# Underscreened Kondo effect in quantum dots coupled to ferromagnetic leads 

$^{1}$Department of Physics, Adam Mickiewicz University, 61-614 Pozna´ n, Poland 

Ireneusz Weymann 1, 2, ∗ and L´ aszl´ o Borda 3, 4 

$^{2}$Physics Department, Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universit¨ at, Theresienstrasse 37, D-80333 Munich, Germany 

$^{3}$Department of Theoretical Physics and Research Group “Physics of Condensed Matter” of the Hungarian Academy of Sciences, Budapest University of Technology and Economics, Budafoki ´ ut 8., H1111 Budapest, Hungary 

$^{4}$Physikalisches Institut and Bethe Center for Theoretical Physics, Universit¨ at Bonn, Nussallee 12, D-53115 Bonn, Germany 

(Dated: November 7, 2018) 

We analyze the equilibrium transport properties of underscreened Kondo effect in the case of a two-level quantum dot coupled to ferromagnetic leads. Using the numerical renormalization group (NRG) method, we have determined the gate voltage dependence of the dot’s spin and level-resolved spectral functions. We have shown that the polarization of the dot is very susceptible to spin imbalance in the leads and changes sign in the middle of the S = 1 Coulomb valley. Furthermore, we have also found that by fine-tuning an external magnetic field one can compensate for the presence of ferromagnetic leads and restore the Kondo effect in the case of S = 1 2 Coulomb valley. However, the underscreened Kondo effect cannot be fully recovered due to its extreme sensitivity with respect to the magnetic field. 

Ever since the observation of the resistivity anomaly in normal metals at low temperatures, 1 the Kondo prob- lem 2 has been investigated perpetually both experimen- tally and theoretically. 3–5 As a result of that intensive research, in many respects, the Kondo effect is well un- derstood now. Very recently the interplay between the Kondo effect and other many-body phenomena, such as superconductivity or ferromagnetism, has attracted a lot of interest. That attention was mainly motivated by the recent advances in nanofabrication which have opened the possibility to attach superconducting 6,7 or ferromag- netic 8–13 leads to molecules or semiconducting quantum dots. In the following we will focus our attention on the case of ferromagnetic leads coupled to a quantum dot exhibiting the Kondo effect. 

From the theoretical side, a consensus was found that single level quantum dots (i.e. relatively small dots with level spacing δ larger than the hybridization Γ) attached to ferromagnetic leads exhibit finite spin asymmetry as a result of the spin imbalance in the leads, which gives rise to a splitting and suppression of the Kondo reso- nance. 14–17 This spin asymmetry can, however, be com- pensated by means of an external magnetic field or by tuning the gate voltage properly. 18,19 In that way the strong coupling Kondo fixed point can be reached, with a fully developed Kondo resonance and a somewhat re- duced Kondo temperature. This expectation has recently been confirmed experimentally. 8–13 

While in many cases the quantum dots can success- fully be modelled by only a single local level, there are few exceptions when the finite level spacing affects the low energy physics of such a device considerably. An iso- lated quantum dot with even number of electrons can 

PACS numbers: 72.25.-b, 73.63.Kv, 85.75.-d, 72.15.Qm 

## I. INTRODUCTION 

have spin triplet S = 1 ground state, if the level spac- ing between the two single particle levels closest to the Fermi energy is anomalously small, i.e. smaller than the exchange coupling between the two electrons situated on those levels. If such a device is attached to only one lead then the system will exhibit the so-called underscreened Kondo effect. 20–24 A single lead can only screen half of the local spin below the Kondo temperature T$_{K}$ while the residual spin-a-half object is left unscreened with a ferromagnetic Kondo coupling to the rest of the conduc- tion electrons. Such a ferromagnetic Kondo coupling is known to be irrelevant in renormalization group sense, as it scales to zero when the energy scale is lowered, re- sulting in a Fermi liquid plus a decoupled S = 1 2 object at zero temperature. This behavior is reached in a non- analytic fashion resulting in singular terms in e.g. the single particle scattering rate. Such kind of behavior was recently termed as singular Fermi liquid . 22,24 

In a real quantum dot subject to a transport experi- ment the situation is slightly more complex as the dot has to be coupled to two leads to drive current through it. Even if the second lead is just a weakly coupled probe, its presence introduces a second energy scale, T ′ K ≪ T$_{K}$ , at which the residual spin-a-half is screened by the other lead mode. The screening takes place in two stages and the separation between the energy scales T$_{K}$ and T ′ K can be tuned by the asymmetry between the coupling to the leads. Since the Kondo temperature is exponentially sen- sitive to the coupling, a realistic value of asymmetry is enough to separate the two stages completely. In such a case, if the other experimentally relevant energy scales, such as temperature T or magnetic field B , lie in between the two Kondo temperatures T ′ K ≪ B, T < T$_{K}$ , then the system is well described by the underscreened Kondo model. In fact, very recently, underscreened Kondo effect 

