The Kondo effect in a quantum confined system

Abraham Kipnis

The Kondo effect in a quantum confined system

Thesis submitted for examination for the degree of Master of Science in Technology.

Supervisor Professor Bernardo Barbiellini School of Engineering Science

Lappeenranta-Lahti University of Technology Finland

Reviewers M.Sc. (Tech) Markus Aapro Department of Applied Physics Aalto University

Finland

Lappeenranta-Lahti University of Technology LUT

Lappeenranta-Lahti University of Technology LUT LUT School of Engineering Science

Computational Engineering Abraham Kipnis

The Kondo effect in a quantum confined system

Master’s thesis 2022

48 pages, 36 figures, 1 table, 1 appendix

Examiners: Professor Bernardo Barbiellini and Markus Aapro (M.Sc. (Tech))

Atomic-scale engineering aims to exploit quantum effects to produce tailored electronic properties. One such effect is the Kondo effect, in which a many-body correlation arises between the spin of an electron in a localized quantum state and the spins of electrons near Fermi energy in its local host environment. The Kondo temperature is the relevant energy scale associated with the electrons involved in the spin-exchange events related to this correlation. The relative strength of surface state and bulk state electron contributions to the Kondo temperature are still under theoretical and experimental debate. Here we use low-temperature scanning tunneling microscopy (LT-STM) and scanning tunneling spec- troscopy to tune and measure the Kondo temperature of single cobalt atoms on Ag(111).

We adjust the surface electron local density of states around a Co atom by confining it within quantum corrals made from Ag atoms on the Ag(111) surface. By building corral walls from Ag atoms rather than Co atoms, we exclude surrounding magnetic impurities affecting the Kondo temperature. We thus change the magnitude of the Co atom coupling to the surrounding conduction electron bath, thereby the effective Kondo temperature.

We measure the width of the Kondo resonance on Co atoms inside corrals as a function of corral radius, which allows to extract the bulk and surface state coupling strengths. Our findings motivate further LT-STM experiments to inform unanswered questions concern- ing the Kondo effect in single magnetic atoms on metals.

Keywords: Kondo physics, quantum corral, scanning tunneling microscopy, nanoscale magnetism, low-temperature, condensed matter, surface science

Acknowledgements

A thesis from the northernmost low-temperature STM group in the world would not have been possible without support from mentors, friends, peers, and family who have made Finland a second home. First, I must thank the Fulbright Finland Foundation and Dr.

Bernardo Barbiellini. They sponsored my move here, beginning this impactful scientific collaboration. I would also like to acknowledge my scientific mentors, Dr. Francis Starr, Dr. Renee Sher, and Dr. Jack DiSciacca, among others. They helped shape my research interests and encouraged me to develop the skills and background knowledge to learn from and contribute to this project. Finally, to Lexi and Alex, and all the others from LUT: thanks for arriving with me in Lappeenranta, being great friends, and helping to make some beautiful memories. As this chapter comes to an end, I look forward to many more!

Thank you to my colleagues from the Aalto Atomic Scale Physics group, the Nanomi- croscopy Center, and the Aalto Applied Physics department. First and foremost, I owe this thesis to Dr. Peter Liljeroth, without whom this project would not have been pos- sible. I also owe many thanks to Markus Aapro, for conceptualizing and designing the experiment, teaching me the ins-and-outs of LT-STM, and endowing me with some of his vision, determination, and patience needed to progress in experimental physics. In addi- tion, Jose, Robert, Guangze, Hector, Chen, Shuning, Shawulienu, Somesh, Ben, Kuhsar, Viliam, Xin, Beatrice, and Antti all contributed to forming a talented, innovative, chal- lenging, inspiring, and encouraging intellectual community that helped me get to this point.

To my parents: even though my scientific career sometimes separates us intellectually and academically, I am lucky to have and always appreciate your unconditional support.

To the rest of my nuclear and extended family and friends, far and wide, who helped raise me to be the person I am now, know that you are also in my heart and mind.

Espoo, January 2022 Abraham Kipnis

Nomenclature

1 Introduction 1

1.1 Surface states . . . 3

1.2 Kondo effect . . . 5

1.3 Atom manipulation . . . 10

1.4 Quantum corrals . . . 11

2 Methods 15 2.1 Scanning tunneling microscopy . . . 15

2.2 Scanning tunneling spectroscopy . . . 17

2.3 Ultra-high vacuum low-temperature STM . . . 19

2.4 Substrate preparation . . . 20

2.5 Magnetic atom deposition . . . 20

2.6 Nonmagnetic atom deposition . . . 21

2.7 Atom manipulation . . . 24

2.8 Analysis . . . 25

3 Results 26

4 Discussion 33

5 Summary and Outlook 35

6 Appendix 36

Nomenclature

Physical constants

e Elementary charge 1.602×10⁻¹⁹C

h Planck constant 6.626×10⁻³⁴J Hz⁻¹

k_b Boltzmann constant 1.380×10⁻²³J K⁻¹

m_e Electron mass 9.109×10⁻³¹kg

Symbols

E₀ Surface state onset energy

E_F Fermi energy

T_K Kondo temperature v_F Fermi velocity

m_{ef f} Electron effective mass Abbreviations

ARPES Angle-resolved photoemission spectroscopy DFT Density-functional theory

EQC Elliptical quantum corral FER Field emission resonance IETS Inelastic tunneling spectroscopy LDOS Local density of states

