exchange, only the first may bear a negative (ferromagnetic) sign.
The two other contributions always have an anti-ferromagnetic nature.
More accurate expression for $J^{\mathrm{eff}}$ is derived in Appendix~\ref{sec:downfolding}
[see \eq{A_J}] by the Hamiltonian down-folding procedure. The relevant terms differ
by factors important only for large $\GS{}/U$.
Finally, it seems worth stressing that normal leads are not necessary for CAR exchange to occur.
At least one of them is inevitable for the Kondo screening though, and two symmetrically coupled
normal leads allow for measurement of the normal conductance.

It is also noteworthy that inter-dot Coulomb interactions
decrease the energy of intermediate states contributing to direct exchange
[\fig{system}(c)], while increasing the energy of intermediate
states causing the CAR exchange [\fig{system}(f)].
This results in different dependence of corresponding terms in \eq{Jeff} on $U'$.
As can be seen in \figs{G-T}(b) and \ref{fig:G-T}(c), it has a significant effect
on the actual values of $T^*$.

\begin{figure}
\includegraphics[width=1\linewidth]{Fig2.pdf}
\caption{(a) Linear conductance $G$ as function of $T$ calculated for
		 $\varepsilon_1=\varepsilon_2=-U/2$, $\Gamma=U/5$, $U'=U/10$ and different situations,
		 as indicated. The quantity $\xi\equiv\sqrt{\GS{}^2+t^2}$ is fixed
		 for different curves drawn with the same dashing style.
		 Note the logarithmic scale on both axes.
		 %
		 (b) Points show $T^*/T_K$ calculated by NRG from curves in subfigure (a).
		 Lines present the fit to \eq{Tstar} with $J^{\mathrm{eff}}$ obtained from \eq{Jeff}.
		 %
		 (c) The same as (b), only for $U'=0$.
		 %
		 (d) and (e) show the residual conductance $G_{\mathrm{min}} \equiv G(T \!=\! 0)$ as a function of
		 $\GS{}$ for $t=0$ (denoted "CAR") and $t=\GS{}$ (denoted "Both").
		 Dotted line is a guide for eyes. $U'=U/10$ in (b) and (d) and $U'=0$ in (c) and (e).
		}
\label{fig:G-T}
\end{figure}

\section{CAR exchange and Kondo effect}
\label{sec:main}

To verify \eqs{Tstar}-(\ref{Jeff}) we calculate $G$ using
accurate full density matrix numerical renormalization group (NRG) technique \cite{WilsonNRG,Weichselbaum,FlexibleDMNRG,fn2}.
We compare $U'=0$ case with experimentally relevant value $U'=U/10$ \cite{Keller2013Dec}.
While for two close adatoms on SC surface RKKY interactions may lead to prominent consequences
\cite{Klinovaja}, the conventional ({\it i.e.} non-CAR) contribution should
vanish rapidly when the inter-impurity distance $r$ exceeds a few lattice constants \cite{RKKYrange,SC_RKKY}.
Meanwhile, the CAR exchange may remain significant for $r$ of the order
of coherence length of the SC contact \cite{Yao}. Therefore, we first neglect the conventional RKKY coupling and analyze its consequences in Sec.~\ref{sec:RKKY}.

The main results are presented in \fig{G-T}(a), showing the temperature dependence of $G$
for different circumstances.
For reference, results for $\GS{}=0$ are shown, exhibiting
the two-stage Kondo effect caused by \emph{direct} exchange mechanism.
As can be seen in \figs{G-T}(b) and \ref{fig:G-T}(c), an excellent agreement of $T^*$ found from NRG calculations and \eq{Tstar}
is obtained with $a=0.42$ and $b=1.51$, the same for both $U'=0$ and $U'=U/10$. Note,
however, that $J^{\mathrm{eff}}$ is different in these cases, cf. \eq{Jeff},
and $U'$ leads to increase of $T^*$.

Furthermore, for $t=0$ and $\GS{}>0$ the two-stage Kondo effect caused solely by the \emph{CAR
exchange} is present; see \fig{G-T}(a).
Experimentally, this situation
corresponds to a distance between the two QDs smaller than the superconducting coherence length,
but large enough for the exponentially suppressed direct hopping to be negligible.
While intuitively one could expect pairing to compete with any kind of magnetic ordering,
the Kondo screening induced by CAR exchange is a beautiful example of a superconductivity
in fact leading to magnetic order, namely the formation of the Kondo singlet.
This CAR-exchange-mediated Kondo screening is our main finding.
For such screening, \eq{Tstar} is still fulfilled with very similar
parameters, $a=0.37$ ($a=0.35$) and $b=1.51$ ($b=1.50$) for $U'=0$ ($U'=U/10$),
correspondingly; see \figs{G-T}(b-c).
Moreover, as follows from \eq{Jeff}, $U'$ reduces CAR exchange, and therefore diminishes $T^*$.
For the same values of $J^{\mathrm{eff}}$, the dependence of $G(T)$ for $t=0$ and $\GS{}>0$ is hardly different
from the one for $\GS{}=0$ and $t>0$ for $T\geq T^*$ (results not shown).
However, $G(T)$ saturates at residual value $G_{\mathrm{min}}$ as $T\to 0$ only for finite
$\GS{}$, which at particle-hole symmetry makes $G_{\mathrm{min}}$
the hallmark of SC proximity and the corresponding CAR exchange processes.
From numerical results, one can estimate it as
\begin{equation}
G_{\mathrm{min}} = \frac{e^2}{h} \cdot c \, \frac{\GS{}^2}{U^2}
	\qquad {\scriptstyle (\GS{1}=\GS{2}=\GS{})} ,
\label{Gmin}
\end{equation}
with $c\approx 2.25$, barely depending on $U'$ and getting smaller for $t>0$.
This is illustrated in \figs{G-T}(d-e), where the dotted line corresponds to \eq{Gmin} with $c=2.25$.

Lastly, in \fig{G-T}(a) we also present the curves obtained for $t=\GS{}$ chosen such,
that the quantity $\xi=\sqrt{t^2+\GS{}^2}$ remains the same
in all the cases.
This is to illustrate what happens when \emph{both} (direct and CAR) exchange interactions are
present. \fig{G-T}(c) clearly shows that $T^*$ remains practically unaltered for $U'=0$.
The comparison with \fig{G-T}(b) proves that in this case it practically does not depend
on $U'$. The enhancement of direct exchange is compensated by the decrease of the CAR one.
On the contrary, $G_{\mathrm{min}}$ decreases for larger $t$ below the estimation given by Eq.~(\ref{Gmin}),
as can be seen in \figs{G-T}(d-e).

