9.5 Surface states

Types of Surface States

Surface states are electronic states localized at material boundaries where the perfect periodicity of a crystal breaks down. They matter because many device properties (Schottky barrier heights, catalytic activity, contact resistance) are controlled not by the bulk but by what happens at the surface. This topic covers the main types of surface states, how they affect electronic properties, and the experimental tools used to study them.

Tamm States

Tamm states appear at abrupt terminations of a periodic crystal potential. When you chop off the crystal, translational symmetry breaks, and new localized states form in the gap.

• Their wave functions decay exponentially in both directions: into the vacuum and into the bulk
• They're modeled by assuming the surface potential differs sharply from the bulk periodic potential
• Commonly found in ionic crystals and semiconductors like NaCl and GaAs

Because the potential change at the surface is treated as a sharp step, Tamm states are sometimes called the "tight-binding" picture of surface states.

Shockley States

Shockley states are intrinsic surface states that emerge from the bulk band structure itself, even without a sharp potential change. They form when bulk bands of different symmetry invert near a band gap, and the surface boundary condition forces new states into that gap.

• Their wave functions extend deeper into the bulk than Tamm states
• They show up prominently on close-packed metal surfaces: Cu(111), Au(111), Ag(111)
• ARPES measurements of these surfaces reveal a nearly free-electron-like parabolic surface band within the projected bulk gap

The distinction between Tamm and Shockley states is partly historical. Tamm states come from a strong, localized surface perturbation; Shockley states arise from the topology of the bulk bands and require only a weak perturbation.

Topological Surface States

Topological surface states are qualitatively different from Tamm and Shockley states. They're guaranteed to exist by the topology of the bulk band structure and are protected by time-reversal symmetry.

• They exhibit a Dirac cone dispersion: linear energy-momentum relationship
• Spin-momentum locking means the electron's spin direction is tied to its momentum, so backscattering off non-magnetic impurities is forbidden
• This protection makes them robust against disorder, enabling nearly dissipationless surface transport
• Found in topological insulators like $\text{Bi}_2\text{Se}_3$ and Weyl semimetals like $\text{Cd}_3\text{As}_2$

The key point: you cannot remove topological surface states by surface roughness or contamination alone. You'd have to close the bulk band gap or break time-reversal symmetry (e.g., with magnetic doping).

Electronic Properties

Surface Band Structure

The surface band structure describes the $E(\mathbf{k}_\parallel)$ relationship for electrons at the surface, where $\mathbf{k}_\parallel$ is the momentum component parallel to the surface. It differs from the bulk because:

• Broken symmetry along the surface normal lifts degeneracies and creates new bands
• Surface reconstruction changes the periodicity, folding bands into a new surface Brillouin zone
• Surface bands can hybridize with bulk states where they overlap in energy and momentum, forming surface resonances rather than true surface states

ARPES is the primary tool for mapping surface band structure. It directly measures photoelectron kinetic energy and emission angle, which convert to binding energy and $\mathbf{k}_\parallel$.

Density of States

The surface density of states (DOS) counts available electronic states per unit energy at the surface. It often shows sharp peaks or features that don't appear in the bulk DOS, corresponding to surface-localized bands.

• These features influence surface chemical reactivity (the d-band center model in catalysis relies on surface DOS)
• Scanning tunneling spectroscopy (STS) measures the local DOS directly: the differential conductance $dI/dV$ at a given bias voltage is proportional to the local DOS at that energy

Work Function

The work function $\phi$ is the minimum energy needed to remove an electron from the Fermi level to the vacuum level just outside the surface.

• It depends on both the bulk chemical potential and the surface dipole layer
• Different crystal faces of the same material have different work functions (e.g., W(110) vs. W(100))
• Adsorbing electropositive species like Cs lowers $\phi$ by creating a surface dipole pointing into the surface; electronegative adsorbates raise it
• Work function differences drive contact potentials and affect Schottky barrier formation at metal-semiconductor interfaces

Experimental Techniques

Angle-Resolved Photoemission Spectroscopy (ARPES)

ARPES maps the occupied electronic band structure of surfaces by exploiting the photoelectric effect.

