Quantum Engineered Kondo Lattices

TL;DR

This paper explores the quantum engineering of nanoscopic Kondo lattices, demonstrating how geometry influences their properties and revealing a new quantum phenomenon called the Kondo echo, advancing understanding of strongly correlated electron systems.

Contribution

It introduces the concept of engineering Kondo droplets with tailored geometries to emulate heavy-fermion materials and uncovers the Kondo echo as a novel quantum signature.

Findings

Geometry controls Kondo temperature in droplets.

Coherent coupling approaches quantum-coherent Kondo lattice.

Discovery of the Kondo echo as a probe of Kondo cloud correlations.

Abstract

Recent advances in atomic manipulation techniques have provided a novel bottom-up approach to investigating the unconventional properties and complex phases of strongly correlated electron materials. By engineering artificial condensed matter systems containing tens to thousands of atoms with tailored electronic or magnetic properties, it has become possible to explore how quantum many-body effects---whose existence lies at the heart of strongly correlated materials---emerge as the size of a system is increased from the nanoscale to the mesoscale. Here we investigate both theoretically and experimentally the quantum engineering of nanoscopic Kondo lattices -- Kondo droplets -- that exemplify nanoscopic replicas of heavy-fermion materials. We demonstrate that by changing a droplet's real space geometry, it is possible to not only create coherently coupled Kondo droplets whose properties asymptotically approach those of a quantum-coherent Kondo lattice, but also to markedly increase or decrease the droplet's Kondo temperature. Furthermore we report on the discovery of a new quantum phenomenon -- the Kondo echo -- a signature of droplets containing Kondo holes functioning as direct probes of spatially extended, quantum-coherent Kondo cloud correlations.

Equations

$$\mathcal{H}=-\sum _{\mathbf{r},{\mathbf{r}}^{\prime },\sigma }{t}_{\mathbf{r},{\mathbf{r}}^{\prime }}{c}_{\mathbf{r},\sigma }^{†}{c}_{{\mathbf{r}}^{\prime },\sigma }+J{\sum _{\mathbf{r}}}^{\prime }{\mathbf{S}}_{\mathbf{r}}^K\cdot {\mathbf{s}}_{\mathbf{r}}^c,$$

$${\mathbf{S}}_{\mathbf{r}}^K=\sum _{\alpha ,\beta }{f}_{\mathbf{r},\alpha }^{†}{\mathbf{\sigma }}_{\alpha ,\beta }{f}_{\mathbf{r},\beta }{\mathbf{s}}_{\mathbf{r}}^c=\sum _{\alpha ,\beta }{c}_{\mathbf{r},\alpha }^{†}{\mathbf{\sigma }}_{\alpha ,\beta }{c}_{\mathbf{r},\beta },$$

$${\hat{n}}_f(\mathbf{r})=\sum _{\alpha }{f}_{\mathbf{r},\alpha }^{†}{f}_{\mathbf{r},\alpha }=1.$$

$$s(\mathbf{r})=\frac{J}{2}\sum _{\alpha }⟨f_{\mathbf{r},\alpha }^{†}{c}_{\mathbf{r},\alpha }⟩.$$

$$s(\mathbf{r})$$

$$n_f(\mathbf{r})$$

$${\hat{G}}_{ff}(\omega )$$

$${\hat{G}}_{cc}(\omega )$$

$${\hat{G}}_{fc}(\omega )$$

$${\hat{g}}_{ff}(\mathbf{r},\mathbf{r},\omega )$$

$${\hat{g}}_{cc}(\mathbf{k},\omega )$$

$$\frac{dI(\mathbf{r},\omega )}{dV}=-\frac{e}{\hslash }{N}_t\sum _{\alpha }\sum _{i,j=1}^2{\left[\hat{t}\mathrm{Im}{\hat{G}}_{\alpha }(\mathbf{r},\mathbf{r},\omega )\hat{t}\right]}_{ij}$$