While analyzing the results concerning $G_{\mathrm{min}}(\GS{})$ plotted in \figs{G-T}(d-e)
one needs to keep in mind that $G_{\mathrm{min}}$ is obtained at deeply cryogenic conditions. To illustrate
this better, $G(\GS{})$ obtained for $t=0$ and $T=10^{-6}U$ is plotted with solid line
in \fig{3}. Clearly, for weak $\GS{}$ the system exhibits rather conventional (single-stage)
Kondo effect with $G=G_{\mathrm{max}}\approx 2e^2/h$, while QD2 is effectively decoupled ($G_{\mathrm{max}}<2e^2/h$
in the proximity of SC lead \cite{KWIW}). Only for larger values of $\GS{}$
the CAR exchange is strong enough, such that $T^*>T$ and the dependence $G(\GS{})$ continuously
approaches the $T=0$ limit estimated by \eq{Gmin} and presented in \figs{G-T}(d-e).

\section{CAR-RKKY competition}
\label{sec:RKKY}

\begin{figure}
\includegraphics[width=0.98\linewidth]{Fig3.pdf}
\caption{Linear conductance $G$ vs. $\GS{}$ calculated
		 for $t=0$, $\Gamma=U/5$, $U'=U/10$, finite $T=10^{-6}U$
		 and different values of RKKY coupling $J$, as indicated.
		 Inset shows QD1 spectral function $\mathcal{A}(\omega)$ as a function of energy $\omega$
		 for points on $J=-0.1U$ curve, indicated with corresponding symbols.
		}
\label{fig:3}
\end{figure}

Let us now discuss the effects introduced by the conventional RKKY interaction.
We choose $t=0$ for the sake of simplicity and
analyze a wide range of $\GS{}$, starting from the case of anti-ferromagnetic
RKKY interaction ($J>0$). Large $J>0$ leads to the formation of a molecular singlet in the
nanostructure. This suppresses the conductance, unless $\GS{}$ becomes of the order of $U/2$,
when the excited states of DQD are all close to the ground state. This is illustrated
by double-dotted line in \fig{3}.
Smaller value of $J>0$ causes less dramatic consequences, namely it just increases $J^{\mathrm{eff}}$ according
to \eq{Jeff}, leading to enhancement of $T^*$, cf. \eq{Tstar}. This is presented with
dot-dashed line in \fig{3}.

The situation changes qualitatively for ferromagnetic RKKY coupling, $J<0$.
Then, RKKY exchange and CAR exchange have opposite signs and compete with each other.
Depending on their magnitudes and temperature, one
of the following scenarios may happen.

For $J^{\mathrm{eff}} > 0$, {\it i.e.} large enough $\GS{}$, and $T<T^*$, the system is in the
singlet state due to the two-stage Kondo screening of DQD spins. $G(T\!=\!0)$ is reduced
to $G_{\mathrm{min}}$, which tends to increase for large negative $J$; see dashed lines in \fig{3}.
In the inset to \fig{3}, the spectral density of QD1 representative for this regime is plotted
as curve indicated by triangle. It corresponds to a point on the $J=-0.1U$ curve in the main
plot, also indicated by triangle. The dip in $\mathcal{A}(\omega)$ has width of order of $T^*$.

For finite $T$, there is always a range of sufficiently small $|J^{\mathrm{eff}}|$, where QD2 becomes effectively
decoupled, and, provided $T<T_K$, $G$ reaches $G_{\mathrm{max}}$ due to conventional Kondo effect
at QD1. This is the case for sufficiently small $\GS{}$ for $J=0$ or $J=-0.01U$, and in the narrow
range of $\GS{}$ around the point indicated by a circle in \fig{3} for $J=-0.1U$ (for $J=0.01U$,
the considered $T$ is close to $T^*$ and $G$ does not reach $G_{\rm max}$). The conventional Kondo effect manifests itself with
a characteristic peak in $\mathcal{A}(\omega)$, as illustrated in the inset in \fig{3} with line denoted by circle.

Finally, large enough $J^{\mathrm{eff}} < 0$ and low $T$, give rise to an effective ferromagnetic coupling of DQDs
spins into triplet state. Consequently, the underscreened Kondo effect occurs
\cite{Mattis,NozieresBlandin} for weak $\GS{}$ and, {\it e.g.}, $J=-0.1U$;
see the point indicated by square in \fig{3}.
This leads to $G=G_{\mathrm{max}}$ and a peak in $\mathcal{A}(\omega)$, whose shape is significantly different from the
Kondo peak, cf. the curve denoted by square in the inset in \fig{3}.

\section{Effects of detuning from the particle-hole symmetry point}
\label{sec:asym}

\begin{figure}
\includegraphics[width=0.98\linewidth]{Fig4.pdf}
\caption{
         (a) Linear conductance between the normal leads $G$ as a function of temperature $T$
         for parameters corresponding to \fig{G-T}(a) with $\xi=U/10$, and additional curves for finite
         detuning from particle-hole symmetry point, $\delta_1=-\delta_2$,
         and two values of $\xi=\sqrt{t^2+\GS{}^2}$, as indicated in the figure.
         (b) $G_{\mathrm{min}} \equiv G(T \!=\! 0)$ as a function of QD1 detuning $\delta_1$ for different
         exchange mechanisms, $\xi=U/10$ and $\delta_2=\pm\delta_1$ (as indicated).
		}
\label{fig:asym}
\end{figure}

At PHS $G_{\mathrm{min}}=G(T \!=\! 0)=0$ in the absence of superconducting lead, making $G_{\mathrm{min}} > 0$ a hallmark
of SC-induced two-stage Kondo effect. However, outside of PHS point $G_{\mathrm{min}} > 0$ even in the case of
the two-stage Kondo effect caused by the direct exchange.
Exact PHS conditions are hardly possible in real systems, and the fine-tuning of the QD energy
levels to PHS point is limited to some finite accuracy.
Therefore, there may appear a question, if the results obtained at PHS are of any importance for the
realistic setups. As we show below --- they are,
in a reasonable range of detunings $\delta_i=\varepsilon_i +U/2$.

In \fig{asym}(a) we present the $G(T)$ dependence in and outside the PHS, corresponding to
parameters of \fig{G-T}(a).
Clearly, for considered small values of $\delta_1=\delta_2=\delta$,
$G_{\mathrm{min}}<10^{-3}e^2/h$ for direct exchange only, while $G_{\mathrm{min}}$ in the presence of a superconductor is
significantly increased and close to the PHS value. Furthermore, for $|\delta_1| \sim |\delta_2|
\sim \delta$, the residual conductance caused by the lack of PHS, $G_{\mathrm{min}} \approx e^2/h \cdot (\delta/U)^2$,
which is a rapidly decreasing function in the vicinity of PHS point, as illustrated in \fig{asym}(b)
with lines denoted by a square. Evidently, in the regime $|\delta_i| < 0.01U$ the residual conductance
caused by SC is orders of magnitude larger, leading to the plateau in $G_{\mathrm{min}}(\delta_1)$ dependence,
visible in \fig{asym}(b).
Taking into account that the realistic values of $U$ in the semiconductor quantum dots are rather
large, this condition seems to be realizable by fine-tuning of QD gate voltages.

Lastly, let us point out that while in the presence of only one exchange mechanism, \emph{CAR} or
\emph{direct}, $G_{\mathrm{min}}(\delta_1)$ dependencies depicted in \fig{asym}(b) are symmetrical with respect
to sign change of $\delta_1$, for \emph{both} exchange mechanisms the dependence is non-symmetric.