1. Photons (typically UV or soft X-ray) eject electrons from the sample
2. An analyzer measures each photoelectron's kinetic energy and emission angles $(\theta, \phi)$
3. Conservation of $\mathbf{k}_\parallel$ lets you reconstruct the in-plane momentum: $k_\parallel = \frac{1}{\hbar}\sqrt{2mE_{\text{kin}}} \sin\theta$
4. Plotting binding energy vs. $\mathbf{k}_\parallel$ gives the band structure directly

Spin-resolved ARPES adds a spin detector (Mott or VLEED type) to measure the spin polarization of surface states, which is essential for studying topological insulators.

Scanning Tunneling Microscopy (STM)

STM provides real-space, atomic-resolution images of surfaces using quantum tunneling.

• A sharp metallic tip is brought within ~1 nm of the surface, and a bias voltage drives a tunneling current
• In constant-current mode, the tip height tracks the surface topography (convolved with the local DOS)
• In spectroscopy mode (STS), sweeping the bias at a fixed position maps out the local DOS vs. energy
• STM can image individual surface states, standing wave patterns from electron scattering, and even manipulate single atoms

Low-Energy Electron Diffraction (LEED)

LEED determines the geometric structure and symmetry of crystalline surfaces.

• Low-energy electrons (20–500 eV) have de Broglie wavelengths comparable to interatomic spacings, making them surface-sensitive (penetration depth of only a few atomic layers)
• Elastically backscattered electrons form a diffraction pattern on a fluorescent screen
• The pattern reveals the surface periodicity, and intensity-voltage (I-V) analysis can determine atomic positions
• LEED is the standard tool for confirming surface reconstructions and checking surface order

Surface Reconstruction

Surfaces often rearrange their atomic structure to lower their total energy. This process is called surface reconstruction.

Mechanisms of Reconstruction

The driving force is the reduction of dangling bonds (unsatisfied bonds left when the crystal is cleaved). Atoms at the surface rearrange to form new bonds, reducing the surface free energy.

• Reconstruction changes the surface symmetry and periodicity relative to the ideal bulk-terminated surface
• The balance between energy gained from new bonds and the elastic strain energy of distortion determines which reconstruction is stable
• Temperature, ambient pressure, and adsorbate coverage can all trigger transitions between different reconstructions

Common Reconstruction Patterns

Reconstructions are described using Wood's notation: if the surface unit cell is $m \times n$ times the bulk-terminated unit cell, you write it as $(m \times n)$. A rotation is noted with R followed by the angle.

• Si(111) 7×7: One of the most complex known reconstructions, involving adatoms, rest atoms, dimers, and stacking faults. It was famously solved by the DAS (dimer-adatom-stacking fault) model
• Au(110) 1×2: A missing-row reconstruction where every other close-packed row is removed
• Si(100) 2×1: Surface Si atoms pair up into dimers to reduce dangling bonds from two per atom to one

Energy Considerations

• Surface stress (force per unit length at the surface) can favor or oppose specific reconstructions
• Temperature-driven phase transitions occur: for example, Si(111) 7×7 transforms to a 1×1 structure above ~1130 K
• Predicting reconstruction patterns from first principles requires density functional theory (DFT) calculations that compare total energies of candidate structures

Adsorption on Surfaces

Adsorption of atoms and molecules onto surfaces underlies catalysis, thin film growth, and sensor operation.

Physisorption vs. Chemisorption

These are two fundamentally different regimes of adsorbate-surface interaction:

Physisorption is non-specific and occurs on virtually any surface. Chemisorption is highly site- and species-dependent, and it often involves charge transfer between the adsorbate and surface.

Binding Sites

On a crystalline surface, adsorbates preferentially occupy specific high-symmetry positions:

• On-top: directly above a surface atom
• Bridge: between two adjacent surface atoms
• Hollow: in the center of three (hcp hollow) or four (fourfold hollow) surface atoms

The preferred site depends on the adsorbate, the surface material, and coverage. STM and DFT are the main tools for identifying binding sites.