$$\frac{dI(\mathbf{r},V)}{dV}=$$

$$N_c(\mathbf{r},\omega )=$$

$$N_f(\mathbf{r},\omega )=$$

$$N_{cf}(\mathbf{r},\omega )=$$

$$\frac{dI(\mathbf{r},V)}{dV}=y_0+cV+B\frac{{\left(\frac{V-\alpha }{\Delta E_K}+q\right)}^2}{{\left(\frac{V-\alpha }{\Delta E_K}\right)}^2+1}$$

$$I(\mathbf{r})={\int }_0^{V_s}dV\frac{dI(\mathbf{r},V)}{dV}=I_0$$

$$\frac{dI(\mathbf{r},V)}{dV}=$$

$$t_c^2(\mathbf{r})=I_0{\left\{\frac{2\pi e}{\hslash }{N}_t{\int }_0^{V_s}dV\left[N_c(\mathbf{r},V)+{\left(\frac{t_f(\mathbf{r})}{t_c(\mathbf{r})}\right)}^2{N}_f(\mathbf{r},V)+2\frac{t_f(\mathbf{r})}{t_c(\mathbf{r})}{N}_{cf}(\mathbf{r},V)\right]\right\}}^{-1}$$

$$\frac{dI(\mathbf{r},V)}{dV}=$$

Full text

Quantum Engineered Kondo Lattices

Supplemental Information

Jeremy Figgins1

  

Laila S. Mattos2,3

  

Warren Mar2,4

  

Yi-Ting Chen2,5

  

Hari C. Manoharan2,3

  

Dirk K. Morr1

1 University of Illinois at Chicago, Chicago, Illinois 60607, USA

2 Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA

3 Department of Physics, Stanford University, Stanford, California 94305, USA

4 Department of Electrical Engineering, Stanford University, Stanford, California 94305, USA

5 Department of Applied Physics, Stanford University, Stanford, California 94305, USA

I Theoretical Formalism

We consider Co adatoms placed on a metallic Cu(111) surface in the form of highly ordered, hexagonal Kondo droplets (see Figs.1 and 2 of the main text). Kondo screening arises from the coupling of the magnetic adatoms to the two-dimensional surface band. Such a systems is described by the Kondo Hamiltonian Kon64 ; Don77 ; Col83 ; Hew93 ; Si03 ; Sen04 ; Oha05 ; Paul07

[TABLE]

where $c^{\dagger}_{{\bf r},\sigma},c_{{\bf r},\sigma}$ creates (annihilates) a conduction electron with spin $\sigma$ at site ${\bf r}$ on the Cu surface. Here, $t_{{\bf rr^{\prime}}}=0.924$ eV is the fermionic hopping element between nearest-neighbor sites in the triangular Cu (111) surface lattice, and $\mu=-5.13$ eV is its chemical potential Gom12 , yielding a Fermi wavelength of $\lambda_{F}\approx 11.5a_{0}$, where $a_{0}$ is the Cu lattice constant. Moreover, $J>0$ is the Kondo coupling, and ${\bf S}^{K}_{\bf r}$ and ${\bf s}^{c}_{\bf r}$ are the spin operators of the magnetic (Kondo) atom and the conduction electron at site ${\bf r}$, respectively. The primed sums run over the locations of the magnetic atoms only. We note that the Ruderman - Kittel - Kasuya - Yosida (RKKY) interaction between magnetic moments Don77 ; Hew93 ; Oha05 for the 2D Cu(111) surface band decays rapidly with increasing distance Sim11 , and is significantly smaller than $k_{B}T_{K}$ for the relevant inter-adatom distances considered in the main text (in particular for the experimental results shown in Figs.4 and 5 of the main text), such that it can be neglected. Finally, to reproduce the line shape and width of the Kondo resonance, $\Delta E_{K}\approx 4.7$ meV, measured experimentally Man00 for a single Co adatom on a Cu(111) surface, we employ $J=3.82$ eV [Ref. Fig10, ; Morr17, , and see SI Sec.II].