STM Scanning tunneling microscopy UHV Ultra-high vacuum

XPS X-ray photoemission spectroscopy

List of Figures

1.1 Surface and bulk state wavefunctions . . . 3

1.2 Surface and bulk band structure of Ag(111) . . . 4

1.3 Ag(111) surface with Co adatom . . . 4

1.4 Line spectrum from Ag(111) step edge . . . 5

1.5 Kondo screening in Anderson impurity model . . . 8

1.6 Kondo scattering and Kondo cloud . . . 8

1.7 Normalized Fano resonance profiles . . . 9

1.8 Xe atom manipulations on Ni(110) . . . 10

1.9 Quantum corrals of Fe atoms on Cu(111) . . . 11

1.10 Circular corral ground states . . . 13

1.11 Kondo mirages on quantum corrals . . . 14

2.1 STM setup diagram . . . 16

2.2 STM system energy diagram . . . 16

2.3 Lock-in amplifier diagram . . . 18

2.4 Field emission resonance . . . 19

2.5 Sputtered Ag(111) STM topography . . . 20

2.6 Ag(111) sample annealing process in UHV . . . 21

2.7 Ag(111) terrace with deposited Co atoms . . . 21

2.8 Inelastic tunneling spectra on Co atom . . . 22

2.9 Inelastic tunneling spectra on Co atom inside quantum corral . . . 23

2.10 Topographic scan of Ag atom storage on Ag(111) . . . 24

2.11 Current trace from lateral manipulation . . . 25

3.1 Single Co atom line spectrum . . . 27

3.8 4.65 nmradius corral line spectra . . . 28

3.2 2.44 nmradius corral line spectra . . . 29

3.3 2.59 nmradius corral line spectrum, large bias range . . . 30

3.4 2.48 nmradius corral line spectrum, large bias range . . . 30

3.5 2.38 nmradius corral line spectrum,300 mVto -200 mVbias range . . . 31

3.6 3.81 nmradius corral line spectrum, large bias rangea . . . 31

3.7 3.65 nmradius corral line spectra . . . 32

3.9 Spectra on Co atoms inside circular quantum corrals . . . 32

3.10 Kondo widthwas a function of corral radiusr . . . 34

6.4 Empty3.66 nmand2.65 nmempty corral line spectra,±100 mV . . . 36

6.1 2.5 nmcorral grid spectrum . . . 37

6.2 Corral radius and Fano lineshape parameter extraction . . . 38

6.3 Constant heightdI/dV maps of4.65 nmradius corral . . . 39

1

1 Introduction

Nano-scale engineering aims to control electronic degrees of freedom to gain new insight into physical phenomena. To this end, experimental methods such as low-temperature scanning tunneling microscopy (LT-STM) in ultra-high vacuum (UHV) allow sub-nmpre- cision placement and study of individual atomic and molecular surface adsorbates, which may inform or enable bottom-up fabrication in conjunction with conventional top-down fabrication methods such as lithography or etching. One particular tunable electronic de- gree of freedom is the electron spin, the origin of bulk magnetism [1]. Here we study the local correlation of spins in the conduction bath around individual magnetic atoms on a surface using LT-STM in UHV.

Singly-occupiedd orf orbitals in a transition metal or rare earth metal atom act as a localized magnetic moment when placed in a nonmagnetic metal host. This local moment causes magnetization in the nearby conduction electrons with oscillatory spatial depen- dence from the atom [2]. Due to short-lived spin-exchange events between the conduction electrons and the localized orbital, below a critical temperature a resonance, called the Kondo resonance, appears nearE_F with a width proportional to local density of states (LDOS) atE_F. The LDOS atE_F at a surface has two major contributors: electrons from bulk states and electrons from surface states.

The relative strength of the bulk and surface state contributions in generating the Kondo resonance is under theoretical and experimental investigation. The physics of the problem involves spin-polarized conduction electrons surrounding individual atoms; whether this spin-polarized cloud is tunable, has a large spatial extent, and couples via surface as well as bulk electrons is still under question in various contexts [3–9]. Answers to these ques- tions may provide opportunities to harness Kondo physics for applications in spintronics and quantum computing with interacting single spins on surfaces.

On Au(111), Ag(111), and Cu(111), surface states contribute substantially to total LDOS and are thus measurable with STM. Bulk LDOS is position independent, but surface LDOS is modulated by surface state electron scattering from perturbations such as a step edges or impurities. In addition, two-dimensional surface state eigenmodes emerge when scattering centers form confining geometry for electrons as understood by quantum particle-in-a-box models or many-body scattering theory; manipulation of sur- face adatoms into a circular ‘quantum corral’ was first demonstrated with LT-STM in 1993 [10]. The surface state eigenmodes in a quantum corral can be used to tune surface state LDOS atEF and thus investigate precisely the degree to which the Kondo effect is caused by surface state electrons in addition to bulk electrons [3, 11].

Using the STM to manipulate single atoms on Ag(111) which hosts a surface state with onset energy close toE_F, we build quantum corrals which modulate surface LDOS atE_F and thereby adjust the width of the Kondo resonance on adsorbed Co atoms. With scan- ning tunneling spectroscopy (STS) we measure spectra of the Kondo resonance, which we fit with a Fano lineshape describing interference between tunneling into a continuous background and discrete singly-occupied Co atom d level. The width of this lineshape allows to extract the Kondo temperature, the relevant energy scale of Kondo physics. By extracting the width of the Kondo resonance as a function of corral radius and thus surface

state density atEF, we provide evidence for and approximate the magnitude of contribu- tions of the surface state density to the Kondo resonance of Co atoms on Ag(111).

The structure of this thesis is as follows: inIntroductionwe describe in more detail sur- face states (1.1), the Kondo effect (1.2), atom manipulation with STM (1.3) and quantum corrals (1.4). We also cite relevant experiments with similar techniques and technologi- cal outlook where STM is used to study artificial band structures, quantum corrals, and atomic scale logic and memory. In Methodswe explain our experimental methods, in- cluding STM and STS (2.1, 2.2), UHV (2.3), substrate preparation (2.4), magnetic and nonmagnetic atom deposition (2.5,2.6), atom manipulation (2.7), and some of our anal- ysis (2.8). InResultswe share our results of quantum corral topography, dI/dV maps, line spectra, grid spectra, and results from extracting the Kondo width by fitting Kondo resonances to a Fano lineshape. InSummary and Outlookwe discuss how we can improve our analysis and propose additional experiments and theoretical work to complement our current understanding of the Kondo effect in confined quantum systems.

1.1 Surface states 3

1.1 Surface states

Electrons in surface states are confined in a quasi-2D electron gas near the surface of a crystal. We derive the existence of Shockley-type surface states by assuming the compo- nent of the wavevectorknormal to the planar interface with the vacuumκto be complex.

The wavefunction can be expressed as

ψ_k = exp(ik·r)u_k(r)withk= (k_x, k_y, k_z)andk_z =κ=a+ib∈C

= exp(ik⊥·r⊥) exp(iκz)u_k(r) (1.1) wherek∥ = (k_x, k_y), r∥ = (r_x, r_y), andu_k(r)is an oscillatory function permitting ψ to satisfy Bloch’s theorem thatu_k(r+R) = u_k(r)for all primitive surface lattice vectorsR.

This ψ is not physically realizable in the bulk, since it increases to infinity in−z, but becomes normalizable when matched withψ_z+ exponentially decreasing into bulk. The imaginaryk component κ leads ψ to oscillate and decay over a few atomic layers into the bulk [12]. Surface states are typically non-degenerate but not completely decoupled from bulk states, existing with quasi-parabolic dispersion in a gap between bulk bands in the projectedE-k bulk band structure (figure 1.2). Surface electrons undergo phonon- assisted scattering into bulk states, thus current can flow in thez-direction and imaging surface states with STM is possible. Surface state onset energyE₀ and effective mass m^∗ is measured with STM from constant height maps of electron standing surface waves (figure 1.3) by fittingE =E₀+_2m^ℏ2

ef f|k_∥|²whereE₀is the surface state onset energy with respect to Fermi energy. Angle-resolved photoemission spectroscopy (ARPES) can also measure these parameters as well as directly measure surface state lifetime. Surface state onset energy can be shifted by changing the geometry of the surface, for example near a step edge (figure 1.4).

Figure 1.1: Surface state and bulk wavefunctions. Both surface states and bulk states decay exponentially into the vacuum and are oscillatory in the bulk, but surface states also decay exponentially into the bulk, making them surface-localized. Reproduced from [13].

Crystal facet m^∗ =m_{ef f}/m_e E₀

Ag(111) 0.38 −67 meV

Cu(111) 0.42−0.46 −435 meV

Au(111) 0.28 −484 meV

Table 1.1: Surface state dispersion parameters for three common coinage metal facets where surface states are readily accessible with low-temperature STM.

Figure 1.2: Projected bulk band structure and surface state of Ag(111) computed using density functional theory (DFT). Surface electrons ex- ists inside a projected gap of the bulk band structure, from −67 meVto 250 meV. The surface state exhibits free electron-like parabolic dispersion E =E₀+ _2m^ℏ2

ef f|k∥|². Figure from [14].