Adsorbate-Induced States

When an adsorbate bonds to a surface, its orbitals hybridize with surface states, creating new electronic features:

• Bonding and antibonding combinations form between adsorbate levels and surface bands
• These new states can appear as peaks in the surface DOS, detectable by STS
• The position of antibonding states relative to the Fermi level is a key predictor of bond strength (the Newns-Anderson model captures this physics)

Surface Plasmons

Surface plasmons are collective oscillations of conduction electrons at a metal-dielectric interface. They confine electromagnetic energy to length scales far below the diffraction limit.

Surface Plasmon Polaritons (SPPs)

SPPs are propagating electromagnetic waves bound to a metal-dielectric interface.

• They arise from coupling between photons and the collective electron oscillation
• The SPP dispersion relation lies to the right of the light line, meaning SPPs have greater momentum than free-space photons at the same frequency. This is why you need a prism (Kretschmann configuration) or grating to excite them
• Field intensity decays exponentially away from the interface, with typical decay lengths of ~100 nm into the dielectric and ~10 nm into the metal
• SPPs enable subwavelength waveguiding along metal nanowires and nanostripes

Localized Surface Plasmons (LSPs)

LSPs are non-propagating electron oscillations confined to metallic nanostructures (nanoparticles, nanorods, nanotriangles).

• They produce strong resonant absorption and scattering at frequencies determined by particle size, shape, and dielectric environment
• Gold nanoparticles (~20 nm) appear red in solution because their LSP resonance falls around 520 nm
• The local electric field near the particle can be enhanced by factors of $10^2$–$10^4$, which is the basis for surface-enhanced spectroscopies

Applications in Sensing

• SPR sensors: A thin gold film in the Kretschmann geometry detects refractive index changes near the surface (sensitivity ~$10^{-6}$ RIU). Binding of biomolecules shifts the resonance angle
• LSPR sensors: Shifts in the nanoparticle extinction peak report on local dielectric changes, enabling label-free detection of molecular binding
• SERS (surface-enhanced Raman spectroscopy): Plasmonic hot spots enhance Raman signals by factors up to $10^{10}$, enabling single-molecule detection in some cases

Quantum Well States

When a thin film is sandwiched between barriers (vacuum, a different material, or a band gap), electrons become confined perpendicular to the film. This quantizes their energy in that direction, producing quantum well states.

Confinement Effects

Confinement becomes significant when the film thickness $d$ is comparable to the electron's de Broglie wavelength $\lambda_F$. For typical metals, $\lambda_F \sim 0.5$ nm, so effects appear in films just a few monolayers thick.

• The perpendicular wave vector is quantized: $k_\perp = \frac{n\pi}{d}$, where $n = 1, 2, 3, \ldots$
• This produces a set of discrete sub-bands, each with its own 2D dispersion in $\mathbf{k}_\parallel$
• The energy spacing between levels scales as $\Delta E \propto 1/d^2$, so thinner films have wider level spacing

Quantum Size Effects

Because the quantized energy levels shift as film thickness changes by even a single atomic layer, many physical properties oscillate with thickness:

• Work function oscillates as quantum well states cross the Fermi level
• Superconducting $T_c$ in ultrathin Pb films oscillates with a period of ~2 monolayers
• Film stability itself oscillates: certain "magic" thicknesses are thermodynamically preferred, leading to flat film growth at those thicknesses

These oscillations are a direct, measurable signature of quantum confinement.

Thin Film Properties

• As thickness increases, discrete quantum well states broaden and merge into bulk-like bands
• Quantum well states can hybridize with true surface states at the film boundaries, creating mixed character states
• In magnetic thin films, quantum well states mediate oscillatory interlayer exchange coupling (the mechanism behind GMR in magnetic multilayers)

Surface Magnetism

Magnetic properties at surfaces differ from the bulk because surface atoms have fewer neighbors, which modifies exchange interactions and electronic structure.

Magnetic Anisotropy

Magnetic anisotropy is the energy cost of rotating the magnetization away from a preferred direction. At surfaces, it's often much larger than in the bulk.