\section{Effects of asymmetry of couplings to superconductor}
\label{sec:x}

\begin{figure}
\includegraphics[width=0.98\linewidth]{Fig5.pdf}
\caption{
		 (a) Linear conductance between the normal leads, $G$, as a function of temperature, $T$,
		 for parameters corresponding to \fig{G-T}(a) with $\xi=U/10$, for different values
		 of asymmetry coefficient $x$ [see \eq{xGS}], in the presence of \emph{CAR} exchange only.
		 %
		 (b) The second-stage Kondo temperature $T^*$ normalized by $T_K$ as a function of $x$,
		 calculated with the aid of NRG (points) and a fit to \eq{Tstar} (lines)
		 with $J^{\mathrm{eff}}$ from \eq{Jeff}.
		 %
		 (c) The zero-temperature conductance $G_{\mathrm{min}}$ as a function of QD1 coupling to SC lead, $\GS{1}$,
		 compiled from data obtained at different circumstances (as indicated in the legend)
		 for different $x$. Dotted line corresponds to \eq{Gmin2} with $c=2.25$.
		}
\label{fig:x}
\end{figure}

Similarly to PHS, the ideal symmetry in the coupling between respective QDs and SC lead is hardly possible
in experimental reality. As shown below, it does not introduce any qualitatively new features.
On the other hand, it decreases the second stage Kondo temperature, which is already small, therefore,
quantitative estimation of this decrease may be important for potential experimental approaches.
To analyze the effects of $\GS{1}\neq\GS{2}$, we introduce the asymmetry parameter $x$ and extend
the definition of $\GS{}$,
\beq
x = \frac{\GS{1}-\GS{2}}{\GS{1}+\GS{2}}, \quad \GS{} = \frac{\GS{1}+\GS{2}}{2}.
\label{xGS}
 \end{equation}
Note, that even for a fixed $\GS{}$, the actual CAR coupling $\GS{\rm X}=\GS{}\sqrt{1-x^2}$ decreases
with increasing $|x|$, which is a main mechanism leading to a decrease of $T^*$ outside the $x=0$ point
visible in \figs{x}(a) and (b). To illustrate this, the curves corresponding to \emph{both} exchange
mechanisms were calculated using $x$-dependent $t=\GS{\rm X}$ instead of $t=\xi/\sqrt{2}$.
Therefore, $\xi$ was generalized for $x\neq 0$ by setting $\xi=\sqrt{t^2(1-x^2)^{-1}+\GS{}^2}$.
Clearly, in \fig{x}(b) the curves for different exchange mechanisms are very similar and differ mainly
by a constant factor, resulting from different influence of $U'$; see \Sec{scales}.
The magnitude of $T^*$ changes is quite large, exceeding an order of magnitude for $x=\pm 0.5$
and $\xi=U/20$. Moreover, $T^* \to 0$ for $x\to\pm 1$. Consequently, for strongly asymmetric
devices one cannot hope to observe the second stage of Kondo screening.

A careful observer can note that the $T^*(x)$ dependency is not symmetrical; note for example different
$T^*$ for $x=\pm 0.5$ in \fig{x}(a). This is caused by the dependence of the first stage Kondo temperature
$T_K$ on $\GS{1}$ \cite{part1,DomanskiIW},
\beq
\widetilde{T}_K(\GS{1}) = T_K \cdot \exp\!\left( \frac{\pi}{2} \frac{\GS{1}^2}{\Gamma U}\right).
 \end{equation}
Here, $T_K$ is, as earlier, defined in the absence of SC, while $\widetilde{T}_K$ is a function
of $\GS{1}$, such that $G(\widetilde{T}_K) = G_{\rm max}(\GS{1})/2$ in the absence of QD2.
As $\widetilde{T}_K$ grows for increasing $\GS{1}$ (or $x$), $T^*$ decreases according to \eq{Tstar}.
Its $\GS{}$ dependence can be accounted for by small changes in the coefficients $a$ and $b$ in \eq{Tstar},
as long as $x$ is kept constant.

To close the discussion of $T^*(x)$ dependence let us point out, that in \eq{A_J}
there appears a correction to \eq{Jeff} for $x\neq 0$. However, it is very small due to additional
factor $\GS{}^2/U^2$ in the leading order. Its influence on curves plotted in \fig{x}(b) is hardly visible.

In turn, let us examine the $x$ dependence of the $T=0$ conductance $G_{\mathrm{min}}$. As can be seen
in \fig{x}(a), it monotonically increases with $x$, as it crosses $x=0$ point. In fact, \eq{Gmin}
can be generalized to
\beq
G_{\mathrm{min}} = \frac{e^2}{h} \cdot c \, \frac{\GS{1}^2}{U^2} ,
\label{Gmin2}
 \end{equation}
with $c\approx 2.25$ (indicated by a dotted line in \fig{x}(c)). Note that $G_{\mathrm{min}}$ is proportional to
$\GS{1}^2=(x+1)^2 \GS{}^2$, instead of simply $\GS{}$, cf. \eq{Gmin}. The values of $G_{\mathrm{min}}$ obtained
from all analyzed $G(T)$ dependencies for different $x$ have been compiled in \fig{x}(c).
It is evident, that \eq{Gmin2} is approximately fulfilled for all the considered cases.

Finally, it seems noteworthy that the normal-lead coupling asymmetry,
$\Gamma_{\rm L}\neq \Gamma_{\rm R}$, is irrelevant for the results except for a constant factor
diminishing the conductance $G$ \cite{KWIWJB-asym}.

\section{The role of CAR efficiency}
\label{sec:coef}

\begin{figure}[tb]
\includegraphics[width=0.98\linewidth]{Fig6.pdf}
\caption{Linear conductance between the normal leads
		 $G$ as a function of coupling to SC lead, $\GS{}$, for indicated values of RKKY exchange $J$
		 and the efficiency of CAR processes reduced by factor (a) $\mathcal{C}=0.9$ and (b) $\mathcal{C}=0.5$.
		 Other parameters as in \fig{3}.
		 Insets: QD1 local spectral density $\mathcal{A}(\omega)$ as a function of energy $\omega$
		 for points on $J=-0.1U$ curve, indicated with corresponding symbols.
		}
\label{fig:C}
\end{figure}

Up to this point we assumed $\GS{\rm X} = \sqrt{\GS{1}\GS{2}}$, which is valid when the two
quantum dots are much closer to each other than the coherence length in the superconductor.
This does not have to be the case in real setups, yet relaxing this assumption does not
introduce qualitative changes. Nevertheless, the model cannot be extended to inter-dot
distances much larger than the coherence length, where $\GS{\rm X}\to 0$.

To quantitatively analyze the consequences of less effective Andreev coupling we define the
CAR efficiency as $\mathcal{C} \equiv \GS{\rm X} / \sqrt{\GS{1}\GS{2}}$ and analyze $\mathcal{C} < 1$
in the wide range of $\GS{1}=\GS{2}=\GS{}$ and other parameters corresponding to \fig{3}.
The results are presented in \fig{C}.