0.0 2.5 5.0 7.5 10.0 12.5 nm

0 2 4 6 8 10

nm

0.0 0.1 0.2 0.3 0.4

Å

Figure 1.3: Constant current STM image of Co atom on Ag(111) im- aged as bright contrast. The bias voltage 92 meV is in the Ag(111) surface state band, so standing waves in image contrast originate from Co atoms due to interference of scattered surface electrons lead- ing to LDOS oscillations (called Friedel oscillations). Setpoint cur- rent: 125 pA.

1.2 Kondo effect 5

0 5 10 15 20

nm 100

50 0 50 100

mV

0 10 20 30

nm 0

5 10 15 20

25 0.00

0.25 0.50 0.75 1.00 1.25

nm

(a) (b)

Figure 1.4: (a) Spectra taken near step edge on Ag(111) along line marked in the topog- raphy scan (b), showingE₀ ≈ −67 meV(horizontal line). The contrast in the spectra at a given bias voltage in (a) correspond todI/dV which is directly proportional to the density of states at the given bias, where 0 bias corresponds to the sample Fermi energy. Spectra acquired with initial bias voltage100 mV, current500 pA, lock-in frequency602 Hz, and lock-in amplitude2 mV. Each spectrum is normalized bydI/dV at−100 mV. Imaging conditions:100 mV,500 pA. The effectiveE0increases near the step edge. Surface state electron phase relaxation length and lifetime can be extracted from this step edge inter- ference pattern [15, 16].

1.2 Kondo effect

In 1934, W.J. de Haas et. al. measured temperature-dependent electrical resistivity ρ(T) in Au wires at historically low temperatures. The resistivity did not monotoni- cally decrease with decreasing temperature but instead hit a minimum and then began in- creasing again logarithmically [17]. This contradicted experimental intuition of the time guided by Matthiessen’s rule which predicts resistivity originates from electron-phonon and electron-impurity scattering which reduce as temperature decreases.

In the 1960s, Myriam P. Sarachik measured correlation between low temperature elec- trical resistivity minima and magnetic moments in Mo_xNb_1–x alloys containing 1% Fe [18], which led Jun Kondo to use perturbative methods to calculate conduction electron scattering rates from local moments of magnetic impurities (equation 1.2) [19]. Kondo’s theory reproduced the observed low-temperature resistivity minimum and logarithmic divergence, thus this effect was named the Kondo effect. Resistance minima in dilute magnetic alloys have been observed in Cu, Ag, Au, Mg, and Zn with Cr, Mn and Fe impurities [2].

R_tot(T) = aT⁵ − c_impR_implog(k_BT

D ) (1.2)

Phonon contribution to resistivity

Electron-impurity scattering contribution

The second term in equation 1.2 describes Kondo’s prediction of the resistance min- imum and logarithmic divergence for T → 0, where D is the half-bandwidth of the metallic band fromE = −D to E = D centered aroundE = E_F = 0 andc_imp is the impurity concentration. The infinity inherent in the logarithm presents a problem which Phil Anderson tried to solve using a model Hamiltonian (equation 1.3) describing spin- conserved scattering between a localized spin1/2electron and a surrounding conduction electron energy band.

H =X

k,σ

c_k,σc^†_k,σE_k+X

σ

E_dd^†_σd_σ+U d^†_↑d↑d^†_↓d↓ +X

k,σ

V_k(d^†_σc_kσ+c^†_k,σd_σ) (1.3) In this modelc(c^†)is the annihilation (creation) operator for an electron in the conduction band,d(d^†)is the creation (annihilation) operator for an electron in the localizeddorbital, Uis the Coulomb repulsion, andV is the hybridization. Numerical renormalization group theory is required to fully resolve the logarithmic divergence and calculate finite ρ as T → 0[11]. The free parameter in equation 1.2 is the Kondo temperatureT_K, related to LDOS at the impurity by equation 1.4:

kBTK =Dexp[−1/J ρ(EF)], (1.4) whereDis the band cutoff, J is the exchange constant, andρ is the density of states at E_F. Another approximation (equation 1.5) accounts for contributions from both surface and bulk states:

k_BT_K =Dexp[− 1

J_bρ_b+J_sρ_s], (1.5)

whereJ_b is adatom–bulk-state exchange coupling,J_s is adatom–surface-state exchange coupling, andρ_bandρ_sare bulk and surface LDOS atE_F.

BelowTK, the magnetic impurity and its surrounding conduction electrons form an an- tiferromagnetic many-body singlet state (the ‘Kondo screening cloud’), effectively screen- ing the local spin of the magnetic impurity (figure 1.6). This many-body correlation made by electrons near EF screening the magnetic impurity spin in short-lived virtual states produces a perturbation of width w ≈ k_bT_K in the LDOS at E_F (figure 1.5b). This spectroscopic signature of the Kondo effect is called the Kondo resonance. The Kondo resonance has been measured with STS on atoms [20–25, 25, 26] and molecules [27–32]

on various metal surfaces with and without decoupling layers. The Kondo effect can be exploited, for example, in tuning electron transport through quantum dots, where a gate voltage determines the number of electrons on the dot and thus whether an unpaired electron spin causes repulsion from the Kondo screening cloud [33, 34]. Although the

1.2 Kondo effect 7

spectroscopic signature of Kondo physics is measurable with STM, the spatial extent of the Kondo screening cloud is nevertheless inaccessible because the spin-polarization of the cloud is short-lived and oscillatory between spin-up and spin-down.

The lineshape of a Kondo resonance measured in STS dI/dV spectrum can be mod- eled via the Fano resonance describing interference between two tunneling paths: elec- trons tunneling between the tip and the continuous conduction electron band and electrons tunneling between the tip and the discrete level on the atom (equation 1.6).

dI

dV ≈ (q+ϵ)²

1 +ϵ² , whereϵ(V, ϵ₀,Γ) = V −ϵ0

Γ is the normalized energy (1.6) In equation 1.6, Γ is the half width at half maximum (HWHM) of the resonance (Γ = k_BT_K/2 = w/2) and ϵ₀ is an energy shift of the scattering process due to level repul- sion between thed level and the Kondo resonance [35]. q is the interference parameter determined by the ratio between tunneling into a continuum of bulk states versus a dis- creted or f level in the atom. For|q| → ∞, the resonance is created by transmission through the discrete state, and transmission through the continuum state is negligible. For

|q| → 0, transmission through the continuum dominates over transmission through the discrete state (figure 1.7).

Several experimental lines of evidence point to contributions from surface states to the Kondo resonance as well as estimate or give bounds on the Kondo screening cloud spatial extent. Kondo temperature measured for Co as a bulk impurity (T_K ≈ 500 K− 1000 K) is larger than for Co adatoms on surfaces (T_K = 30 K−100 K), likely due to increased hybridization with the bulk electronic system due to greater number of nearest- neighbor atoms [20]. Remote probing of the Kondo effect through the surface state was demonstrated in experiments where the Kondo resonance was projected7 nm across an elliptical quantum corral made from Co atoms on Cu(111) (figure 1.11) [22]. Shown in [29] was that adsorbed TBrPP-Co molecules on reconstructed Si(111)-√

3×√ 3Ag, which has no bulk state contribution nearE_F, exhibit the Kondo resonance with spatial extent> 1 nm from the molecules on the surface. The characteristic Kondo screening lengthζ_Kis approximated byτ_K ≈ℏ/k_bT_Kas the mean spin flip time andv_Fτ_Kthe mean distance quasiparticles travel travel at Fermi velocityv_F before spin-flipping [36], thus ζ_K ≈ _k^ℏvF

BTK. For Co on Ag,v_F = 1.39×10⁸cm s⁻¹ [2] andT_K = 80 K(table 1.2) yields ζ_K ≈132 nm, a crude approximation becausev_F is not well-defined; quasiparticles do not propagate withv_F but with a renormalizedv^∗affected by the magnetic impurity [36]. The Kondo resonance in STS measurements on Co/Au(111), Co/Cu(111), and Co/Cu(100) [9, 22, 35] vanishes over a distance on the order of1 nmfrom the Co atom. Low temperature transport measured through a single-electron GaAs/AlGaAs quantum dot connected to a linear Fabry-Perot cavity showsζ_K ≈ 4µm, close to that expected for a GaAs/AlGaAs 2DEG [4].

