Spin-polarized quantum confinement in nanostructures: Scanning tunneling microscopy

Experimental investigations of spin-polarized electron confinement in nanostructures by scanning tunneling microscopy (STM) and scanning tunneling spectroscopy (STS) are reviewed. To appreciate the experimental results on the electronic level, the physical basis of STM is elucidated with special emphasis on the correlation between differential conductance, as measured by STS, and the electron density of states, which is accessible in ab initio theory. Experimental procedures which allow one to extract the electron dispersion relation from energy-dependent and spatially resolved STM and STS studies of electron confinement are reviewed. The role of spin polarization in electron confinement is highlighted by both experimental and theoretical insights, which indicate variation of the spin polarization in sign and magnitude on the nanometer scale. This review provides compelling evidence for the necessity to include spatial-dependent spin-resolved electronic properties for an in-depth understanding and quantitative assessment of electron confinement in magnetic nanostructures and interaction between magnetic adatoms.

Figure 1

Schematic of the tip apex as used in the mathematical description of the tunneling process. The STM tip is assumed to be locally spherical with radius of curvature $R$, where it approaches closest to the sample surface (shaded). The distance of nearest approach is $d$. The center of curvature of the tip is located at ${\overrightarrow{r}}_0$. Note that this schematic is idealized as it does not consider the atomic structure of a tip apex. Adapted from [319].

Figure 2

Simplified picture of spin-polarized tunneling within a hypothetical spin-split density of states model in parallel (P) and antiparallel (AP) magnetization orientations. The spin orientation of the tunneling electrons is assumed to be conserved during tunneling, i.e., spin-up electrons always tunnel into spin-up states and spin-down electrons always tunnel into spin-down states. Arrows (bottom) indicate the DOS of spin-up and spin-down electrons. The spin direction is antiparallel to the magnetic moment ([60]). $M_T$ and $M_S$ (top arrows) denote the magnetization orientation of tip and sample, respectively. (a) P and (b) AP alignment of tip and sample magnetization. Adapted from [352].

Figure 3

Ultrahigh-vacuum system for low-temperature STM in magnetic fields. The STM is cooled by liquid ${\mathrm{N}}_2$ and liquid He in a bath cryostat. This allows for STM measurements at 7 K in fields of up to 8 T. (a) Schematic drawing of the system. Photos of the system can be found in [337]. (b) Head of the STM with tip and sample inserted. (c) STM tip attached on a tip holder. (d) Cu(111) single crystal mounted on a sample holder.

Figure 4

Differential conductance hysteresis loops of Co islands on Cu(111) at 8 K. (a) Constant-current STM image of Co islands on Cu(111) ($V_S=-0.1\mathrm{V}$, $I=1.0\mathrm{nA}$). (b) Differential conductance ($dI/dV$) spectra measured at the center of the Co island shown in (a) (dashed circle A) at different external magnetic fields, $B=0.0\mathrm{T}$ and $+0.6\mathrm{T}$ ($V_{\mathrm{stab}}=+0.5\mathrm{V}$, $I=1.0\mathrm{nA}$). The broken line indicates the bias voltage where the corresponding $dI/dV$ hysteresis loop of (c) was taken. (c) $dI/dV$ hysteresis loops at the center of the Co islands marked by the dashed circles in (a) ($V=-0.58\mathrm{V}$). The colors correspond to upward and downward sweeps of the magnetic field, respectively. The dashed line presents a $dI/dV$ hysteresis loop of the smaller Co island marked by the dashed circle B in (a). The dots correspond to measurement conditions of the $dI/dV$ spectra in (b), the markers identify the measurement conditions of the $dI/dV$ images shown in Fig. 12 corresponding to P and AP states.

Figure 5

Magnetic hysteresis loops measured with different tips. The thin black arrows show the sequence of data acquisition while sweeping the magnetic field. The shorter, thicker arrows represent schematically the magnetization directions of tip and sample, respectively. For loop ③, a fixed magnetization direction is found. However, its direction cannot be determined (up or down), and the two possibilities are shown. Adapted from [273].