Clearly, decreasing $\mathcal{C}$ from $\mathcal{C}=1$ causes diminishing of $\GS{\rm X}$, and consequently of CAR
exchange. For a change as small as $\mathcal{C}=0.9$, the consequences reduce to some shift of the
conventional Kondo regime, compare \fig{C}(a) with \fig{3}. Stronger suppression of CAR may,
however, increase the SC coupling necessary to observe the second stage of Kondo screening caused
by CAR outside the experimentally achievable range, see \fig{C}(b). Moreover, the reduced $T^*$
leads to narrowing of the related local spectral density dip, while the
increased critical $\GS{}$ necessary for the observation of the second stage of screening leads to the
shallowing of the dip. This is visible especially in the inset in \fig{C}(b).

\section{Conclusions}
\label{sec:conclusions}

The CAR exchange mechanism is present in any system comprising at least
two QDs or magnetic impurities coupled to the same superconducting contact
in a way allowing for crossed Andreev reflections.
In the considered setup, comprised of two quantum dots in a T-shaped geometry
with respect to normal leads and proximized by superconductor,
it leads to the two-stage Kondo
screening even in the absence of other exchange mechanisms.
This CAR induced exchange screening is characterized by a residual
low-temperature conductance at particle-hole symmetric case.
We have also shown that the competition between CAR exchange and RKKY
interaction may result in completely different Kondo screening scenarios.

The presented results bring further insight into the low-temperature
behavior of hybrid coupled quantum dot systems, which hopefully could be verified
with the present-day experimental techniques.
Moreover, non-local pairing is present also in bulk systems such as non-$s$-wave superconductors.
The question if an analogue of discussed CAR exchange may play a role there
seems intriguing in the context of tendencies of many strongly correlated materials
to possess superconducting and anti-ferromagnetic phases.

\begin{acknowledgments}
This work was supported by the National Science Centre in Poland through project no.
2015/19/N/ST3/01030.
We thank J. Barna\'{s} and T. Maier for valuable discussions.
\end{acknowledgments}

Answer: It tends to suppress the Kondo effect.


Input: How many underclassmen are on the NBA Draft Early-Entry List?