Starting from the Hamiltonian in Eq.(S1), a systematic large-$N$ expansion Col83 ; Hew93 ; Sen04 ; Paul07 ; Read83 ; Bic87 ; Mil87 ; Aff88 can be achieved by generalizing the spin operators to $SU(N)$ and representing them using Abrikosov pseudofermions

[TABLE]

where $\alpha,\beta=1,...,N$ and ${\bm{\sigma}}_{\alpha,\beta}$ is a vector whose $(N^{2}-1)$ elements are the generators of $SU(N)$ in the fundamental representation (each generator is represented by an $(N\times N)$ matrix with indices $\alpha,\beta$). Here, $f^{\dagger}_{{\bf r},\alpha}$ ($f_{{\bf r},\alpha}$) creates (annihilates) a pseudofermion in the magnetic $d$-orbitals of the Co adatom characterized by the spin quantum number $\alpha$. To ensure the existence a magnetic moment, one needs to satisfy the constraint that each adatom site is singly-occupied, i.e.,

[TABLE]

Inserting the representations of Eq.(S2) into the Hamiltonian, Eq.(S1), yields quartic fermionic interaction terms. On the mean-field level, we decouple these terms by introducing the expectation value

[TABLE]

where $s({\bf r})$ describes the local hybridization between the conduction electron states and the magnetic $f$-electron states. $s({\bf r})$ is a measure of the strength of the Kondo screening, with $s({\bf r})=0$ representing an unscreened magnetic moment at site ${\bf r}$. The constraint in Eq.(S3) is enforced on the mean-field level, yielding $n_{f}({\bf r})=\langle{\hat{n}}_{f}({\bf r})\rangle=1$, by adding a Largrange multiplier in the form of the term $\sum_{{\bf r},\alpha}\varepsilon_{f}({\bf r})f^{\dagger}_{{\bf r},\alpha}f_{{\bf r},\alpha}$ to the Hamiltonian in Eq.(S1), where $\varepsilon_{f}({\bf r})$ represents the on-site energy of the $f$-electron states. The resulting Hamiltonian is quadratic and can therefore be diagonalized in real space. However, since the lifetime of the conduction and magnetic $d$-orbital states plays a major role in determining the $dI/dV$ lineshape, we account for them by rewriting the above equation for $s({\bf r})$ [Eq.(S4] and the constraint $n_{f}({\bf r})=1$ in the form

[TABLE]

where

[TABLE]

Here, ${\hat{G}}_{\alpha\beta}(\omega)\ (\alpha,\beta=c,f)$ are Greens function matrices in real space with $G_{\alpha\beta}({\bf r},{\bf r},\omega)$ being the $(r,r)$ element of the matrix ${\hat{G}}_{\alpha\beta}$. ${\hat{g}}_{ff}$ and ${\hat{g}}_{cc}$ are the unhybridized Greens function matrices, and the spin degeneracy has been accounted for by dropping the spin index in the matrices. The hybridization matrix ${\hat{s}}$ is only non-zero at the sites where a magnetic adatom is located. A finite lifetime, $\tau_{c,f}$, of the conduction and $d$-orbital states is then introduced in the above Greens functions via the scattering rate $\Gamma_{c,f}=\hbar/\tau_{c,f}$ yielding

[TABLE]

with ${\hat{g}}_{cc}({\bf r},{\bf r},\omega)$ being obtained from ${\hat{g}}_{cc}({\bf k},\omega)$ via Fourier transform. To reproduce the experimentally measured $dI/dV$ lineshapes shown in Fig.4 of the main text, we used $\Gamma_{f}=4$ meV and $\Gamma_{c}=65$ meV. Eqs.(S5) - (S8) are a closed set of equations that can now be self-consistently solved to obtain the local hybridizations, $s({\bf r})$ and $f$-electron energies $\varepsilon_{f}({\bf r})$. We note that for a single Kondo impurity, this formalism is identical to the saddle-point approximation of the path-integral approach of Ref. Read83 in the large-$N$ approximation which becomes exact in the limit $N\rightarrow\infty$.

