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A positive electric potential difference between gate and substrate attracts Electrons towards the insulating interface and repels the mobile electron holes of the p-type semiconductor substrate. The exposed, negatively charged acceptor impurities form a depletion region; the thin mobile Electron layer at its surface is the semiconductor inversion layer. The conduction band and valence band bend downwards towards the positively biased gate, because the Electron potential energy is $-e\psi$.

To specify the conventional MOS inversion threshold, let $N_A$ be the ionized acceptor density, $n_i$ the intrinsic carrier concentration, andIn the neutral bulk the intrinsic level lies $e{\varphi }_F$ above the semiconductor chemical potential. A positive surface potential ${\psi }_s$ lowers both band edges by $e{\psi }_s$, giving, in the nondegenerate approximation,At ${\psi }_s={\varphi }_F$, the surface is intrinsic: $n_s=p_s=n_i$. The usual onset of strong inversion instead meansThus the surface electron density has grown to the bulk majority-carrier density. The chemical potential need not actually enter the conduction band. The phrase “just forms” can also refer to the earlier weak-inversion crossover; the diagram uses the conventional strong-inversion threshold and labels it explicitly.

Within the depletion approximation the negative space charge is approximately $-e{N}_A$ over a widthwhere ${ϵ}_s$ is the absolute semiconductor permittivity. Its outer edge is where the bands become flat and mobile electron holes recover their bulk density. The electrical insulator blocks steady carrier transport while sustaining an electric field; the metal gate supplies positive surface charge balancing depletion and inversion charge. Under gate bias, the gate and substrate electrochemical potentials are different, so a single equilibrium Fermi level should not be drawn through the whole biased stack.

In a weak applied field, magnetic susceptibility relates magnetization to magnetic field intensity, $M=\chi H$. Its sign, temperature dependence and nonlinear field dependence distinguish several mechanisms.

Diamagnetism is the orbital response opposing an applied magnetic field, with $\chi <0$. Closed-shell atoms and filled bands generally give a weak, nearly temperature-independent response. In a simple free Electron metal, Landau diamagnetism contributes ${\chi }_L=-{\chi }_P/3$; the factor is specific to the isotropic parabolic-band model and is not universal in crystals.

Paramagnetism gives $\chi >0$. Localized magnetic moments align statistically with the field against thermal disorder. For independent moments of angular momentum $J$,which is the Curie law. Exchange interactions often lead approximately to the Curie–Weiss law, $\chi =C/(T-\Theta )$, sufficiently far above any ordering temperature. The sign of $\Theta$ suggests predominantly parallel or antiparallel exchange, though it is not a complete classification. Conduction Electrons instead exhibit Pauli paramagnetism:for $g\simeq 2$ and a spin-summed density of states per unit volume. Only states close to the Fermi energy change occupation, making this response nearly temperature independent when $k_{B}T\ll E_F$.

Ferromagnetism has spontaneous parallel magnetic order below a Curie temperature. Exchange interaction, not simply classical dipole attraction, provides the dominant alignment mechanism. In the convention $H=-J_{\mathrm{ex}}{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$, positive $J_{\mathrm{ex}}$ favours alignment. Magnetic domains can make a ferromagnet's total magnetization vanish before an external field aligns its domains. Domain motion and anisotropy produce saturation, hysteresis, remanent magnetization and a reverse field measured by magnetic coercivity.

Antiferromagnetism has oppositely ordered sublattices below a Néel temperature; equal opposing moments give zero net magnetization while retaining a nonzero staggered order parameter. A susceptibility anomaly and magnetic neutron diffraction reveal this order even when bulk magnetization vanishes. Ferrimagnetism also has opposing sublattices but unequal contributions, giving a nonzero spontaneous net moment. Above their ordering temperatures both can become paramagnetic. Measurements with a magnetometer as a function of field and temperature, supplemented by magnetic diffraction, therefore distinguishes a compensated ordered material from an ordinary paramagnet.