The parameterqhas been modeled as [35]

q= Re[G(r)] +t(r)

Im[G(r)] (1.7)

Figure 1.5: Kondo effect in the Anderson impurity model. a: Left: density of states in the metal substrate, with states filled up to Fermi levelEF. Right: density of states in the magnetic Kondo impurity, with half-occupied d or f shell with energy ϵ_f below E_F, FWHM broadening∆, and on-site Coulomb repulsionU which is the energy cost of adding a second electron into the impurityd or f shell. b: Tunneling resulting in spin exchange can occur by doubly occupying the impurity (path 1) or vacating the impurity state (path 2). Energy is conserved but spins atϵ_f and E_F are flipped. Image adapted from [8]

Figure 1.6: (a) At temperatures belowT_K, a conduction electron in the substrate is elas- tically scattered and exchanges spin with the localized spin of the magnetic impurity. (b) The impurity magnetic moment is screened by these spin-flipping events over the Kondo screening lengthζ_K. Figure from [20, 37].

1.2 Kondo effect 9

6 4 2 0 2 4 6

V 0.0

0.2 0.4 0.6 0.8 1.0

Fano pr ofile

q = |1|

q = 0

q = | |

Figure 1.7: Normalized Fano profiles (equation 1.6),Γ = 1for limiting values ofq.

TK(K) ϵ0 (meV) q Reference

92±6 3.1±0.5 0.0±0.1 [38]

83 5.8 0 [25]

56.1±0.9 7.39±0.04 0 [7]

96.3±2.3 7.8±0.2 −0.09±0.01 [39]

89 4.71 −0.05 [6]

Table 1.2:Reported values from Fano resonance fits to STS spectra on Co/Ag(111).

where G(r) is the modified conduction electron Green function as seen by the tip and t(r)is proportional to the matrix element for direct tunneling into the localized state and depends on tip-adatom distanced(r)with decay lengthαas equation 1.8

t(r) = exp[−d(r)

α ] (1.8)

Limot and Berndt found constant Kondo temperature of Co on Ag(111) with spatially varying LDOS near step edges, pointing to a low contribution from surface states [25].

Li et al. modeled the effect of surface state LDOS variations on the Co/Ag(111) Kondo resonance [39], evaluatingJ as a sum of contributions from surface statesJ_Sρ_Sand bulk statesJ_Bρ_B whereρ_S,B is LDOS from surface and bulk, respectively: dI/dV = Bρ_b + Sρ_s(x). They used equation 1.9 to describe variation in surface LDOS as a function of distance from an impurity:

ρ_s(x) =ρ_s0[1 +Acos(2kx+δ₁)/(kx)^α] (1.9) wherex is distance from another Co atom or step edge, B andS are tunneling factors for bulk and surface electrons,ρ_s0 is the surface LDOS on a clean wide terrace,Ais the oscillatory amplitude,δ1 is a phase shift,δ2 is an additional phase shift to accommodate the additional perturbation caused by the Co adatom, andαis a decay constant. They put

equation 1.9 describing LDOS variation due to distance from an impurity into equation 1.5 to obtain equation 1.10 for the Kondo width:

w=Dexp[ 1

Jbρb+Jsρs0[1 +Acos(2kx+δ1 +δ2)/(kx)^α]] (1.10) They then used equation 1.10 with measurements of w as a function of corral radius, distance from other Co atoms, and distance from step edges to extract valuesD, J_b, J_s, andδ, usingρ_b = 0.27 eV⁻¹, J_b = 0.51 eV, J_s = 0.225 eV, andD = 4.48 eVfor Co on Ag(111) reported elsewhere (other values areρb = 0.135 eV⁻¹andρs= 0.0466 eV⁻¹[11]

for Ag(111)). They found agreement in values from fits to data in the corral, atom-atom, and step-edge cases and estimate coupling between the Co atom and bulk states is twice as strong as coupling between the Co atom and surface states (Jb = 0.51 eV,Js = 0.25 eV).

1.3 Atom manipulation

Single atom manipulation with STM was first reported by Eigler and Schweizer in 1990 (figure 1.8). By positioning the tip over an atom of interest, applying bias voltage and current, moving the tip while in constant current feedback, and then retracting the tip, an atom or molecule can be dragged, pushed, or pulled to a desired location on a clean surface. Key to this process is selecting appropriate tip-sample bias and current setpoint to overcome energetic barriers for adatom diffusion. Difficulty of this process depends on substrate-adsorbate coupling strength, system stability, and environmental noise. The mode (drag, push, pull) can be distinguished by the STM current trace. Another atom manipulation mode is vertical manipulation, by which an adsorbate is picked up by the tip and dropped in another location using bias voltage and current setpoints that overcome kinetic barriers for adsorption and desorption of a species from the surface under study.

Stroscio et. al. review atom manipulation with STM with field-assisted diffusion, lateral manipulation, and vertical manipulation processes [40]. Resolution of the lattice of sur- face atoms while imaging a metal surface with surface states (i.e. Ag(111)) is typically difficult due to scattering of tunneling electrons into surface states but atomic resolution imaging is achievable by scanning while an atom is being manipulated with the tip [41].

Figure 1.8: In this world-first demonstration from 1990, Xe atoms were manipulated on Ni(110) at4 K at IBM Research Division in San Jose, California.

Each letter is50Å from top to bottom. The⟨1ˆ10⟩di- rection runs vertically [42]. Properties of substratet- adsorbate systems can be modified by manipulating atoms and molecules with an STM tip at low temper- ature.

Atom manipulation with STM has been used for creation and study of arbitrary band structures, including topological insulators believed to host Majorana fermions [43–45],

1.4 Quantum corrals 11

Figure 1.9: Constant current STM scan of quantum corral (r = 71.3Å), 48 Fe atoms on Cu(111) (4 K, tunnel- ing current 1 nA, bias 10 mV). The l = 1,2,3eigenenergies from the cir- cular hard wall model agree well with experimental energies extracted from

dI

dV spectra peaks. Eigenstate overlap due to level broadening makes LDOS a superposition of eigenstates within a range around tip-sample bias. A com- bination of J₀²(k_5,0ρ), J₂²(k_4,2ρ), and J₇²(k_2,7ρ)eigenmodes agrees well with the topography [10].

artificial graphene [46], hexagonal Kondo lattices [37], kagome lattices [47], Lieb lat- tices [48], fractal lattices [49], and two-dimensional quasicrystals [50]. Atomic scale logic [51–54], memories [52, 55–57], surface state modulations for remote sensing [22], switching [58], and control of adsorbate motion [51,59,60] have been demonstrated in ex- periments using STM atom manipulation. Scientists have also automated scanning probe microscope tip preparation [61], tip functionalization [62], atom manipulation [63, 64], and molecular structure discovery [65].