Context:
The Official 2006 NBA Draft Early-Entry List. 30 International Players, 62 underclassmen and one lone 5th year high school player make up this year's list, for a total of 93 early-entrants. Extensive commentary and early draft status projections are included.. For comparison, in 2005, 108 players declared (61 NCAA, 12 high school, 35 internationals), up from 94 in 2004, and 73 in 2003.
There were no major surprises on the early-entry list, besides a few mid-major, Division II and NAIA players that decided to enter, as well as 5th year high school player Clarence Holloway. Amongst the International players, Hrvoje Peric, Renaldas Seibutis, and Kyrylo Fesenko are considered mild surprises who could end up cracking the 2nd round. NCAA Lottery prospects Joakim Noah, Al Horford, Corey Brewer, Josh McRoberts, Brandon Rush and Tyler Hansbrough decided to sit this one out as expected, as did Marco Belinelli, Uros Tripkovic and Goran Dragic on the international front.
Akbar Abdul-Ahad 6-0, PG, Idaho State Junior No Undrafted Averaged under 6 points in 20 minutes per game playing in the Big Sky. Being the first player on the NBA Draft Early-Entry list will likely go down as the highlight of his basketball career.
Arron Afflalo, 6-5, SG, UCLA Sophomore No Undrafted Afflalo initially told the LA media hes returning to school, but after a deep run in the NCAA tournament-- more in spite of his play than because of it--Afflalo will be testing the waters. Afflalo has very average size, athleticism, perimeter shooting and ball-handling skills. Hes clearly receiving bad advice on where his stock lies.
LaMarcus Aldridge, 6-11, PF/C, Texas Sophomore Yes Top 5 pick Aldridge made his announcement official to enter the draft some weeks ago. He will hire an agent soon (Arn Tellem?) and is considered a lock for the top 5 and a strong candidate for #1 overall.
Morris Almond, 6-6, SG, Rice Junior No ??? Almond announced hell be entering the draft, without an agent. He might be the best scorer in the NCAA youve never heard about. His stats are terrific, despite being the sole focal point of opposing defenses, and hes capable of scoring in a variety of ways, particularly with his jumper. Hes hoping for an invite to Orlando.
Renaldo Balkman, 6-8, PF, South Carolina Junior No Undrafted After winning the NIT MVP award, Balkman has decided to see where he stands in the eyes of the NBA by testing the waters. Hes likely to find them downright freezing, as hes a skinny and undersized power forward with little to no skills who came off the bench for a very average team.
Larry Blair,6-1, SG, Liberty Junior No Undrafted The 22 point per game scorer Blair is attempting to get some exposure for himself by testing the waters.
Will Blalock, Iowa State, 5-11, PG, Junior No Second round pick? Declared for the draft together with Curtis_Stinson after Iowa States coach was fired. Size is a big question mark. Will likely hope to attend the pre-draft camp in Orlando and try to show scouts hes a 1st rounder. Likely returns for his senior year.
Jahsha Bluntt, 6-6, SG, Deleware State Junior No Undrafted Puts up fairly average numbers (14.6 ppg, 41% FG) in one of the worst conferences in America. Looking for exposure at the Orlando pre-draft camp but its highly unlikely to receive it.
Josh Boone, 6-10, PF/C, UConn Junior No First round pick? Boone announced hell be entering the draft without an agent. An up and down season has left his stock in the air, and will likely force him to prove himself at the Orlando pre-draft camp. Would greatly benefit from a productive senior season as an offensive focal point now that UConn has lost almost all of its firepower from last year.
Ronnie Brewer, 6-6, PG/SG, Arkansas Junior No Lottery pick? After initially wavering a bit on his decision, Brewer announced hell be entering the draft without an agent in a press conference. Brewer is considered a likely late lottery pick to mid-first rounder pick, as his physical attributes and array of versatile skills on both ends of the floor are highly sought after.
Bobby Brown, 6-1, PG, Cal-State Fullerton Junior No First round pick? DraftExpress exclusively reported that Brown will be testing the waters. Still considered a bit of a sleeper because of the school he plays for, he will not be hiring an agent at this point. Some scouts are very high on his quickness and perimeter shooting ability and feel he will help his stock tremendously in private workouts.
Shannon Brown, 6-4, SG, Michigan State Junior No First round pick? As exclusively reported by DraftExpress, Brown will be testing the waters. He will likely conduct a number of workouts and attend the Orlando pre-draft camp to attempt and gauge where his stock lies. Scouts compare him to Celtic guard Tony Allen, but with a better attitude. Hes a very borderline first rounder in a draft that is stacked with shooting guards.
Derek Burditt, 6-7, SG, Blinn Junior College Sophomore No Undrafted Unknown Junior College prospect. Not ranked as one of the top 25 JUCO players in the country, averaged around 17 points per game. Not burning his draft card as hes not yet an NCAA player, so really doesnt have much to lose, or gain.
Leroy Dawson, 6-2, SG, Emporia State Junior No Undrafted Anonymous Division II player from the MIAA conference. 2nd team all conference, averaged 20 points per game. Like MANY on this list, only declaring because he can and has nothing to lose.
Travis DeGroot, 6-4, SG, Delta State Junior No Undrafted Plays in a strong Division II conference, but is at best only the 3rd best prospect on his own team after Jasper Johnson and Jeremy Richardson, and is therefore not a prospect at all.
Guillermo Diaz, 6-2, PG/SG, Miami Junior Yes First round pick? As reported by DraftExpress all year long, Diaz decided to forgo his senior year of college by hiring an agent, Miami based Jason Levien. One of the top athletes and shooters in the draft, which makes for an intriguing combination.
Cem Dinc, 6-10, SF/PF, Indiana Freshman No Undrafted As exclusively reported by DraftExpress, Dinc will be testing the waters. The coach that recruited him and then never played him, Mike Davis, resigned, so it would not shock anyone to see Dinc return to play in Europe and become automatically eligible next year after pulling out of this years draft.
Quincy Douby, 6-3, PG/SG, Rutgers Junior No First round pick As exclusively reported by DraftExpress, Douby sent out his paperwork to enter the draft. NBA scouts are all over the board on him, with some saying they consider him a 2nd round pick and others saying they would not be surprised if he ended up in the lottery. Terrific shooter and shot creator, averaged 28 ppg in the Big East conference. A real sleeper who will likely play in Orlando.
Mike Efevberha, 6-5, SG, Cal State Northridge Junior ??? Undrafted Ramona Shelburne of the LA Daily News reported that Efevberha will be testing the waters. Efevberha was the leading scorer in the country until he had a falling out with his coach and saw his playing time reduced significantly. Hell likely be looking for an invite to the Orlando pre-draft camp, and does not appear to be likely to head back to school.
Carl Elliot, 6-4, PG, George Washington Junior No Undrafted Elliot is using his use it or lose it draft card as a junior to get some exposure for himself through workouts and try to figure out where he stands in the eyes of the NBA. Elliot has excellent size for the PG position, but is still lacking plenty of all-around polish. His senior year will be essential to his development as a player. Reportedly has a family to support, which makes his decision tough considering how old he is already, despite only being a junior.
Jordan Farmar, 6-2, PG, UCLA Sophomore No First round pick? Farmar was the engine that led his team to the Finals of the NCAA tournament, and the only player that showed up once they got there. He is one of the top playmakers in the country, a Steve Nash type point guard, but his average athleticism, defense and outside shooting means hes only a bubble first-rounder. DraftExpress has been on his bandwagon since day one at UCLA, but is the NBA on it too?
Nick Fazekas, 6-11, PF, Nevada Junior No First round pick? Fazekas announced hell be entering the draft without an agent and will likely return to Nevada if it looks like hes not going to be a first round pick. If hes not a first rounder this year, its hard to imagine him ever being one since there isnt much left for him to accomplish individually in the NCAA. An interesting candidate for the pre-draft camp in Orlando.
Thomas Gardner, 6-5, SG, Missouri Junior No Second round pick? The St. Louis Post Dispatch reported that Gardner will enter the draft. Firing of underachieving Missouri coach Quin Snyder appeared to be the straw the broke the camels back. Gardner will have to hope to get invited to Orlando, but moving into the first round appears unlikely without an incredible performance there.