A solid stores thermal energy in lattice vibrations and, when available, electronic and other excitations. For a nonmagnetic insulating crystal at sufficiently low temperature, the gap suppresses mobile electronic excitations and the leading bulk contribution normally comes from acoustic phonons. The three-dimensional Debye heat capacity isThe vibrational density of states scales as ${\omega }^2$, so only a fraction proportional to $T^3$ of the modes is thermally active at low temperature. The two limiting forms areThe high-temperature value is the Dulong-Petit law, or $3R$ per mole of atoms. Optical phonons add thermally activated contributions as their characteristic energies become accessible. A multiatom cell has more branches, but still three vibrational degrees of freedom per atom.

A normal metal has an additional low-temperature electronic heat capacity. The Pauli exclusion principle restricts excitations to a shell of width about $k_{B}T$ around the Fermi energy. The number of thermally active Electrons is therefore proportional to $T$, rather than the total Electron number contributing a classical heat capacity. With spin-summed electronic density of states $g(E_F)$ for the whole specimen,For a three-dimensional free-electron gas, $g(E_F)=3N_e/(2E_F)$ and $C_{\mathrm{el}}=({\pi }^2/2)N_e{k}_B(T/T_F)$. Only a small fraction of the classical electronic value is accessible when $T\ll T_F$, even though the electronic term dominates the phonon term as $T\to 0$ in a normal gapless metal.

Plotting $C/T$ against $T^2$ gives a straight line with intercept $\gamma$ and slope $\beta$ in the appropriate low-temperature range. This separates the electronic density-of-states information from the acoustic-phonon scale. In a Fermi liquid, the interacting quasiparticle density of states replaces the free value, so an enhanced effective mass enhances $\gamma$.

The qualifications matter experimentally. Superconductivity can gap electronic excitations, magnetic order adds magnetic excitations, defects can generate additional low-temperature terms, and low-dimensional or very small specimens need not obey the bulk cubic law. The Debye expression is for constant-volume heat capacity; constant-pressure heat capacity differs through thermal expansion, although the difference is often small at low temperature. Thus the normal-metal and ordinary insulating limits are useful leading behaviors, not universal formulas for every solid in every phase.

A Peierls distortion is a spontaneous lattice modulation that can lower the energy of a one-dimensional metal by opening a gap at its Fermi points. The associated change from metallic to gapped electronic behavior is a Peierls transition. The key wavevector is $2k_F$: it couples states near $+k_F$ and $-k_F$, which otherwise have the same energy.

Consider a spin-degenerate chain with one orbital per atom at spacing $a$ and one Electron per atom. The band is half filled and $k_F=\pi /(2a)$. A distortion of period $2a$ alternates short and long bonds, doubles the primitive cell, and folds the original Fermi points onto the new zone boundary. The distortion can therefore mix the degenerate states precisely where the occupied and empty states meet.

An explicit dimerized tight-binding chain has alternating hopping amplitudes $-(t+\delta )$ and $-(t-\delta )$, where $t>0$. In the reduced zone $|k|\le \pi /(2a)$,There are two Electrons per doubled cell, including spin, so the lower band is filled and the upper one empty. Splitting the degenerate states pushes occupied levels downward and empty ones upward, giving an electronic energy gain.

In weak coupling the electronic gain per length has the characteristic form $-C{\delta }^2\ln (t/|\delta |)$, with $C>0$. The elastic cost of moving the ions is proportional to $+K{\delta }^2$. The logarithm makes the undistorted ideal one-dimensional zero-temperature metal unstable to a small distortion even when the electron-lattice coupling is weak. A self-consistent balance selects a finite distortion amplitude. Two opposite dimerization patterns have the same energy for a translation-symmetric isolated chain.

The basic picture links structural modulation to electronic bond order or charge-density modulation, and can open an insulating gap. It does not assert that every real chain has exactly one Electron per atom or period doubling: away from half filling, a modulation at $2k_F$ need not be commensurate with two lattice spacings. Temperature, finite chain length and fluctuations cut off the ideal logarithmic response. Interchain coupling and the real lattice dynamics determine whether a macroscopic finite-temperature ordered phase actually forms. The strictly one-dimensional zero-temperature instability alone is not a proof of finite-temperature long-range order.