Figure 6

Observation of standing waves near a step on Cu(111) by STM and STS. (a) A three-dimensional (3D) view of a constant-current STM image of the step edge of Cu(111). $I=1.0\mathrm{nA}$, $V=+0.02\mathrm{V}$, $28\times 28{\mathrm{nm}}^2$. (b) Differential conductance ($dI/dV$) map of the step edge simultaneously measured with the STM image in (a). (c) Line profile of the $dI/dV$ map along the arrow in (b), showing a spatial oscillation of the $dI/dV$ signal. (d) Dispersion relation of electronic states forming standing waves in $dI/dV$ maps taken at different energies. The data points are extracted by a Fourier transform of the differential conductance map and a Bessel-function fit [Eq. (24)] in (c). The data points are compared to a parabolic fit [Eq. (22)]. (e) $dI/dV$ spectrum measured on Cu(111) in a region of negligible spatial modulation.

Figure 7

Dispersion of the surface state of Cu(111) in the bulk band gap obtained by multiphoton photoemission (mPPE), Fourier transform (FT) STS, and theory along the $\overline{\mathrm{\Gamma }M}$ direction. The solid lines display the calculated dispersion of the surface state and the bulk band edge. The experimental results are compared to the model of a quasi-free-electron-like parabolic dispersion (dashed curve). A deviation from the parabolic dispersion is apparent at higher energies above $+0.1\mathrm{eV}$. From [326].

Figure 8

Electron confinement for a monatomic-high stripe structure on Cu(111). (a) A 3D view of an STM image of the Cu stripe. Scan size: $15\times 15{\mathrm{nm}}^2$. (b) Differential conductance as a function of the energy and the position in the Cu stripe. The dashed lines show the edges of the stripe of (a). Schematics of a “particle-in-a-box” model for (c) the probability density $|\psi {|}^2$ and (d) the wave function $\psi$. The arrows indicate the mode assignment $n_i$ for the different energies shown in (b). (e) Fourier transform map of the wave pattern taken at $-0.3\mathrm{V}$. (f) Dispersion relation of the electronic states, where the wave vector $k$ has been extracted from the Fourier analysis. The continuous line is a parabolic function with $m^{*}=0.40m_e$ and $E_0=-0.43\mathrm{eV}$. Note that only discrete $k$ values are observed, which obey the quantization rule $k_n=n\pi /d$. The inset shows the $k_n$ vs $n$ relation. A linear fit $k=n\pi /d$ indicates $d=8.2\mathrm{nm}$. Adapted from [337].

Figure 9

Modeling a hexagonal vacancy island on Cu(111) for ab initio calculations. (a) STM image of a hexagonal vacancy island on Cu(111). ($V=-0.02\mathrm{V}$, $I=0.5\mathrm{nA}$). (b) Three line scans along the lines in (a). The line scans reveal the monolayer-deep depression (0.21 nm) and two apparent modulations (arrows) due to the impact of the electron standing waves on the topographic image. The depression has a size of $d=7\mathrm{nm}$. (c) A corral of Cu adatoms on Cu(111). The same size $d$ as the depression is used in the calculations to model the vacancy island of (a). From [213].

Figure 10

Comparison of experimental maps of differential conductance of a hexagonal vacancy island with calculated LDOS maps of a corral structure. (Left column) $dI/dV$ images of the hexagonal vacancy island in Fig. 9 at the indicated sample-bias voltage at 1 nA. The solid hexagon indicates the vacancy island rim. (Center column) Calculated LDOS maps of the corral structure in Fig. 9 at various energies. (Right column) Line scans passing through the center of the vacancy island and the corral structure. Dotted line: experiment; solid line: calculation. From [213].

Figure 11

Confinement of surface-state electrons by a 20 nm hexagonal vacancy island. (a) Differential conductance ($dI/dV$) map of the hexagonal vacancy island of 20 nm size. ($V=-0.16\mathrm{V}$, $I=1.0\mathrm{nA}$). (b) Map of the Fourier transform of the $dI/dV$ map in (b). (c) Dispersion relation obtained by the FT analysis of the $dI/dV$ maps. (d) Quantization rule followed by the wave vector $k$. The line is a fit using $k=n\pi /d$ with $d=20.1\mathrm{nm}$. Adapted from [272].

