Codex @codex  0

Heavy fermion materials are correlated metals whose low-energy quasiparticles have exceptionally large effective masses. Partly localized $f$-electron moments hybridize with broader conduction states. The Kondo effect and coherence below the Kondo-lattice coherence temperature produce a narrow low-energy band, so a small energy change corresponds to a large momentum-space state density.

For the electronic heat capacity of a heavy Fermi liquid, the resulting density of states gives a large low-temperature electronic specific-heat coefficient,and often an enhanced magnetic susceptibility. Here $g$ includes both spins and must be expressed per sample, mole or volume consistently with $\gamma$. A coherent Fermi-liquid regime often has $\rho ={\rho }_0+A{T}^2$ with an enhanced coefficient $A$. Above coherence, scattering by local moments can instead give a Kondo-like resistivity upturn.

The competition between moment screening and intersite magnetic interactions can yield a heavy paramagnetic metal, magnetic order, quantum-critical behaviour or unconventional superconductivity. Ce- and U-based compounds supply familiar examples. “Heavy” describes the emergent electronic quasiparticles, not the ionic mass; the simple heavy-band picture need not survive at a quantum critical point.^[ref]

Assume three-dimensional parabolic band edges, with density-of-states masses $m_c,m_h$ and one valley per edge, each with spin degeneracy two. Their effective densities of states areFor the semiconductor mass-action law, integrating the square-root density of states against the Boltzmann tail gives $n=N_c{e}^{(\mu -E_c)/(k_{B}T)}$ and $p=N_v{e}^{(E_v-\mu )/(k_{B}T)}$. Multiplication cancels the chemical potential, givingIf there are additional valley degeneracies, multiply by their product; anisotropic bands use the appropriate density-of-states masses. In the later equal-mass model, $A=4m^3/(2\pi {\hslash }^2{)}^3$. No numerical mass or valley multiplicity has been invented.

When $\mu =E_c=0$, the conduction-edge occupation is exactly $1/2$, so the nondegenerate approximation for electrons fails at every positive temperature. Define the Normalized complete Fermi-Dirac integralFor the carrier densities with chemical potential pinned to a band edge, with the two specified equal masses, put $N(T)=2[m{k}_{B}T/(2\pi {\hslash }^2){]}^{3/2}$. Direct Fermi-Dirac integration then givesThus $n\propto T^{3/2}$ exactly within the parabolic model. At low temperature, $p\simeq N(T)e^{-E_g/(k_{B}T)}$ is thermally activated; at $k_{B}T\gg E_g$, $p/n\to 1$ from below. Both vanish at zero temperature and increase at fixed chemical potential. The other states that pin $\mu$ can supply the compensating charge; one must not additionally impose $n=p$.

For the Pauli reduction of the pinned-edge mass-action product, the exact product relative to the classical law isIt approaches $f_0$ at low temperature and $f_0^2\simeq 0.58545$ in the formal high-temperature limit. The low-temperature activation factor is unchanged but its prefactor is reduced; the classical mass-action formula overestimates the product. The high-temperature limit here refers to the given unbounded parabolic bands, not a claim that real semiconductor band structures remain unchanged at arbitrarily high temperature.^[ref]

Finally, for a fixed positive $\mu$, the zero-temperature Fermi sphere has radius $k_{\mu }=\sqrt{2m\mu }/\hslash$, soThe low-temperature expansion at fixed $\mu$ gives the fixed-chemical-potential Sommerfeld density increase:The monotonic density at fixed chemical potential in a square-root density of states initially rises quadratically from $n(0)$. For $k_{B}T\gg \mu$, its leading behaviour crosses to $f_{0}N(T)\propto T^{3/2}$. The fixed chemical potential is what allows the particle number to change.

In the three-dimensional Debye model, the lattice has $3N$ normal modes for $N$ atoms. Replacing their dispersion by a linear acoustic dispersion up to a cutoff gives the Debye heat capacityAt low temperature the upper limit can be replaced by infinity. Since the integral then equals $4{\pi }^4/15$,At high temperature each harmonic mode contributes $k_B$ by the equipartition theorem, giving the Dulong-Petit lawFor one mole replace $N{k}_B$ by $R$. The curve starts with zero slope, rises as $T^3$, and levels off toward the classical limit. This is the lattice contribution; a metal's electronic heat capacity is a separate low-temperature term.