II Differential Conductance $dI/dV$, the Fano-fit, and the width of the Kondo resonance

To compute the differential conductance, $dI/dV$ Mal09 ; Fig10 ; Wol10 ; Morr17 , measured in STS experiments on Kondo systems Mad01 and heavy fermion materials Sch09 ; Ayn10 ; Ern11 , we define the spinor $\Psi^{\dagger}_{{\bf k},\alpha}=(c^{\dagger}_{{\bf k},\alpha},f^{\dagger}_{{\bf k},\alpha})$ and the Green’s function matrix ${\hat{G}}_{\alpha}({\bf k},\tau)=-\langle T_{\tau}\Psi_{{\bf k},\alpha}(\tau)\Psi^{\dagger}_{{\bf k},\alpha}(0)\rangle$. With $t_{c}$ and $t_{f}$ being the amplitudes for electronic tunneling from the tip into the conduction electron bands of the Cu(111) surface and the magnetic $d$-orbitals of the Co adatom at site ${\bf r}$, respectively, one obtains Fig10

[TABLE]

where ${\hat{t}}=\begin{pmatrix}t_{c}&0\\ 0&t_{f}\end{pmatrix}$, and $N_{t}$ is the STS tip’s density of states, which is taken to be constant. In the weak-tunneling limit, $t_{c},t_{f}\rightarrow 0$, we obtain

[TABLE]

where

[TABLE]

with $N_{c}$ and $N_{f}$ being the density of states of the conduction and $f$-electron ($d$-orbital) states, respectively. To reproduce the line shape and width of the Kondo resonance, $\Delta E_{K}\approx 4.7$ meV, measured experimentally Man00 for a single Co adatom on a Cu(111) surface, we employ $J=3.82$ eV and $N=4$ [Ref. Fig10, ; Morr17, ]. Moreover, the theoretical fits to the experimental $dI/dV$ results in Figs.4 and 5 of the main text were obtained with the same set of parameters, $J,\Gamma_{c}$, and $\Gamma_{f}$. Note that the experimental data in Fig.4a,b of the main text show a weak scattering peak around $V=-8$ mV, which arises from the presence of proximal step edges, and is not directly related to the Kondo resonance. This mode, in the form of a Lorentzian peak centered at $E=-8$ meV with half-width $\Delta E=5.75$ meV as well as a sloping background were added to the theoretically computed $dI/dV$ lineshape, yielding the results shown in Figs.4c,d.

To extract the width, $\Delta E_{K}$ of the Kondo resonances (shown in Figs.2e and 3a of the main text), we have fitted our theoretically obtained $dI/dV$ lineshapes (shown, for example, in Fig.3b of the main text) by the Fano formula Fano61

[TABLE]

where $\Delta E_{K}$ measures the width of the Kondo resonance, $q$ its asymmetry, and $cV$ represents a sloping background. We find that changing the tunneling ratio, $t_{f}/t_{c}$, changes the values for $\alpha$ and $q$, but has only a weak effect on the width of the resonance $\Delta E_{K}$.

III Quantum Interference and Kondo screening

To demonstrate how quantum interference of the Kondo screening clouds associated with different magnetic adatoms can give rise to variations in the strength of the Kondo screening, as evidenced by the magnitude of the hybridization, $s$, or the width of the Kondo resonance, $\Delta E_{K}$ (as shown in Figs.2 and 3 of the main text) we begin by