1.4 Quantum corrals

Crommie, Lutz, and Eigler demonstrated the first quantum corral using atomic manip- ulation with STM to arrange 48 Fe adatoms in a 14 nm diameter ring on Cu(111) in 1993 (figure 1.9) [10]. Quantum corrals, sometimes referred to as ‘artificial atoms’, are thus closed geometries harboring coherent electron eigenstates exhibiting the wave na- ture of electrons accessible via STM imaging of surface states. Quantum well states in confined surface structures have been thoroughly studied with STM and made via atom manipulation techniques [7, 10, 26, 54, 66–69], organic molecules and molecular frame- works [70–72], and vicinal surfaces such as islands and vacancies [16, 73–79]. Electronic surface state resonators have been studied in various substrate-adsorbate systems in lin- ear, triangular, rectangular, circular, elliptical, and hexagonal geometries, alone and in coupled formations. Thorough reviews of surface state engineering using quantum cor- rals built from atom manipulation, supramolecular grids and vicinal surfaces have been conducted [80–83]. Several notable experimental demonstrations have defined the direc- tion of research in the field of quantum corrals engineered via atom manipulation with STM, such as gating of superpositions [68], phase extraction and isospectrality [84], pro- jection of surface state features [54] such as the Kondo resonance [22] (figure 1.11), and sensing of the interaction between quantum corral and scanning probe tips [67].

Theoretical formalisms, ranging in complexity, inform measurements made during ex- periments on quantum corrals. Simple hard-walled particle-in-a-box models can predict

eigenmodes and energies but are unable to predict level broadening. Finite lifetime of surface state electrons due to scattering into bulk bands causes broadening of eigenmodes which increases at higher energies [5]. Particle-in-a-box models also fail to predict stand- ing wave patterns for open geometries [10]. To predict standing surface wave patterns for open geometries, more robust models use many-body scattering [24, 26, 54, 73, 85–88], muffin tin models [46, 48, 64, 81], density-functional theory [89], or tight-binding the- ory [90], particularly in the case of Kondo physics where the spin degree of freedom is involved.

Standing wave patterns due to electron confinement within a circular geometry are approximated with eigenstates of a quantum particle in a circular well of infinitely high potential barrier of radiusr[91]:

ψ_n,l ≈J_l(k_n,l, r)e^ilϕwherek_n,l = zn,l

r ,E_n,l = ℏ²k_n,l²

2m^∗ , (1.11)

wherel ∈Zis the angular momentum quantum number,J_lis thel^thorder Bessel function of the first kind, andz_n,lis then^thzero crossing ofJ_l(z)(figure 1.10). Analytical solutions to eigenstates of a quantum particle confined in an elliptical geometry require use of Mathieu functions [92, 93].

Ag(111) was used in this thesis because its surface state onset energy E₀ is closest to EF out of the common coinage metals hosting surface states (Au, Ag, Cu). This allows to easily tune the LDOS at Fermi by building small corrals with radius on the order of a couplenms. Occupied corral eigenenergies were solved with the finite element method with Matlab’ssolvepdeeig()function on a circular mesh with an additional circular potential of radius0.6 nm, representing the central adatom, and potential energy0.9 eV at the center. Vertical lines in figure 1.10 are target radii (2.5 nm,3.8 nm, and4.5 nm) for construction of circular corrals on Ag(111) with then = 0, l = 0eigenmode above, at, and belowE_F = 0.

1.4 Quantum corrals 13

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75

Eigenenergy (eV)

n=0n=1 n=2

2.0 2.5 3.0 3.5 4.0 4.5 5.0

Circular corral radius (nm) 0.05

0.00 0.05 0.10 0.15 0.20

First eigenmode energy (eV)

E0,r= 2.5nm= 0.12eV

E0,r= 3.8nm= 0.00eV

E0,r= 4.5nm= 0.02eV (a)

(b) Unoccupied corrals

Occupied corrals Radii of interest

Figure 1.10: (a) First three eigenmodes using equation 1.11 for unoccupied circular cor- rals withr ≈ 2 nmto5 nm. (b) Ground state energies of occupied and unoccupied cor- rals. When ‘empty’ corrals are ‘occupied’ by a central adatom, the additional confinement pushes eigenmodes to higher energies. For unoccupied corrals, eigenenergies were plot- ted with equation 1.11 with Ag(111) surface state dispersion (table 1.1). Occupied corral eigenenergies were solved with a numerical PDE solver. Confirmation of these predic- tions is made by comparison with line spectra in figures 6.4 and 6.4 which depict LDOS as a function of location across the diameter of two unoccupied corrals. For ther∼ 3.8 nm corral, the first eigenmode is slightly below E_F; the addition of a central atom to this corral will push the first eigenmode aboveEF. For the unoccupied ∼ 2.5 nm, the first eigenmode is slightly aboveE_F, and additional confinement due to a central Co atom will further shift the eigenmode up in energy.

Figure 1.11: Kondo mirages for EQCs made from Co on Cu(111). (a) Topography of EQC, eccentricity 0.5. (b) Topography of EQC, eccentricity 0.786. There is one Co atom at the focus in each ellipse. (c) and (d) are associated differential conductance maps with the standing wave LDOS subtracted to remove the background electronic contribution and emphasize the Kondo component. IndI/dV spectra of Co on Ag, the Kondo resonance appears as a suppression nearE_F spatially centered at Co atoms and vanishing laterally over∼10Å. Figure from [22].

15

2 Methods

Development of new surface science methods enables synthesis and characterization in increasing detail. These methods have corresponding benefits and drawbacks. Optical methods, e.g. interferometry, microscopy, and spectroscopy measure quickly and reliably for surfaces with tall features and may use phase extraction methods to gather information about thin films without visible step edges [94] but are nevertheless laterally diffraction- limited by source λ ∼ 10 nm−1000 nm. Methods using higher energy particles such as x-rays or accelerated electrons (i.e. ARPES, scanning electron microscopy, transmis- sion electron microscopy) overcome optical diffraction limits to resolve topographic or crystallographic sample details.

Below, we describe scanning tunneling microscopy, a non-contact method by which its inventors Gerd Binnig and Heinrich Rohrer won the Nobel Prize in physics in 1986 and which allows to measure local topographic and electronic properties with lateral resolu- tion free from the diffraction limit. STM can achieve<0.5 nm px⁻¹ resolution to resolve objects as small as individual atoms and their electronic structure. The ability to mea- sure spectral properties makes this surface analysis method crucial to understanding the nano-scale.

2.1 Scanning tunneling microscopy

STM operation involves bringing the tip of a sharp wire to a fewnm(10Å) distance from a conducting sample. Bias voltage applied between tip and sample causes electrons to quantum mechanically tunnel through the separating barrier (figure 2.2) while this tun- neling currentI is measured (figure 2.1) [95, 96]. STM tip wire is typically electrically conductive, non-reactive, thermally stable, resistant to decomposition by pressure, and mechanically strong and rigid W or PtIr of diameter∼250µm. Tip position is controlled atpmprecision with fine and coarse piezoelectric stages which mechanically deform in response to applied voltages and voltage pulses. For constant-current measurements, tun- neling current and tip-sample bias are held constant by a proportional-integral-derivative (PID) feedback system controlling tip-sample distance via aZ-piezo scanner whileXY- piezoelectrics scan the tip across the surface (figure 2.1); tip extension is measured as the tip follows a zigzag trajectory along the surface, providing data correlated to sample topography. Constant-height measurements, which only work well for very flat surfaces, hold tip extension constant and measure tunneling current while the tip rasters the surface.