A normal Fermi liquid describes an interacting fermion system whose low-energy excitations remain continuously connected to adding particles to or removing particles from a Fermi gas. The elementary long-lived excitation is a Landau quasiparticle: adding a particle also changes the surrounding many-body state. Its energy, velocity and interactions are renormalized, even though its electric charge and spin remain those appropriate to the excitation.

Near an isotropic Fermi surface,Residual interactions make the quasiparticle energy depend on the occupations of other quasiparticles, expressed by the Landau quasiparticle energy functional. Its angular components give the Landau parameters controlling density and spin responses. Thus the successful particle picture does not mean that interactions have become zero.

Pauli blocking restricts the final states available for low-energy collisions. In an ordinary three-dimensional Fermi liquid with regular interactions, a representative decay law isAt $T=0$, the width vanishes faster than $|\xi |$, so an excitation becomes increasingly sharp as it approaches the Fermi surface. Its coherent spectral weight is the quasiparticle residue $Z$, and the physical Electron occupation has a discontinuity of height $Z$ at zero temperature. Incoherent spectral weight accounts for the remainder. The quasiparticle occupations themselves obey a Fermi-Dirac distribution.

Several leading thermodynamic signatures retain their gas-like form but have interaction-dependent coefficients. In particular,for an isotropic spin-one-half model, with $N^{\ast }(0)$ the spin-summed quasiparticle density of states per volume. effective mass changes $N^{\ast }(0)$, while the dimensionless interaction parameters $F_0^a,F_0^s$ separately change the spin susceptibility and compressibility. Collisionless collective density motion can propagate as zero sound, distinct from the single-particle excitations.

The $T^2$ quasiparticle scattering rate often underlies a low-temperature $T^2$ contribution to metallic electrical resistivity, but a mechanism relaxing electrical electric current is needed. In a clean Galilean-invariant continuum, electron-electron collisions conserve total momentum and do not alone give a finite dc electrical resistivity. Crystal Umklapp scattering or other current-relaxing processes change that conclusion.

Normal liquid helium-3 and many metals motivate the Fermi-liquid framework. Pairing, magnetic ordering or other transitions can reconstruct the low-energy states. Generic interacting one-dimensional fermion systems instead form a different low-energy state, so the three-dimensional quasiparticle conclusions should not simply be transferred to a chain.

Why degenerate perturbation theory is needed. If several unperturbed states have equal or nearly equal energy, a perturbation can mix them at leading order. In that subspace, diagonalize the perturbation matrix rather than using nondegenerate formulas with small energy denominators. The resulting eigenvectors identify the combinations selected by the coupling, and the eigenvalues give the leading splitting. Other remote states can then be included perturbatively.

For the linear combination of atomic orbitals description of molecular hydrogen, equivalent atomic $1s$ states on the two nuclei supply the nearly degenerate starting functions. In an orthonormal two-state model, equal diagonal energy $E_a$ and real hopping $t$ give $E_a\pm t$ and symmetric/antisymmetric combinations. For actual overlapping atomic functions with overlap $S$ and Hamiltonian coupling $h$, one instead solves a generalized eigenvalue problem:The bonding state has enhanced density between the nuclei and the antibonding state a separating node. Two opposite-spin Electrons occupying the bonding molecular orbital describe the simplest closed-shell hydrogen-molecule bond. Electron-electron interaction and nuclear repulsion are needed to calculate a realistic total binding energy; the two-state splitting alone is not the full molecular energy.

In the nearly-free electron model, the analogous degenerate states are plane waves $|\mathbf{k}⟩$ and $|\mathbf{k}-\mathbf{G}⟩$ at a Bragg scattering. A Fourier coefficient $U_{\mathbf{G}}$ of the periodic potential couples them. At exact degeneracy $E_B$, diagonalizing their two-by-two matrix givesThe associated standing waves place electronic density differently relative to the ionic potential. These are the same gap-opening combinations illustrated in Q1(b). The labels of the two starting basis states survive as mixed orbital or plane-wave character, rather than independent crossing eigenstates.