Rudy Gay, 6-8, SF, UConn Sophomore Yes Top 10 pick Gay announced hes leaving UConn at a press conference on campus, with Coach Calhoun by his side. He will hire an agent eventually. Size, length, incredible talent and athleticism means he might have the most upside of any player in this draft. Does he have the fire to capitalize on it though?
Reggie George, 6-10, PF, Robert Morris Chicago (NAIA) Junior No Undrafted Transfer from Iowa State had a nice season in the NAIA and is looking to capitalize on it by gaining some exposure for himself.
Daniel Gibson, 6-2, PG/SG, Texas Sophomore ??? Second round pick As exclusively reported by DraftExpress, Gibson will be entering the draft. There appears to be a conflict between Gibson and Texas regarding what his role will be next year, specifically whether or not hell be playing the point, meaning its unclear whether or not hell be returning. Gibson will likely go to Orlando to help him decide what his next step is. Showing off some PG skills will be essential there.
Aaron Gray, 7-0, Center, Pitt Junior No First round pick? After a disappointing end to his season, being outplayed by Patrick OBryant in the NCAA tournament, Gray has put that behind him and entered his name in the draft without an agent. Hes yet another underclassmen with huge questions marks about his pro potential that will likely have to go to the Orlando pre-draft camp to show he is worthy of a first round pick. Made some great strides this year, but still has a ways to go, especially conditioning-wise.
LeShawn Hammett 6-0, PG, St. Francis Junior No Undrafted Undersized combo guard played only 7 minutes in the mighty Northeast Conference before being suspended indefinitely for conduct detrimental to team. The NBA is clearly the only goal left for him to achieve.
Brandon Heath, 6-3, PG/SG, San Diego State Junior No Second round pick? Streaky shooting combo guard Heath announced the he will test the NBA draft process this summer, and is hoping for an invite to the Orlando pre-draft camp. MWC player of the year; has a lot of wrinkles to his game that need to be ironed out before he can legitimately think about the NBA.
Tedric Hill, 6-10, PF, Gulf Coast Community College Sophomore Yes Undrafted Ineligible to return to school after flunking out of college once again. Has bounced around over the past few years, and received some early hype from wannabe draftniks such as Gregg Doyel (CBS-Sportsline) and Sam Smith (Chicago Tribune) who compare him to Kevin Garnett. Very athletic we're told, but has absolutely no idea how to play the game. Has no chance of being drafted without an amazing showing at the Orlando pre-draft camp.
Clarence Holloway 7-0, Center, IMG Academy (Prep School) 5th year High School No Undrafted Lone high school player in this years age-limit depleted draft. Former Louisville commit never got eligible for college and was always considered too slow and heavy to make much of an impact anyway. Reportedly lost weight and improved his grades this past year at IMG and is currently being recruited by UConn, Kansas State and Oklahoma, amongst others.
Ekene Ibekwe, 6-9, PF, Maryland Junior No Undrafted Sources told DraftExpress exclusively that Ibekwe will be testing the waters. Likely only making this move because he can, as his chances of being drafted are very low. Athletic and long, but still lacking any type of polish.
Donald Jeffers, 6-8, PF, Roxbury Community College Sophomore No Undrafted Anonymous junior college player.
Alexander Johnson, 6-9, PF, Florida State Junior Yes First round pick? Sources told DraftExpress, that Johnson will be hiring an agent, mainly because he is already 23 years old. Hes considered intriguing because of his strength, raw offensive tools and freakish athleticism at the 4 position, and could work his way into the 1st round with strong workouts.
David Johnson, 6-7, PF, Clinton Junior College Sophomore No Undrafted 6-7 JUCO power forward who averaged 2 points and 3 rebounds per game.
Trey Johnson, 6-5, SG, Jackson State Junior No Undrafted Small school prolific scorer and one of the most accurate perimeter shooters in the country will attempt to draw some more attention to himself by testing the waters this summer. Johnson is hoping for a chance to prove himself in the Orlando pre-draft camp in June.
Coby Karl, 6-4, PG/SG, Boise State Junior No Undrafted Son of Denver Nuggets head Coach George Karl put up nice numbers (17 ppg, 5 rebs, 4 assists, 39.5% 3P) in the underrated WAC conference. Had surgery in March to remove a cancerous lump from his thyroid.
Mark Konecny, 6-10, Center, Lambuth (NAIA) Junior No Undrafted Transfer from Syracuse with mediocre production is looking for any type of exposure he can get before he graduates next season.
Kyle Lowry, 6-1, PG, Villanova Sophomore No First round pick NCAA tournament performance showed that he definitely needs another year, but regardless, Lowry is in. For now its without an agent. Considering the lack of quality point guard prospects in this draft, Lowry is likely a first round pick. Says he will attend the Orlando pre-draft camp if invited.
Aleks Maric, 6-11, Center, Nebraska Sophomore No Undrafted As exclusively reported by DraftExpress, Maric will be testing the waters. What may have played a role in this is the fact that the assistant coach that recruited him at Nebraska, Scott Spinelli, just moved on to Wichita State. Maric is considered a very average athlete who is still very raw and is therefore likely to go undrafted should he decide to stay in. Thanks to his Croatian passport, there is money waiting for him overseas if he chooses to take it.
Japhet McNeil, 5-10, PG, East Carolina Junior No Undrafted Severely undersized PG averaged 4 points and 5.6 assists in watered down Conference USA.
Paul Millsap, 6-8, PF, Louisiana Tech Junior Yes First round pick? As expected, Millsap has declared his intentions to enter the NBA draft, and according to sources hired an agent as well. Millsap has likely achieved just about everything he can in college at this point, and will land somewhere in the 20-40 part of the draft depending on workouts and measurements.
Matt Mitchell, 6-0, PG, Southern University-New Orleans Junior No Undrafted Anonymous NAIA player.
Adam Morrison, 6-8, SF, Gonzaga Junior Yes Top 5 pick As DraftExpress exclusively reported that Morrison will be declaring for the draft and hiring Chicago based agent Mark Bartelstein. Morrison, the top scorer in college basketball, is expected to be a top 5 pick and potentially the #1 pick overall. Questions linger about his athleticism and defense, but no one questions his passion, talent or feel for the game.
Patrick O'Bryant, 7-0, Center, Bradley Sophomore Likely First round pick NBA sources in Portsmouth told DraftExpress exclusively that OBryant will be testing the waters without an agent, but is likely to go all the way once he hears that hes a lock for the 1st round. His steady improvement, strong sophomore season, outstanding NCAA tournament and considerable upside means hes probably gone. O'Bryant since confirmed both DraftExpresss reports, particularly the one about hiring an agent in the Tri-State area (Andy Miller) should he decide to go all the way.
Evan Patterson, 6-7, SF, Texas Wesleyan Junior No Undrafted Mediocre numbers (11 ppg, 2 rebs) in a mediocre Southland conference.
Danilo Pinnock, 6-5, SG, George Washington Junior No Undrafted The extremely athletic Pinnock has told GWs student paper hell be testing the waters. Pinnock will attempt to capitalize on his teams success this year by potentially attending the NBA pre-draft camp in Orlando. Pinnock will have to show better ball-handling and perimeter shooting ability than he did during the regular season.
Leon Powe, 6-7, PF, Cal Sophomore No Second round pick Powe announced hell be testing the waters in a statement released by Cal. Where he ends up being projected depends heavily on how his knee checks out. Powe is already considered a serious tweener by NBA scouts, and had a hard time this season gaining back much of the explosiveness he had earlier in his career. Could realistically go undrafted should he decide to stay in.
Richard Roby, 6-5, SG, Colorado Sophomore Likely Second round pick As first indicated by DraftExpress Roby has decided to test the waters. Disappeared against any major competition he went up against, particularly towards the end of the season. Roby will likely have to put on weight in the next few months and show off his perimeter stroke in the Orlando pre-draft camp. Sources tell us that he is on the verge of making a huge mistake by hiring an agent.