• Reduced coordination at the surface breaks cubic symmetry, enhancing the spin-orbit contribution to anisotropy
• This surface anisotropy can force the magnetization perpendicular to the film plane, even when the bulk prefers in-plane alignment
• Co/Pt and Co/Pd multilayers exploit strong interface anisotropy for perpendicular magnetic recording media

Exchange Coupling

Exchange interactions between magnetic moments determine the type of magnetic order (ferromagnetic, antiferromagnetic) and the ordering temperature.

• At surfaces, reduced coordination typically weakens exchange coupling, lowering the local Curie temperature
• At interfaces between ferromagnetic and antiferromagnetic layers, exchange bias shifts the hysteresis loop, pinning the ferromagnet's magnetization direction. This effect is used in spin-valve read heads

Spin-Polarized Surface States

In ferromagnetic materials, exchange splitting separates majority and minority spin bands. Surface states inherit this splitting.

• Spin-resolved ARPES directly measures the spin polarization of surface bands
• High spin polarization at the Fermi level is critical for efficient spin injection in spintronic devices
• Magnetic tunnel junctions (MTJs) rely on spin-polarized tunneling through a thin insulating barrier between two ferromagnetic electrodes. The tunneling magnetoresistance depends on the spin polarization of the interface states

Surface Superconductivity

Superconducting behavior at surfaces and interfaces can differ from the bulk due to modified electron-phonon coupling, reduced dimensionality, and proximity to other materials.

Proximity Effect

When a superconductor (S) is placed in contact with a normal metal (N), Cooper pairs leak into the normal metal over a characteristic length scale called the coherence length $\xi_N$.

• In the normal metal, superconducting correlations decay exponentially with distance from the interface
• This enables Josephson junctions (S-N-S or S-I-S structures) where a supercurrent tunnels through a thin non-superconducting barrier
• SQUIDs (superconducting quantum interference devices) use two Josephson junctions in a loop to detect magnetic flux changes as small as a fraction of $\Phi_0 = h/2e$

Surface Superconducting Gap

The superconducting gap $\Delta$ can be modified at surfaces:

• Reduced coordination can change the electron-phonon coupling strength, altering $\Delta$
• STS measures the gap directly: the $dI/dV$ spectrum shows coherence peaks at $\pm\Delta$ and suppressed DOS inside the gap
• In some materials, surface states themselves can become superconducting with a gap distinct from the bulk value

Vortex States

In type-II superconductors, magnetic flux penetrates as quantized vortices, each carrying one flux quantum $\Phi_0$.

• Near surfaces, vortices can bend, creating "pancake" vortices in layered superconductors
• Surface pinning sites (defects, steps) trap vortices and influence the critical current
• Scanning SQUID microscopy and magnetic force microscopy (MFM) image individual vortices at surfaces, providing direct information about pinning landscapes and vortex dynamics

Technological Applications

Catalysis

Surface electronic structure directly controls catalytic performance. The d-band model (Hammer and Nørskov) predicts that the position of the d-band center relative to the Fermi level determines how strongly molecules adsorb, and therefore how active a catalyst is.

• Surface defects (steps, kinks) often serve as the most active catalytic sites because of their low coordination
• Pt nanoparticles are widely used in fuel cells and automotive catalytic converters, where the high surface-to-volume ratio maximizes the number of active sites
• Alloying and strain engineering tune the surface d-band center to optimize activity for specific reactions

Nanoelectronics

• Topological surface states offer a path toward low-dissipation interconnects and spin-based logic
• Graphene's surface-like 2D electron system provides extremely high carrier mobility (~200,000 $\text{cm}^2/\text{V·s}$) at room temperature
• In nanoscale transistors, surface and interface states create trap states that affect threshold voltage and reliability. Passivation of these states (e.g., with $\text{HfO}_2$ on InGaAs) is a major engineering challenge

Surface Acoustic Wave (SAW) Devices

SAW devices use acoustic waves confined to the surface of a piezoelectric substrate (commonly $\text{LiNbO}_3$ or quartz) for signal processing.

• Interdigitated transducers (IDTs) convert electrical signals to surface acoustic waves and back
• SAW filters are used extensively in mobile phones for frequency selection
• SAW sensors detect mass loading or viscosity changes on the surface, with applications in chemical and biological sensing
• Surface reconstruction and contamination can degrade SAW device performance, making surface preparation critical