Figure 12

Co islands on Cu(111). (a) Constant-current STM image of a triangular Co island on Cu(111). $V_S=-0.1\mathrm{V}$, $I=1.0\mathrm{nA}$. (b) Line profile along the arrow in the STM image. The Co island is two atomic layers high, $\approx 0.4\mathrm{nm}$, and has a base length of 12 nm. Adapted from [221].

Figure 13

Energy dependences of the measured differential conductance $dI/dV$-asymmetry maps and calculated spin-polarization maps of Co islands. (a) Calculated spin-resolved LDOS of a two atomic-layer Co film on Cu(111). (b)–(e) Experimental $dI/dV$-asymmetry maps measured on the Co island of Fig. 12. The $dI/dV$-asymmetry maps are calculated from two $dI/dV$ images measured at AP and P states from Eq. (20). Measurement conditions of $dI/dV$ images: $B=-1.1\mathrm{T}$, $V_{\text{stab}}=+0.5\mathrm{V}$, and $I=1.0\mathrm{nA}$. (f)–(i) Calculated spin-polarization maps of the triangular Co island. The spatial dependence of the spin polarization as defined by Eq. (28) is shown by the maps, which are calculated from two LDOS maps for the majority and the minority states. Vertical green lines in (a) correspond to the energy positions where the $dI/dV$ asymmetry maps are obtained. A color map in (a) indicates the energy area where the experimental results for the inner part of the Co island show only positive (blue), only negative (red), or both signs (lattice pattern with blue and red) of the $dI/dV$ asymmetry in the $dI/dV$-asymmetry maps. Yellow arrows in (c) and (g) give line profiles in Fig. 14. From [221].

Figure 14

Line profiles along the yellow arrows in Figs. 13 and 13. The plots correspond to the profiles of the $dI/dV$-asymmetry map and the spin-polarization map, respectively.

Figure 15

Energy dependences of the measured differential conductance asymmetry and the calculated spin polarization. Plots of the $dI/dV$ asymmetry averaged over the inner part of the Co island and of the spin polarization of a two atomic-layer Co film on Cu(111).

Figure 16

(a) Sketch of the band structure and the shifts induced by an electric field (EF). The dashed lines show the zero-external-field case. The solid lines show data for a field of $0.6\mathrm{V}/\AA$. Majority and minority bands are plotted with solid and dotted curves, respectively. The shaded area shows a continuum of bulk states. (b) The binding energies ($E_0$) and curvatures as given by the effective masses ($m^{*}$) of majority (circles) and minority (squares) surface bands are plotted. Adapted from [144].

Figure 17

Electron density of states of a bilayer Co island at the Fermi energy $E_F$ in (a) zero external electric field and (c) in an external field of $0.6\mathrm{V}/\AA$ calculated for majority and minority (insets) electrons. The color map is given on the right of the figure. The numbers next to the electron density of states distributions denote maximum values of the majority (red) and minority (blue) electron densities. The minimal values are always zero $1/\mathrm{eV}$ (white). (b), (d) The TMR distribution maps for the above cases, calculated with Eq. (30). The numbers next to the maps denote the boundaries of the color scale. White always corresponds to zero TMR. The Co island has a base length of $\sim 12\mathrm{nm}$, which is the same size as was considered in Fig. 13 ([221]). Adapted from [144].

Figure 18

Calculated minority-spin (black line) and majority-spin (gray line) $s$-like DOSs at $E-E_F=0.012\mathrm{Ry}\approx 0.16\mathrm{eV}$ as a function of the distance from a single Fe impurity on a Cu(111) surface. The DOS was calculated at the first vacuum layer. The distance is expressed in units of the nearest-neighbor distance in Cu(111), $a_{2\mathrm{D}}=2.55\AA$. Inset: $s$-like magnetization density of states (MDOS), which is defined as the difference of the spin-projected $s$-like DOSs. Adapted from [182].