Rajon Rondo, 6-2, PG, Kentucky Sophomore Yes First round pick As expected, Rondo has decided to enter the NBA draft, and has also hired an agent, Bill Duffy. Despite an inconsistent sophomore season, most scouts weve spoken to still had him as at least the #2 point guard on their board because of his intriguing upside. Workouts will be huge for him.
Blake Schilb, 6-7, SG/SF, Loyola Chicago Junior No Undrafted Declared his intentions to enter the draft, without an agent, and is hoping for an invite to Orlando. Schlib is sorely lacking in the quickness and explosiveness departments that scouts demand from swingman prospects, but he makes up for it with his skill set to a certain extent. Regardless, sources tell us he wont be invited to Orlando, meaning he has to go back to school.
Mustafa Shakur, 6-4, PG, Arizona Junior No Second round pick? According to the Arizona Star, Shakur will likely enter his name in the draft, without an agent. Lute Olson confirmed it, saying he is not concerned about it. Shakur is hoping for an Orlando invite to show what he thinks he couldnt at Point Guard U.
Cedric Simmons, 6-9, PF/C, NC State Sophomore No First round pick? Simmons is reportedly "exploring his options," in regards to the 2006 NBA draft, but will do so without an agent. Nice size, frame, length, athleticism and defensive skills make him a very intriguing prospect.
Marcus Slaughter, 6-8, PF, San Diego State Junior Yes Second round pick? After burning his lone draft card a year early last June, despite being considered a marginal prospect, Slaughter has announced that he will be hiring agent Dan Fegan and forfeiting his remaining college eligibility. Slaughters father thinks that There was nothing else for Marcus to do at San Diego State. Many would disagree with that.
Curtis Stinson, 6-3, PG/SG, Iowa State Junior Yes Second round pick After swearing up and down last month that he has no intention on entering the draft, Stinson did just that. His coach Wayne Morgan, who he was very close to, was fired, resulting in him hiring agent Kevin Bradbury. The 23 year old combo guard will have to go to the Orlando pre-draft camp and impress if he wants to come close to being a 1st rounder.
Tyrus Thomas, 6-9, PF, LSU Freshman Yes Top 5 pick As DraftExpress exclusively reported Thomas called a press conference to announce his intentions to enter the 2006 NBA draft, as well as hire agents Brian Elfus and Mike Siegel. SEC Freshman of the year could be the most athletic player in the draft, as well as the player with the most overall upside.
PJ Tucker, 6-5, SF, Texas Junior No Second round pick As reported all year long by DraftExpress, Tucker will be entering the draft without an agent. Considering that hes a 6-5 combo forward with tremendous skills, his stock widely fluctuates depending on who is being asked. Phenomenal basketball player, but is severely lacking in 2-3 inches of height. Will likely need a strong showing at the Orlando pre-draft camp to have a legitimate shot at the 1st round. Some scouts compare him to Bonzi Wells.
Junior No Undrafted Undersized Division II post player has no chance of being drafted despite 20+8 averages.
Ian Vouyoukas, 6-10, Center, St. Louis Junior ??? Undrafted Vouyoukas declared his intentions to enter the draft, supposedly without an agent. Sources in Europe tell us he is likely to return to Greece to take a large contract offer from a first division team once he realizes he has no chance of being drafted. Vouyoukas is a nice mid-major big man who has improved somewhat in his junior season, but does not possess the necessary combination of athleticism and size required of an NBA center.
Darius Washington, 6-2, PG, Memphis Sophomore Likely First round pick? DraftExpress exclusively reported that Washington will be in the draft. It appears that hell be hiring an agent as well, despite not being anywhere near a lock for the first round.
Albert Weber, 6-3, SG, Connors State Sophomore No Undrafted Transfer from Alabama led his conference in scoring and is considered one of the top Junior College players in the country. Not officialy an NCAA player yet, and has not committed to any school yet, so really doesn't stand much to lose (or gain) from this move.
Marcus Williams, 6-3, PG, UConn Junior Yes Late Lottery-Mid-First As expected, Williams is expected to announce that hes hired Calvin Andrews of BDA Sports Management as his agent at a press conference next week. A strong junior season and outstanding NCAA tournament, establishing himself as one of the purest playmakers in the nation, means hes likely one of the first PGs taken.
Andriy Agafonov, 6-8, PF, Khimik 1986 Ukraine Undrafted Ukrainian power forward played 15 minutes and scored 6 points with 4.4 rebounds per game playing for FIBA EuroCup participants, and is declaring in hopes of getting his name out as he has one more draft card to burn after this before becoming automatically eligible.
Nemanja Aleksandrov, 7-0, SF/PF, KK Reflex 1987 Serbia & Montenegro ??? American agent has been telling us all year that hes likely to enter. Still hasnt played a game this year after taking slow recovery process from torn ACL. Once regarded as a prodigy and potential #1 overall pick, but injuries mean he hasnt played in nearly two years and is now considered damaged goods. Might just look for an attractive team to guarantee him in the 2nd round and develop him in the NBDL.
Pape-Philippe Amagou, 6-1, PG, Le Mans 1985 France ??? Amagous American agent has informed us that he will enter the NBA Draft this year, and participate in the Reebok Eurocamp in Treviso. Shares playmaking duties and spotlight with fellow early-entrant Yannick Bokolo.
Andrea Bargnani, 7-0, PF, Benetton Treviso 1985 Italy Top 5 pick Bargnani's Italian agent Stefano Meller told DraftExpress in Portsmouth that the Italian star power forward will definitely be entering the NBA draft. Bargnani is in the process of hiring an American agent and the only question is how long will it take for him to make it over to the US after Benetton finishes up in the Italian playoffs, which could last as far as mid-June. He is expected to be a top 5 pick with a shot at going #1 depending on how the lottery plays out. Considered a phenomenal talent thanks to his excellent size, perimeter skills and athleticism relative to height.
Yannick Bokolo, 6-3, PG/SG, Le Mans 1985 France ??? Terrific athlete who is still making the transition to playing the point full time.
Carlos Cedeno, 6-5, SG, Guaiqueries 1985 Venezuela Undrafted Relatively unknown Venezuelan player. Has some international experience at the junior levels.
Tadija Dragicevic, 6-8, PF, Red Star Belgrade 1986 Serbia & Montenegro Undrafted Undersized power forward barely played in the Adriatic League this past season.
Lior Eliyahu, 6-9, SF/PF, Galil Elyon 1985 Israel Second round pick? Prolific and athletic Israeli combo forward will be entering the NBA draft this year looking for certain guarantees from an NBA team in the 1st or 2nd round. Eliyahu is still in the Israeli army and will stay overseas for another year regardless of what happens. He'll be represented by the American agency Entersport in the United States. A midseason injury set him back from being the top Israeli player in the league despite his youth.
Rudy Fernández, 6-5, SG, DKV Joventut 1985 Spain First round pick? Has some minor buyout issues to deal with to make sure he can stay in the draft. Excellent season in Spain has him projected as a pretty solid first round pick. Improved outside shooting, and still the same excellent athlete, passer, defender and all-around player hes always been. Still very skinny too.
Kyrylo Fesenko, 6-11, PF, Azovmash 1986 Ukraine Second Round Pick More to come.
Rafael Hettsheimeir, 6-9, Center, Akasvayu Girona 1986 Brazil Undrafted Undersized Brazilian center did not overly impress at the Nike Hoop Summit, showing that he will likely lack mobility until he takes off some weight.
Marko Lekic, 6-11, PF, Atlas 1985 Serbia & Montenegro ??? American agent Marc Cornstein, Lekic told us hell be putting his name in the draft this year once again. Still a bit of an unknown, numbers are fairly average in the Serbian YUBA league.
Damir Markota, 6-11, SF/PF, Cibona Zagreb 1985 Croatia Second round pick American agent Marc Cornstein told us Markota will definitely be putting his name in the draft once again. He had a breakout season in the Euroleague and Adriatic league before a groin injury slowed him down and eventually forced him to have minor surgery. Likely wont be able to come to the States until very late in the process. Does not have a buyout.
Mickael Mokongo, 5-11, PG, Chalon 1986 France ??? DraftExpress was exclusively informed hell be in the draft. Considered a talented athlete, but lack of size and the fact that he missed a large chunk of the season due to injury means his draft stock is very much up in the air still.