Origin of broad and narrow bands. Spatially extended s/p-derived orbitals have substantial overlap and therefore large hopping amplitudes and broad bands. More localized d orbitals overlap less strongly and produce narrower bands. A multiorbital tight-binding calculation assigns the on-site orbital energies and hopping matrices, forms Bloch sums, and diagonalizes the resulting matrix at each wavevector. Different hopping ranges and orbital symmetry determine the bandwidths and allowed interband coupling. The two-band description retains one broad combination and one nearly flat d-derived combination; a real transition metal contains several d orbitals and more complicated dispersions.

Hybridized eigenvalues and eigenstates. Let $|{\psi }_{\mathbf{k}}⟩=a|{\varphi }_{\mathbf{k}}⟩+b|{\chi }_{\mathbf{k}}⟩$, in an orthonormal basis. Hermitian operator fixes the reverse coupling to be $V^{\ast }$. Projection onto the two components givesSolving this quadratic yields the band hybridization dispersionFor nonzero $V$, one eigenvector representation is $(a,b)\propto (V,E-E_1)$, followed by normalization. Thus the wavefunctions exchange broad-band and d-band character through the avoided crossing; the levels are equal mixtures at the uncoupled crossing.

At $V=0$, the parabola crosses the horizontal level at $k=\pm k_0$, with $k_0=\sqrt{2m{E}_d}/\hslash$. For small nonzero $|V|$, the curves avoid crossing with minimum direct separation $2|V|$. The lower branch follows the parabola below the crossing and flattens toward $E_d$ above it; the upper does the opposite. The lower branch satisfies $E_{-}<E_d$ everywhere and tends to $E_d$ from below as $k\to \infty$, while the upper branch always lies above $E_d$. This is the perfectly flat-band continuum approximation, with zone-boundary intersections excluded as requested.

Fixed particle number and the larger Fermi sphere. Denote the uncoupled broad-band Fermi radius by $k_0$ to distinguish it from the new one $K$. With two spin states, the broad-band number density at $V=0$ is $k_0^3/(3{\pi }^2)$. The stipulated equal population in the flat sector makes the total density twice that value. At zero temperature and any nonzero hybridization, all occupied states fit into the monotonic lower branch below $E_d$, while the upper branch is empty. ThereforeThis is the stated factor increase. It uses fixed total particle number, equal spin counting, an exactly flat unhybridized band and enough available states below any Brillouin-zone restriction. In generic pairs of dispersive bands both hybridized pockets can remain occupied; then Fermi-volume conservation in two hybridized electron bands fixes a sum of volumes rather than this one radius. The exactly flat band is degenerate at $V=0$, so its initial occupation must be specified, as it is here.

Fermi-velocity suppression. Set $\Delta (k)=E_1(k)-E_d$. The exact derivative of the lower branch isAt $k=K$, $E_1(K)=2^{2/3}{E}_d$ and $\Delta (K)=(2^{2/3}-1)E_d>0$. For $|V|\ll \Delta (K)$, the square bracket divided by two is $|V{|}^2/\Delta (K{)}^2+O(|V{|}^4/\Delta (K{)}^4)$. Since group velocity is ${\hslash }^{-1}dE/dk$ and the broad-band reference velocity is $v_F^{(1)}=\hslash k_0/m$,so the flat-band hybridization Fermi-velocity suppression isThe reference velocity is that of the original broad-band Fermi sphere, not the broad-band slope at the larger $K$; omitting the radius change loses the factor $2^{1/3}$.

The low-temperature qualification becomes more stringent as $V\to 0$: the new chemical potential approaches $E_d$ with $E_d-\mu \simeq |V{|}^2/\Delta (K)$, and the relevant thermal energy must be small compared with this hybridization scale. At fixed nonzero temperature the singular flat-band redistribution is not automatically resolved for arbitrarily small coupling. A small but finite narrow-band dispersion can also change the occupation topology.