considering the effects of a single, Kondo-screened Co atom on the electronic structure in its vicinity. To this end, we present in Fig. S1a the local density of states (LDOS) of the conduction $c$-electrons for several distances $\Delta$r from a single, Kondo screened Co adatom located an a Cu(111) surface. The non-monotonic dependence of the LDOS on $\Delta$r arises from $2k_{F}r$ scattering of the screening conduction electrons. Let us now consider the effects on the hybridization when a second Co adatom is added to the system by noting that the magnitude of the hybridization is in general determined by the value of the conduction electron LDOS – the screening band – near the Fermi energy. This LDOS at $E=0$ as a function of $\Delta$r is shown in Fig. S1b. When a second Co adatom is now placed at a distance $\Delta$r from the first impurity where the LDOS is smaller (greater) than in the unperturbed metal, one expects that the resulting hybridization (which by symmetry is the same for both Co adatoms) is also smaller (greater) than that of a single Co adatom, which is placed on the unperturbed Cu surface. This expectation is confirmed by the explicit calculation of the hybridization of a 2 Co adatom system, as a function of inter-adatom distance, $\Delta$r, shown in Fig. S1c, which shows the same non-monotonic dependence on $\Delta$r as the LDOS in Fig. S1b. In Kondo droplets consisting of multiple Co adatoms, it is the same effect as discussed above for the 2 Co adatom system that leads to the non-monotonic dependence of the hybridization on $\Delta$r: it is the quantum interference of the Kondo screening clouds associated with each of the Co adatoms that determines the resulting spatial distribution of the hybridization.

IV Position dependence of $dI/dV$ in closed loop STS experiments

To compare our theoretical calculations for the spatial dependence of the differential conductance to the experimental results, shown in Figs.5c,d of the main text, we note that the STS experiments are performed in closed loop mode. This implies, that at every position, the height of the tip from the sample is adjusted to keep the total current

[TABLE]

for a given set point voltage, $V_{S}$, constant. A change in height, however, implies that the tunneling amplitudes become position dependent, and that therefore Eq.(S10) needs to be generalized to

[TABLE]

Inserting Eq.(S15) into Eq.(S14) then yields

[TABLE]

and hence

[TABLE]

It is this “normalized” differential conductance, which is shown in Figs.5g,h of the main text and accounts for the experimental closed loop mode that needs to be compared to the experimentally measured $dI/dV$ linecuts through a Kondo hole droplet, shown in Figs.5c,d of the main text. For our theoretical results, we used $t_{f}/t_{c}=0.03$ at the sites of the Co atoms, and $t_{f}=0$ otherwise.

We note that when the height of the tip is varied, both $t_{c}({\bf r})$ and $t_{f}({\bf r})$ can in general change, which can lead to variations in $t_{f}/t_{c}$ not only between the experimentally studied droplets, but also within the same droplet. From a comparison of our theoretical results with the experimental $dI/dV$ lineshapes shown in Figs.4a,b of the main text for the Kondo droplets with $\Delta r_{2}=4\sqrt{3}a_{0}$ (droplet 2) and $\Delta r_{1}=4a_{0}$ (droplet 1), respectively, we find that $t_{f}/t_{c}$ is slightly larger in droplet 2 than in droplet 1. Specifically, we obtain for droplet 1 $t_{f}/t_{c}=0.025$ both at the center and edge sites, while for droplet 2 we obtain $t_{f}/t_{c}=0.0327$ at the center site, and $t_{f}/t_{c}=0.03$ at the edge site. Finally, for an isolated Co atom, we obtain $t_{f}/t_{c}=0.0275$. To show that these small changes in $t_{f}/t_{c}$ are related to changes in the tip height (due to the closed loop mode in which the STS experiments are performed) between the different spatial locations for which $dI/dV$ lineshapes are shown in Fig.4 of the main text, we computed the $dI/dV$ lineshapes for all spatial positions with the same value of $t_{f}/t_{c}$, as shown in Fig. S2. Given the experimentally used set point voltage of $V_{s}=10$ meV, it immediately follows that if $t_{f}/t_{c}$ were constant, that then the total current [see Eq.(S14)] flowing through the STS tip were larger for droplet 1 than for an isolated Co atom, and larger for an isolated Co atom than for droplet 2. This implies that in order to measure the same total current in the droplets and for an isolated Co atom, the tip height needs to be the smallest for droplet 2, followed by the isolated Co atom and then droplet 1. Given our analysis above, this suggests that the ratio $t_{f}/t_{c}$ systematically increases with decreasing tip height, likely due to the increasing overlap of the tip orbitals with the Co d-orbitals.