Both measurement types are sensitive to topography and sample LDOS modulations. For spectroscopy measurements, the tip position is fixed inXY position and bias voltage is ramped between two values withZ position feedback turned off while an additional bias modulation is added and a lock-in amplifier measures dI/dV which is proportional to sample LDOS (equation 2.4) [13, 96].

STM tips may be superconducting or antiferromagnetic for improved energy resolu- tion or spin-selective tunneling current, respectively. STM topography of a sample with known lattice constant and where atomic resolution is easily achievable, e.g. graphene monolayers on hexagonally-oriented pyrolitic graphite, are used to calibrateXY-piezo

Figure 2.1: The STM tip is held with sub- Å stability a few Å from the sample surface by an electronic PID feedback system while the tip rasters in a zigag formation across the surface, measuring tip extension, collecting information correlated to sample topography and LDOS. Figure from [13].

290 20 Scanning Tunneling Microscopy

Fig. 20.5 Energy diagram of tip and sample states for the case of positive sample bias voltageV. Tunneling with energy conservation can only occur within the bias window (blue arrows). Above the bias window the initial states are empty and below the final states are occupied. All energies are referred relative to the sample Fermi level

eV

E Φ ε( )

E_F,sample E_F,tip

Sample Tip

E_i

ε ^f 0

ε-eV

E

tip and sample. When a positive voltage is applied to an electrode, the energy of the states in this electrode is decreased. In the reverse case, a negative voltage at an electrode leads to an upward shift of the energy levels (for electrons it is energetically unfavorable to hop onto a negatively charged electrode). Thus Fig.20.5corresponds to the case in which a positive voltageV is applied to the sample or equivalently a negative voltage is applied to the tip. Furthermore, we use the zero temperature limit in which all levels are filled up to the (tip or sample) Fermi level and are empty above.³This means that tunneling (or scattering from one electrode to another) can only occur in the bias window betweenEF,sampleandEF,tip.

In Bardeen’s approach, the transition rate from one electrode to the other is calcu- lated using the time-dependent perturbation theory assuming weak coupling between the two electrodes. Specifically, a variant of Fermi’s golden rule is applied in order to calculate the transition rate. Since Fermi’s golden rule is often not a part of the intro- ductory courses in quantum mechanics, Bardeen’s variant for tunneling is derived in Appendix B with emphasis on the case of scanning tunneling microscopy.

Applied to the case of tunneling, Fermi’s golden rule shows that scattering from a particular (initial) tip stateiatEtip,ito a particular (final) sample state f atEsample,f

results according to (B.19) at a transition rate (number of electrons per time) of wtip,i→sample,f = 2π

!

!!M_{f i}!!²δ(Esample,f −Etip,i), (20.37)

with the matrix elementMf i calculated according to (B.24) as M_{f i} = !²

2m

"

S_tip/sample

#ψtip,i(r)∇ψ^∗_sample,f(r)−ψ_sample,^∗ _f(r)∇ψtip,i(r)$

·dS.

(20.38)

3This arises because electrons are fermions and only one electron can occupy an electronic state due to the Pauli principle.

Figure 2.2: STM tip sample energy dia- gram, positive sample bias. Electrons tunnel from tip to sample or vice versa over the en- ergy window set by the bias E = eV. Pos- itive bias voltage corresponds to measuring empty energy states (states aboveE_F) in the sample. Negative bias voltage measures oc- cupied energy states (states belowE_F) in the sample. Figure from [96].

actuators by applying corrective coefficients in the feedback control software. To cali- brate theZ-piezo, samples with known step height are measured and coefficients applied to make measurements conform to the expected value.

Tunneling current is written in Bardeen’s tunneling Hamiltonian formalism (equa- tion 2.3) as a convolution of tip density of states with sample density of states.

I = It→s−Is→t

= 4πe ℏ

Z ∞

−∞

ρ_t(E−eV)ρ_s(E)(f(E−eV)−f(E))

:constant

|M(E−eV, E)|² dE (2.1) AssumingM = 1andρ_t(E−eV)is constant, we have:

I ∼ Z ∞

−∞

ρt(E−eV) ρs(E)f(E−eV)dE − Z ∞

−∞

ρt(E−eV)ρs(E)f(E)dE.

Tip LDOS

Sample LDOS

Taking the derivative with respect to applied bias voltage and bringing the derivative

2.2 Scanning tunneling spectroscopy 17

inside the integral, we have:

dI

dV ∼ d dV

Z ∞

−∞

ρ_S(E)f(E−eV)dE (2.2)

= Z ∞

−∞

ρ_S(E) d

dV f(E−eV)dE, wherefis the Fermi-Dirac distributionf(T, E, EF) = ¹

1+exp^E−EF

kbT

andMis the tunneling matrix element specifying tip-sample wavefunction overlap.

By assuming small bias voltage, constant tip DOS, and low temperature, the derivative of the Fermi-Dirac distribution turns into a Dirac delta centered atE−eV, and differential conductance_dV^dI becomes

dI

dV ∼ ρ_t Z ∞

−∞

ρ_s(E)δ(E−eV)dE =

>

ρ_t 1ρ_s(E−eV) (2.3) dI

dVb

∼ ρ_s(eV_b) (2.4)

Assuming constant tip density of states results in tunneling conductance directly propor- tional to sample density of states at the energy given by the bias voltage. Thus via lock-in amplification the STM becomes a tool for resolving spectroscopic features of a sample in addition to topography.

2.2 Scanning tunneling spectroscopy

The standard method of taking STS spectra is via lock-in amplification of the tunneling current while applying a modulation to tip sample bias. This is less noisy than signals acquired by numerical integration of anI-V curve. The way to understand this method is by modeling theI(V) curve as a Taylor series centered around V_b +V_m. The Taylor series forf(x)centered aroundx^′ +δxis

f(x^′+δx) = f(x^′) +f^′(x^′)δx+f^′′(x^′)(δx)²

2! + f^′′′(x^′)(δx)³

3! +... (2.5)

ExpandingI(V)aroundV =V_b +V_msin(ωt), we have equation 2.6:

I(V_b +V_msin(ωt) )≈I(V_b)+dI(V)

dV |_V=VbV_msin(ωt)+1 2

dI(V)²

dV² |_V=VbV_m²sin²(2ωt)+...

(2.6) Bias voltage

Bias modulation

whereV_bis DC bias,V_m is modulation amplitude, andωis modulation frequency.

A lock-in amplifier extracts a signal with a known carrier wave from a source by ex- ploiting orthogonality of sinusoidal signals. The inner product of two sinusoids over a

Figure 2.3: A lock-in amplifier extractsdI/dV from the oscillating current signal. Cur- rent from STM is ‘signal in’ and driving lock-in amplifier frequency is ‘reference in’.