Density of states and magnetism. For an isotropic three-dimensional branch with spin degeneracy two,A small Fermi velocity therefore produces a large density of states. The localized, partially filled d-derived bands in iron, cobalt and nickel place many states close to the Fermi level; broad s/p-derived conduction bands in copper, silver and gold coexist with filled d bands lying below it. This explains the substantially larger Fermi-level density of states of the former group within the band picture. An exact factor of ten requires the actual band parameters and is not fixed by the two-band toy model.

A high density of states lowers the kinetic-energy cost of spin polarization. A repulsive exchange interaction can then satisfy the Stoner criterion and stabilize ferromagnetism. With a single-spin density of states $g_s$ and an on-site interaction normalization, the criterion is $U{g}_s(E_F)>1$; with spin-summed $g=2g_s$, it is $Ug(E_F)>2$. The large d-state contribution favors this instability in Fe, Co and Ni. The noble metals do not satisfy the corresponding instability condition in the same ordinary state; this does not imply that they have no weak magnetic response whatsoever.

Use a representative conduction-electron density $n\sim 10^{29}{\mathrm{m}}^{-3}$, roughly one mobile Electron per atom in a dense metal, and the bare Electron mass for this estimate. The plasma frequency is thenThis is an order-of-magnitude answer because no particular metal or carrier density is specified. It corresponds to an ordinary frequency $f_p\simeq 2.8\times 10^{15}\mathrm{Hz}$, excitation energy $\hslash {\omega }_p\simeq 12\mathrm{eV}$, and free-space wavelength near $1.1\times 10^{-7}\mathrm{m}$.

The physical oscillation is collective motion of the electron gas against the positive ionic background. Displacing the Electrons relative to the ions produces a restoring electric field. In the simplest uniform-displacement picture, $\ddot{x}+{\omega }_p^{2}x=0$: the frequency increases with the density of mobile electric charge and decreases with carrier inertia. Quantizing this longitudinal collective oscillation gives a plasmon.

The same scale controls the ideal collisionless Drude model optical response. Its cold collisionless plasma dielectric response is ${ϵ}_r(\omega )=1-{\omega }_p^2/{\omega }^2$. Below ${\omega }_p$, the negative relative permittivity makes a transverse electromagnetic wave evanescent, producing strong reflection; above it, the ideal model admits propagation. Real-metal collisions, interband photon absorption and background polarization modify this simple optical edge. Thus ${\omega }_p$ is a collective charge-response scale, not a single-electron scattering rate or a sharp transparency threshold for every real metal.

For state counting in a Bloch band, take a primitive cubic lattice of spacing $a$ and impose periodic boundary conditions on lengths $L_i=N_{i}a$. The allowed wavevectors are $k_i=2\pi l_i/L_i$. The first Brillouin zone has side $2\pi /a$, so it contains $N_i$ distinct values in each direction and hence $N_x{N}_y{N}_z=N$ wavevectors. Each orbital-derived energy band has one state per wavevector per spin orientation. ThereforeThe phrase “per atom” attached to $2N$ in the PDF is a counting error: $2N$ is the total, not the per-atom count.

By the Pauli exclusion principle, a monovalent crystal supplies $N$ valence Electrons and half-fills a spin-degenerate band; its Fermi level lies within allowed states, giving a metal in the independent-electron picture. Two electrons per atom can fill one band. If a band gap separates it from all empty bands, the crystal is a band insulator. An even valence count only permits insulation: overlapping bands can still produce a metal. Conversely, electron interactions can produce a Mott insulator, and symmetry breaking can permit insulation outside this simple spin-degenerate electronic band structure criterion. The relevant count in a crystal with a basis is per primitive unit cell.