Figure 19

Quantum mirage in a corral. (a) STM image of an elliptical quantum corral with a Co atom at the left focus. $V=8\mathrm{mV}$, $I=1\mathrm{nA}$, $15.4\times 15.4{\mathrm{nm}}^2$, and $T=4\mathrm{K}$. (b) Differential conductance $d/dV$ difference map. This map was obtained via the difference between differential conductance maps of the quantum corral with and without the Co adatom. The difference map shows a projected electronic signature at the empty upper-right focus, originating from the Co adatom at the lower-left focus. This projected feature is termed a “mirage” in a corral. The ellipse with a dashed white line indicates the boundary of the corral. Adapted from [197].

Figure 20

Spin-polarization map of surface-state electrons inside an elliptic Co corral on a Cu(111) surface. One focus of the corral is occupied by a Co adatom (white dashed circle). The mirage in the empty focus is highlighted by the red dashed circle. The geometrical parameters of the corral are the same as in the experimental setup of Fig. 19 ([197]), i.e., semiaxis $a=71.3\AA$ and eccentricity $\epsilon =0.5$. Adapted from [303].

Figure 21

(a) Constant-current STM image of two Co adatoms on Cu(111), which interact via the electronic surface state. ($I=2\mathrm{nA}$, $V=50\mathrm{mV}$, and $T=6\mathrm{K}$.) (b) Experimental and calculated interaction energies between two Co adatoms on Cu(111). From [301].

Figure 22

The exchange interaction between magnetic adatoms inside Cu corrals of different eccentricities. The distance between foci is fixed. The exchange interaction on an open Cu(111) surface is presented by the black line. The exchange interaction in Cu corrals with eccentricities $\epsilon =0.5$ and 0.74 is shown by the gray lines. The inset shows the corresponding models used for the calculations. Negative energies mean that the spins of both adatoms are ferromagnetically coupled, while positive energies correspond to an antiferromagnetic coupling. Adapted from [303].

Figure 23

(a) Hard sphere model of the system used in the calculations of the spin-dependent Smoluchowski effect. Note that all atomic positions were allowed to relax. (b) Plots of the energy dependence of the total density of states (top) and of the spin-resolved density of states of minority (center) and majority (bottom) electrons as calculated at the position of the vacuum spheres identified in (a). From [258].

Figure 24

Calculation of the energy dependence of the spin polarization $P$ at the vacuum spheres across the step edge. The spin polarization $P$ is defined as Eq. (28). The position of the vacuum spheres is presented in Fig. 23. Adapted from [258].

Figure 25

Constant-current image ($V_{\text{gap}}=+0.1\mathrm{V}$, $I_t=1\mathrm{nA}$, $T=10\mathrm{K}$, bulk Cr tip) at the edge of a two-atomic-layer-high Co nanoisland on Cu(111). Inset: line profile along the line indicated in the figure. The highlighted area indicates the extension of the Co island step edge region of the $dI/dV$ asymmetry line profiles in Fig. 26. From [258].

Figure 26

$A_{dI/dV}$ line profiles for different gap voltages applied to the sample of the tunnel junction with reference to the tip. Gap voltage (a) $-0.5\mathrm{V}$, (b) $-0.05\mathrm{V}$, and (c) $+0.5\mathrm{V}$. The line profile is averaged over six adjacent lines next to the arrow in Fig. 25. From [258].

Figure 27

Schematics of dispersion relations and constant-energy contours of [(a), (b)] Rashba spin-split surface states and [(c), (d)] topological surface states. The constant-energy contours [(c), (d)] correspond to a cut at the dashed line in (a) and (b), respectively. The arrows on the constant-energy contours indicate the spin direction at corresponding wave vectors. In (b), scattering between states with the same spin direction (solid arrows) is allowed, but scattering between those with opposite spin (dashed arrow) is prohibited. In (d), backscattering (dashed arrow) never occurs since states at $k$ and $-k$ have opposite spin directions.

Figure 28

FT-STS measurements on ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ alloys. (a) Experimental FT-STS map of ${\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ alloys at the Fermi energy along the $\overline{\mathrm{\Gamma }}$-$\overline{M}$ direction. (b), (c) Simulated FT-STS maps with and without spin-dependent scattering effect, respectively. In (c), two additional high-intensity points (A and B) appear in comparison with (b). They originate from scattering between states with opposite spins. Adapted from [275].