Reference and signal are multiplied and low-pass filtered; output is the amplitude of the STM signal at the reference frequency, giving differential conductance and LDOS of the sample near the tip location. Image from [97].

range longer than both their wavelengths equals zero if the frequencies are different and equals half the sum of their amplitudes if the frequencies are equal. Lock-in amplifiers use this effect by multiplying and integrating a ‘studied’ signal with a pure sinusoidal reference signal. This allows to extract ^dI(V_dV ⁾ from the oscillating I(V_b), which from equation 2.4 is proportional to the local density of states of the sample at the bias voltage ρ(eV_b).

For a sinusoidal reference signal and input waveform U_in(T), the DC output signal U_out(t)is calculated as

U_out = 1 T

Z t

t−T

sin[2πf_refs+ϕ]U_in(s)ds (2.7) whereϕ is the lock-in amplifier phase angle (zero by default) andf_ref is the reference signal frequency. IfT is much larger than the signal period the output is

U_out = 1

2V_sigcosθ (2.8)

whereV_sigis the signal amplitude at the reference frequency andθis the phase difference between signal and reference. The reference signal phase offset is adjusted to maximize the lock-in. Modulation amplitude and bias voltage are chosen based on different prin- ciples and thus can be selected independently. As a rule of thumb,V_m should be< ₁₀₀¹ of the total bias range for good resolution of spectral features. For measurements of the Kondo resonance on Co, we target a bias range of−20 mV−20 mV withV_m ≈ 1 mV.

Typical modulation frequencies are hundreds ofHzand spectrally far from environmental noise sources.

Electrical sources with harmonics of the50 HzEU grid frequency and mechanical res- onances in100s of Hzcouple to the STM, which can be measured in a power spectrum of the tunneling current. Low-pass filters can reduce noise above∼ 1 kHz. Lock-in am-

2.3 Ultra-high vacuum low-temperature STM 19

(a)FER energy diagram. The first differential conductance peak is from the sample work

function. Image from [98].

2 4 6 8 10

Voltage 0.985

0.990 0.995 1.000 1.005 1.010 1.015

Current (nA)

Field emission resonance spectrum

(b)FER spectrum on Ag(111) surface verifying a spectroscopically sharp STM tip. The first peak at the expected work function of silver and lack of

noisy features indicate STM tip DOS is flat.

Figure 2.4: Field emission resonance. The work function of silver is4.26 eV−4.73 eV.

plifier reference frequency is set to a frequency with small amplitude in the system noise spectrum. Tip sharpness and LDOS flatness must be verified before acquiring high con- trast topography maps and spectra. Low quality scans and high quality spectra, or vice versa, can occur with the same tip. To shape the tip, the tip is crashed into a metallic sample, rearranging tip atoms and energy states. Bias pulses may also be applied. Field- emission resonance (FER) measurements can be used to spectroscopically assess the con- stant tip LDOS. In FER measurements (figure 2.4), bias is ramped from1 Vto10 Vwhile piezoZ feedback maintains constant current. Broad _dV^dI(V)peaks signify the tip is well- conditioned and tip DOS is flat. The first _dV^dI(V)peak corresponds to the sample work function. Peaks at higher biases appear due to confined modes in the vacuum barrier between tip and sample.

2.3 Ultra-high vacuum low-temperature STM

STM is carried out in ultra high vacuum (10⁻⁸Torr to 10⁻¹¹Torr) since gas particles between sample and STM tip increase measurement noise; electron tunneling between tip and sample is more coherent in UHV with fewer loss and noise mechanisms. STM does not always occur in UHV but is easier than ambient condition atomic-resolution STM because of need for a conducting surface, which is difficult with atmospheric O₂ which quickly forms insulating oxide films. The vacuum in stainless steel chambers is maintained with cryogenic, ion, turbomolecular and roughing pumps in series. UHV is also necessary for deposition and subsequent imaging of individual atoms since it assures long mean free path for evaporants [13].

We used a Createc LT-STM at ∼ 4.5 K with a ∼10 L outer liquid nitrogen cryostat and an ∼5 L inner liquid helium cryostat with holding time ∼ 72 h. Low temperature improves energy resolution by reducing thermal broadening of states and allows for easy imaging of adsorbates due to reduced thermal diffusion. There are three chambers in the system: a load-lock by which samples are taken in and out of vacuum, a preparation

Figure 2.5: STM topographic im- age of sputtered Ag(111) without subsequent annealing. Atomically flat terraces are too small to manip- ulate a large number of atoms in close proximity. Imaging parame- ters: 1 V,30 pA. The Ag(111) step height is∼ 0.236 nm, compared to the∼9 nmrange inZin the image, thus many Ag(111) steps comprise the topography visible here.

chamber where samples are prepared for characterization, and the main STM chamber which contains the STM and cryostat. The tip is PtIr wire of diameter on the order of 500−1000 mm.

2.4 Substrate preparation

We used a MaTecK GmbH8 mm diameter hat-shaped 99.999% pure Ag single crystal sample grown using the Czochralski method in the (111) direction to a precision of0.1°

polished to < 0.03µmroughness. The sample was spot welded with tantalum foil to a tantalum flag-style sample holder for UHV transfer. We carried out several sample sput- tering and annealing (figure 2.6) cycles in UHV to prepare the sample for atom deposition and manipulation. X-ray photoemission spectroscopy as well as quadrupole mass spec- trometry were briefly used to measure elemental composition of the sample surface and gas composition in the chamber to check if contamination seen in STM scans was due to remnant materials from previous MBE growth experiments in the chamber.

To sputter, we introduced Ne through a precision leak valve into the preparation cham- ber to5×10⁻⁵mTorr, then used a Varian Ion Bombardment Gun (Model 981-2043) con- nected to the chamber to ionize Ne atoms and accelerate them towards the sample with

∼ 1 kV for 15 min. This removes several surface layers from the sample, resulting in uneven topography and small terraces not adequate for atom manipulation and formation of a surface state (figure 2.5).

2.5 Magnetic atom deposition

We sublimated Co atoms from Co wire wrapped around W filament heated with 2.8 A from a DC power supply. The wire was heated near a shutter to the STM which was opened for ∼ 30 s− 60 s with the STM tip lifted away from the Ag(111) crystal at ∼ 5 K. Co atom coverage was checked by measuring surface topography (figure 2.7). Co atoms are imaged as 1Å high protrusions with radius ∼ 0.6 nm. If Co coverage was low, more Co was deposited. Reliably measuring Kondo resonances on Co atoms was occasionally difficult; some spectra on Co did not show a typical Kondo resonance dip in

2.6 Nonmagnetic atom deposition 21

Figure 2.6: Ag(111) sample in heating stage viewed through win- dow into UHV preparation cham- ber. To form large terraces, the Ag(111) was annealed to 450 − 600°C by ∼2.8 A current for for 10-15 min with additional high voltage (1000 V) e-beam heating.

Sample temperature was measured by a stage-mounted thermocouple.

During some annealing cycles the manipulator was cooled with liquid nitrogen to prevent unwanted mate- rial degassing onto the sample.

dI/dV but rather resembled measurements of hydrogenated Ce on Ag(111) (figures 2.8 and 2.9) [99]. In these cases we suspect we measured inelastic tunneling events in a CoH complex formed by residual hydrogen gas that is difficult to remove with turbomolecular or ion pumps. If Co coverage was too high to build Ag atom corrals around a single Co, or10 V pulses applied with the STM tip at contaminated Co atoms did not reliably remove contamination so Kondo spectra could be measured, the Ag(111) was annealed to room temperature in the STM chamber transfer arm, which causes deposited Co atoms to sublimate or mobilize and cluster.