In a simple metal, conduction Electrons can move collectively in response to an incident electromagnetic wave. The Drude model gives, with time dependence $e^{-i\omega t}$,In the low-loss range $\gamma \ll \omega <{\omega }_p$, the relative permittivity is approximately $1-{\omega }_p^2/{\omega }^2<0$. The wavevector inside is predominantly imaginary, so the field is evanescent and penetrates only a short distance. Surface currents reradiate the field; little energy enters a good conductor. The normal-incidence Fresnel equations giveMany ordinary metals have their plasma frequency in the ultraviolet, placing visible light below this reflection edge. For $\omega >{\omega }_p$ the free-carrier contribution permits propagation; at $\omega \gg {\omega }_p$, ${ϵ}_r\to 1$ and $R\sim {\omega }_p^4/(16{\omega }^4)\to 0$. The electrons' inertial response becomes too small to screen the rapidly oscillating field.

Ultraviolet transparency is a simple-metal model prediction, not a universal property of real metals. Electronic transitions between bands and bound-electron absorption can remain strong, shift the plasma edge and prevent transparency; transmission also depends on thickness and attenuation. The robust explanation is the change from a reflective negative-permittivity range to a possible propagating range above the plasma edge.

The Wiedemann-Franz law applies to a normal degenerate metal when the same itinerant Electrons carry heat and charge, scattering is approximately elastic over the thermal shell, and the transport probability varies slowly across the energy interval $k_{B}T$ around the Fermi level. The comparison uses the electronic thermal conductivity $K_e$, with any lattice contribution removed.

It works particularly well as $T\to 0$ when elastic impurity or defect scattering dominates. It also often holds approximately for ${\Theta }_D\ll T\ll T_F$, where thermally populated phonons act as a nearly elastic classical scattering field. Room temperature need not lie above the Debye temperature for every metal.

At intermediate temperatures, inelastic small-angle electron-phonon scattering can relax heat much more efficiently than charge. Then ${\tau }_Q\ne {\tau }_{\sigma }$ and $K_e/(\sigma T)$ can be smaller than $L_0$. Strong energy dependence of scattering, additional neutral heat carriers, or a transition out of the normal metallic state also invalidates the simple derivation. An observed deviation alone does not identify which assumption failed.

An unbound conduction Electron has no local restoring force, so set ${\omega }_0=0$ while retaining inertia and momentum relaxation time $\tau =1/\gamma$. This is the Drude model:Separating the response givesFor a good metal, $\gamma \ll {\omega }_p$. The real part is strongly negative at low frequency, crosses zero at $\sqrt{{\omega }_p^2-{\gamma }^2}\simeq {\omega }_p$, and approaches one from below. The imaginary part diverges as ${\omega }_p^2/(\gamma \omega )$ at low frequency, reflecting the finite dc electrical conductivity ${\sigma }_0={ϵ}_0{\omega }_p^2/\gamma$, then decays as ${\omega }^{-3}$. The low-frequency imaginary response describes conduction loss, not a finite-frequency bound-electron absorption line. Core-electron and interband responses can add a background in a real metal.

The residual resistivity is the low-temperature electrical resistivity remaining in a normal metal when thermally excited scattering becomes negligible. Static disorder, impurities and boundaries can relax carrier momentum without requiring a thermal population of phonons. Its value depends on sample purity and structure; a superconducting transition is a different phenomenon.

In the 1982 experiment, separate drives using piezoelectricity scanned a metal tip laterally and adjusted its height. A control unit held electric current fixed at constant tip-sample electric potential difference; calibrated drive voltages supplied the displacement readout. Their Fig. 1 explicitly contrasts a surface step with apparent height from contamination changing the work function.^[ref]

Gerd Binnig and Heinrich Rohrer describe a later vibration isolation system using two spring stages and eddy current damping inside a vacuum chamber, with sputtering and annealing for surface preparation. These measures addressed unwanted motion and surface condition separately.^[ref]

Sintering joins particles into a coherent solid by heating, sometimes with applied pressure, without completely melting the material. It is used to consolidate powders of ceramics and metals.