Figure 29

Tunneling current, its magnetization dependence, and TMR spectrum measured on a bilayer Co island on Cu(111) at 8 K. (a) $I(V)$ curves measured at the center of the Co island shown in the inset at different magnetic fields. $V_{\text{stab}}=+0.5\mathrm{V}$, $I_{\text{stab}}=1.0\mathrm{nA}$. The dashed line at $-0.27\mathrm{V}$ gives the bias voltage of the measurement of the hysteresis curve in (b). The inset shows a constant-current STM image of the Co island on Cu(111). $V_S=-0.1\mathrm{V}$, $I=1.0\mathrm{nA}$. (b) Hysteresis loop of the tunneling current measured at the center of the Co island. The gray sections of the hysteresis loop correspond to upward and downward sweeps of the magnetic field. Dots in the loop indicate measurement conditions of the $I(V)$ curves shown in (a). Schematics indicate the magnetization orientations of the Co island and the magnetic tip. (c) TMR spectrum measured at the center of the Co island, which is obtained from the two $I(V)$ curves at $-1.1\mathrm{T}$ in (a) using Eq. (31). From [222].

Figure 30

Spatial modulation of TMR within a single Co island and its energy dependence. (a) TMR spectra as a function of bias voltage $V$ taken at three different positions on the Co island of Fig. 29 shown in the inset. Corresponding positions are indicated by crosses. Each TMR spectrum is obtained from two $I(V)$ curves measured at parallel (P) and antiparallel (AP) magnetization configurations at $-1.1\mathrm{T}$, using Eq. (31). (b)–(e) Maps of TMR ratio obtained at the indicated voltages. ($V_{\text{stab}}=+0.5\mathrm{V}$, $I_{\text{stab}}=1.0\mathrm{nA}$, and $-1.1\mathrm{T}$). The selected voltages are indicated by green dashed lines in (a). (f)–(i) Line profiles, averaged over six adjacent lines for improved signal-to-noise ratio, of the TMR ratio images along the yellow arrows in (b)–(e). Adapted from [222].

Figure 31

Constituent measurements for a TMR spectrum and a TMR map of a Co island. (a) $I$ ($V$) curves measured at the center of the Co island shown in the inset at $-1.1\mathrm{T}$ for antiparallel (AP) and parallel (P) magnetization configurations. $V_{\text{stab}}=+0.5\mathrm{V}$, $I_{\text{stab}}=1.0\mathrm{nA}$. These $I(V)$ curves are the basis for the TMR spectrum shown in Fig. 30. The inset shows a constant-current STM image of a Co island on Cu(111). $V_S=-0.1\mathrm{V}$, $I=1.0\mathrm{nA}$. (b), (c) Tunnel-current images (constant height) of the Co island measured at $V=+0.06\mathrm{V}$ in the AP and P states, respectively. $V_{\mathrm{stab}}=+0.5\mathrm{V}$, $I_{\mathrm{stab}}=1.0\mathrm{nA}$, and $-1.1\mathrm{T}$. Both images show a spatial modulation of the tunneling current. The TMR map presented in Fig. 30 is calculated from these two images using Eq. (31).

Figure 32

Line scan of the TMR ratio measured along the arrow in Fig. 25 at $-0.3\mathrm{V}$. From [258].

Figure 33

Energy dependence of spin-dependent transport properties. (a) Calculated energy-dependent TMR ratio. The inset shows the model of the tip and sample used in our calculations. (b) Spin-resolved transmission coefficient as a function of energy for both P and AP configurations at different bias voltages. The colored areas denote the bias window within which the transmission function is integrated to obtain the current. (c) $s\text{−}p$ projected density of states (PDOS) of a Cr atom at a tip apex and a Co atom under the apex in P and AP configurations at bias voltages of $V=-0.2\mathrm{V}$ (upper panel) and $V=0.0\mathrm{V}$ (lower panel). Up and down arrows indicate spin-up and spin-down channels, respectively. From [222].