0 50 100 150

nm 0

20 40 60 80 100 120 140

nm

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

Å

Figure 2.7: Ag(111) terrace with de- posited Co atoms (small bright contrast points). Clustering of Co atoms (or other abdsorbates) occurs at step edges.

Residual subsurface neon atoms from the sputtering process are visible as dark contrast circles. Imaging parameters:

1 V,500 pA.

.

2.6 Nonmagnetic atom deposition

We used Ag atoms to form corral walls from nonmagnetic atoms rather than Co atoms which might interact with the central Co atom via magnetic interaction through surface state electrons. Another option was to use CO molecules deposited by leaking CO gas

Figure 2.8: Likely inelastic tunneling events seen in dI/dV spectra on Co atom, before and after a 10 V pulse was applied on top of the atom with the STM tip. In inelastic tunneling processes, the tunneling electron loses energy to (or gains energy from) an additional tunneling channel, which can be a vibrational, translational, or spin degree of freedom. This additional channel causes a step-like feature in the tunneling spectrum [99].

into the STM, but we found these molecules to be highly mobile on Ag(111) which caused difficulties for stable scanning and manipulation.

Various options exist for depositing Ag atoms for manipulation. One method is via sublimation from a heated wire or rod. The following two options use the STM tip as a source for Ag atoms. The first is a method in which the tip is crashed ∼ 4 nm into the sample, creating a10 nmwide depression and scattering Ag atoms across the nearby surface [100]. One drawback to this method is subsequent Co atom deposition demands explicit time-consuming spectroscopic or topographic differentiation between Ag and de- posited Co atoms to construct corrals. It is also quite imprecise in placement of Ag atoms for manipulation.

A second option for atom deposition assumes the STM tip contains a practically infi- nite number of Ag atoms and uses it to deposit single atoms on the surface with gentle tip indentations. We follow a procedure whereby we measure tip extension in constant current mode, indent the tip, and measure tip extension again. The distance we indent the tip into the sample is increased in50 pm steps from6Å until the difference between tip height before and after indentation is > 20 pm [58]. Topography measurement checks whether the crash resulted in deposition of a single atom, dimer, trimer, or larger cluster.

Following this method we create ‘silver atom storage’ (figure 2.10) by depositing arrays of Ag atoms before manipulating them around a Co atom using a template.

2.6 Nonmagnetic atom deposition 23

0 1 2 3 4

100 nm 50 0 50 100

Bi as ( m V )

0.0 2.5 5.0 7.5 10.0 nm

0 2 4

6 0.00

0.05 0.10

nm

(a) (b)

Figure 2.9: (a): Atypical line spectrum over the diameter of quantum corral with r = 2.25 nmmade from 8 Ag atoms with a central Co atom showing inelastic tunneling events, in contrast to the spectrum in figure 3.2 which shows the expected Kondo resonance.

(b): Topography of the quantum corral. The horizontal line is the line over which STS spectra were taken. Spectra over the Co atom did not show the typical asymmetric Kondo resonance centered at low bias nearE ∼4 meV(i.e. figure 3.3) but rather a step indI/dV symmetric aroundE_F [99]. A potential cause of this difference is contamination of the Co atom with hydrogen, which creates an additional relaxation or excitation pathway for electron tunneling. Additional noise between the spectra are likely due to changes in the STM tip or changes in the inelastic tunneling pathway during a spectrum acquisition.

Applying large voltage (i.e.10 V) pulses with the STM on top of the Co atom occasionally reverted the state of the atom so that typical Kondo spectra could be measured, but results were inconsistent with respect to long-term stability of the ‘pure’ Kondo state of the atom.

0 10 20 30 40 50 nm

0 5 10 15 20 25 30

nm

0.0 0.2 0.4 0.6 0.8 1.0

Å

Figure 2.10: Constant current topography of Ag adatoms dropped from the STM tip on Ag(111). Atom topographies (bright contrast) have dark contrast to their right due to high scan speed and constant-current feedback; the DSP overshoots as it passes the atoms at high velocity trying to maintain constant current. The image is forward scan data recorded as the tip goes left to right. In ‘backward scan’ data, the depression is to the left. On the top from left to right we deposited: cluster, single atom, cluster, cluster, single atom, cluster, single atom, single atom. Some tip form operations create dark-contrast features;

three are in this image in the second row, for example. Scanning conditions1 V, 500 pA, scanning speed200 nm s⁻¹.

2.7 Atom manipulation

Manipulation was carried out on Co and Ag atoms on large clean Ag(111) terraces. Pa- rameters for manipulation are∼ 5 mV−20 mV, ∼ 60 nA, and2.5 V−2.8 V, ∼ 60 nA for Ag adatom and Co adatom manipulation, respectively. Manipulation of Co atoms on Ag(111) has previously been studied in detail [101]. Manipulation parameters were hand-adjusted depending on tip condition. In some cases manipulation is possible in one crystallographic direction but not others. To manipulate atoms with the STM we per- formed tip crashes until manipulation was successful. An example current trace recorded during successful manipulation of an atom∼6 nmdistance is shown in figure 2.11.

2.8 Analysis 25

Figure 2.11:Current trace from lateral manipulation. Spikes in current indicate the atom is being manipulated across the surface. During this trace, the manipulation mode shifts at∼5 nm, as evidenced by a change in the pattern of tunneling current spikes. Setpoint current is60 pA.

2.8 Analysis

We fit topography data to find corral radii which allows to accurately model w as a function of corral radius r. Atoms are located by pixels where h > 1.5σ_h where σ_h is standard deviation of topographyh. These are then used as first guesses to Gaussian fitsh(x, y) = Hexp (−(^(x0_w^−x)

x

2 +^y0_w^−y

y

2)/2), where H is Gaussian peak amplitude and (x₀, y₀)is peak center. Deviation between first-guess location and Gaussian-fit location are on average < 0.5Å (see figure 6.2). The Gaussian fit locations are used to extract corral radii.

To fit Fano resonances we used equation 1.6 with an additional linear background (fig- ure 6.2). Fitting parameters depend strongly on initial parameters and the data range for fitting. We adjust fit range manually until good fit is achieved by eye using Python and Scipyoptimize.curve_fit(). The outputq,w,ϵ₀,a,b, andcare then used as first guesses for a second fit using optimize.least_squares(), in which the resid- ual function to be minimized is the sum of squares residual multiplied by |q|, pushing q towards 0 in agreement with previous measurements on Co/Ag(111) (table 1.2). This assumes surface state modulations do not affectq, although previous studies demonstrate decay ofq as a function of distance from Co atoms on Cu(100) and Cu(111) [35]. To check if broadening of the typically sharp tip DOS band edge affects our fit results we con- volve the Fano function with the Fermi-Dirac distributionf(T, V) = 1/(1 + exp (^E−E_k ^F

bT )) whereT is temperature in the STM at the time of the spectrum acquisition. Since T ∼ 5 K≪T_K ∼80 K, neglecting thermal broadening does not greatly alter fit results. In fit- ting, attention was paid to the ratio between fit residuals and Kondo resonance amplitude A. Large fit residual compared toAmay indicate the fit may return a biased value forw.

Code is available at https://github.com/abekipnis/Small-Kondo-Corrals.