Brad Newley, 6-6, SG, 1985 Australia Second round pick Newely has told the Australian media that hes entering the draft. Hired Philadelphia based agent Leon Rose. Scouts who saw him play in Argentina last summer like his athleticism. Desperately lacking exposure, but agent appears to be unwilling to provide him with it.
Oleksiy Pecherov, 6-11, PF, Racing Basket 1985 Ukraine Second round pick DraftExpress received indication that Pecherov will be entering his name in the draft after a nice 2nd half regular season in France. Pecherov has his draft card in hand one year before he becomes automatically eligible, meaning he has nothing to lose. Has some nice skills facing the basket, but is still very soft and underdeveloped.
Hrvoje Peric, 6-8, SF, KK Split 1985 Croatia Second round pick? Good athlete who is still coming into his own as a basketball player. Did not play in the Adriatic League this season. Definitely needs at least another year in Europe, but could use the exposure that declaring for the draft provides.
Kosta Perovic, 7-2, Center, Partizan 1985 Serbia & Montenegro Undrafted? DraftExpress has been told that Partizan needs Perovic to be drafted this year to relieve them of his 500,000$ salary next year as well as help them financially with buyout money for their budget. Unfortunately this is happening about 3 years too late as weve seen little to no improvement from Perovic over that span.
Georgios Printezis, 6-9, PF, Olympiakos 1985 Greece Undrafted Greek power forward played 9 minutes and scored 4 points per game playing for a Euroleague team, and is declaring in hopes of getting his name out before he becomes automatically eligible next year.
Milovan Rakovic, 6-10, PF, Atlas 1985 Serbia & Montenegro ??? American agent Marc Cornstein told us Rakovic will be putting his name in the draft. Still an unknown player, puts up nice numbers on occasion in the fairly weak Serbian YUBA league.
Alexandr Rindin, 7-5, Center, Gala Baku 1985 Azerbaijan Undrafted Huge body, complete unknown. 5 points, 5 rebounds per game in FIBA Europe Cup.
Sergio Rodríguez, 6-3, PG, Estudiantes 1986 Spain First round pick Rodríguezs agent in the States told DraftExpress exclusively hell be in the draft, likely for good if he gets a commitment in the 1st round. A disappointing start to his season both in Spain and the ULEB cup made this European prodigy point guard fall on most teams draft boards, but Rodríguez picked things up substantially towards the end of the year and is now playing terrific basketball. Weak NCAA PG crop could put him in the lottery with good workouts.
Dusan Sakota, 6-10, SF/PF, Panathinaikos 1986 Greece Undrafted Fairly unathletic perimeter oriented big man was in the draft last year already. Plays for one of the best teams in Europe and rarely sees the floor for meaningful minutes.
Renaldas Seibutis 6-5, SG, Olympiakos 1985 Lithuania Undrafted One of the most productive players in Europe in his age group considering the level he plays at. Important cog on an excellent team, but lacks athleticism and isnt as good of a shooter as you would hope at this point in his career.
Saer Sene, 7-0, Center, Pepinster 1986? Senegal First round pick? Freakishly long and athletic African prospect who played extremely well at the Nike Hoop Summit. Many question his age and lack of productivity in the very average Belgian league A player teams will want to look at closely.
Sidiki Sidibe, 7-1, Center, Levallois 1985 France ??? 7-1, 265 pound volleyball player and former Kansas State commit will be in this years draft according to his American agent. Too raw to get any playing time whatsoever in French 2nd division.
Tiago Splitter, 7-0, PF/C, Tau Vitoria 1985 Brazil Lottery pick Splitters American agent Herb Rudoy told DraftExpress exclusively hes entering the draft Splitter is having a terrific season in both the ACB Spanish League and the Euroleague, but lack of buyout in his contract means he might not be able to stay in. CBA rules allow him to withdraw and become automatically eligible next season. Tau Vitorias president was quoted saying Splitter will be back in Spain next season.
Sun Yue, 6-9, PG/SF, Aoshen 1985 China Second round pick? Super talented tall point guard with decent athleticism and nice defensive skills. Lacks strength and outside shooting ability. Level of competition is mediocre in American semi-pro ABA league, which makes him an intriguing candidate for Orlando pre-draft camp.
Ali Traore, 6-9, PF, Roanne 1985 France ??? Puts up nice numbers in France. Will participate at the Reebok Eurocamp in Treviso.
Ejike Ugboaja, 6-8, PF, Union Bank Lagos 1985 Nigeria Undrafted Plays for Nigerian National Team.
Goran Dragic, 6-4, PG, Geoplin Slovan 1986 Agent initially notified us that Dragic will be entering the draft, but in the end decided to keep him out. His buyout was always a question mark.
Leigh Enobakhare, 6-10, Center, Oostende 1986 Agent Ugo Udezue from BDA Sports Management told us that Enobakhare will be entering the draft. In the end he must have heard that he is not considered a prospect at all, and decided to keep him out of the draft.
Cartier Martin, 6-8, SF/PF, Kansas State Junior Martin pondered entering his name in the draft, especially after the firing of Kansas State coach Jim Wooldridge.
Nick Young, 6-6, SG, USC Sophomore Young told the LA Daily News in February that hes staying at USC for another year.
D.J. Strawberry, 6-5, SG/SF, Maryland Junior Strawberry initially intended to test the waters, but eventually ended up not doing so once he found out that his chances of being drafted are almost non-existent.
Al Thornton, 6-7, SF/PF, Florida State Sophomore Implied earlier on in the year that he might put his name in, but sources recently told us it appears that he will return for his senior year. Tallahassee media backs this up.
Marcus Williams (AZ), 6-8, SG/SF, Arizona Freshman After initially appearing to be gone after numerous definitive reports, Williams surprised everyone and thrilled Arizona fans by announcing in a press conference hell be returning for his sophomore year.
Josh McRoberts, 6-11, PF, Duke Freshman After being upset by LSU in the Sweet Sixteen, McRoberts was quoted saying Ill be at Duke next year.. Duke issued a press release a month later confirming this.
Yi Jianlian, 7-0, PF, Guangdong 1987? International Jianlian announced in a press conference that hell be staying in China. A CBA official was also quoted on this matter, sounding as if they were the main factor for him staying put.
Acie Law, 6-3, PG, Texas A&M Junior After a fantastic showing in the NCAA tournament, Law helped his NBA draft stock considerably but will return for his senior year where A&M is expected to make a run at possibly winning the Big 12.
Joakim Noah, 6-11, PF/C, Florida Sophomore Huge 2nd half regular of the regular season and NCAA tournament boosted his stock into as high as the top 5. Noah came out and said afterwards hes staying regardless.
Al Horford, 6-9, PF, Florida Sophomore Horford indicated all season long that hes staying at least one more year, but playing extremely well in winning the national championship gave him a realistic chance at being a lottery pick. Regardless, Horford announced he'll return.
Corey Brewer, 6-8, SF, Florida Sophomore Brewer indicated all season long that hes staying at least one more year, but a terrific performance in the NCAA tournament gave him a realistic chance at being a top 20 pick. Regardless, Brewer announced he'll return.
Glen Davis, 6-8, Center, LSU Sophomore Davis announced hell be returning to LSU immediately after an absolutely horrendous showing in the Final Four which exposed all of his glaring weaknesses. Made it official as an LSU press conference alongside Tyrus Thomas.
Jason Smith, 7-0, PF/C, Colorado State Sophomore Smith announced that hes returning for his junior year, stating that "a little further down the road, it [the NBA] might be in my plans. I'm continuing to concentrate on my academics and see how I can help CSU as much as possible."
Jermareo Davidson, 6-10, PF, Alabama Junior > After burning his lone draft card a year early last June, Davidson considered entering the draft again, but eventually made the right decision in announcing hell be returning for his senior year.
Richard Hendrix, 6-8, PF, Alabama Freshman Told Alabama media after NCAA tournament loss that hell be back in Tuscaloosa next year.
Ja'Vance Coleman, 6-3, SG, Fresno State Junior Testing the waters according to the Fresno Bee. Whoops, no hes not.
Sean Singletary, 5-11, PG, Virginia Sophomore Singletary told The Daily Progress in early February that hes returning.

Answer: 62.