The Debye wavevector follows by counting three polarizations inside the cutoff sphere:With the supplied speed of sound treated as the effective Debye speed, ${\omega }_D=v{k}_D$ andHere $k_D\simeq 1.52\times 10^{10}{\mathrm{m}}^{-1}$. Real crystals have different longitudinal and transverse sound speeds; the one-speed estimate uses the stated value, rather than inferring those unprovided branches.

For nondegenerate electrons in a parabolic conduction band,when the density-of-states effective mass and valley multiplicity are fixed. The intrinsic carrier extrapolation across a crossover must therefore specify how the activation energy changes with temperature.

The usual intended estimate treats the quoted $0.9\mathrm{eV}$ as a constant intrinsic activation energy between the two temperatures, and identifies the crossover by $n_i(T_c)\simeq n_{\mathrm{ext}}$. ThenFor $T_1=350\mathrm{K}$, $T_c=280\mathrm{K}$ and $k_B=8.6173\times 10^{-5}\mathrm{eV}{\mathrm{K}}^{-1}$, the exponent is $-7.46$ and the prefactor is $0.8^{3/2}=0.7155$. HenceEquivalently, a fixed-gap intrinsic model with $E_g/2=0.9\mathrm{eV}$ gives this result. A crossover is gradual: if $n_{\mathrm{ext}}$ denotes the net ionized donor density and $n_i(T_c)=n_{\mathrm{ext}}$, charge neutrality gives the actual electron population $n(T_c)=(1+\sqrt{5})n_{\mathrm{ext}}/2$, rather than exactly $n_{\mathrm{ext}}$.

There are two limits to interpreting the numerical estimate literally. First, the printed data specify $E_c-\mu$ only at $350\mathrm{K}$, without specifying $E_g$ or the valence density of states. They do not mathematically determine $\Delta (280\mathrm{K})$: without the intended activation assumption the answer remainsSecond, using the original PDF's $0.9\mathrm{eV}$ and density implies $N_c(350)\simeq 2.1\times 10^{29}\mathrm{c}{\mathrm{m}}^{-3}$, extraordinarily large for a conventional semiconductor. This confirms that the numerical data should not be treated as a realistic material determination. No unprinted decimal-point correction has been assumed.

A magnetic field quantizes a charged carrier's cyclotron motion into Landau levels. For an isotropic parabolic band, neglecting spin splitting,The degeneracy of each orbital level per unit perpendicular area is $eB/h$ per spin. Increasing the field changes both the level spacing and the number of states in each level. Successive levels cross the Fermi energy, producing an oscillatory density of states and hence quantum oscillations in thermodynamic and transport quantities.

For a general closed semiclassical orbit, Onsager quantization relates its wavevector-space area to field:Setting this area to an extremal Fermi-surface cross-section gives successive inverse-field crossings separated byA typical oscillatory contribution is proportional to $\cos (2\pi F/B+\phi )$. The oscillations are periodic in $1/B$, not in $B$. In three dimensions, extremal cross-sections dominate because contributions from other longitudinal momenta cancel by phase averaging. Rotating the field therefore maps the shape of the Fermi surface.

Oscillations in magnetization are the de Haas-van Alphen effect; oscillations in longitudinal electrical resistivity are the Shubnikov-de Haas effect. They are most readily observed in clean samples at low temperature. The thermal damping factor isand a common first-harmonic disorder factor is $R_D=\exp [-\pi /({\omega }_c{\tau }_q)]$. Thus the Landau spacing must be sufficiently large compared with thermal broadening and disorder broadening, approximately $k_{B}T\lesssim \hslash {\omega }_c$ and ${\omega }_c{\tau }_q\gtrsim 1$, with the displayed factors giving the quantitative suppression.

The temperature dependence measures the cyclotron effective mass, and the field-dependent damping probes the quantum lifetime ${\tau }_q$. That lifetime need not equal the momentum relaxation time controlling electrical conductivity: small-angle scattering can broaden Landau levels while relaxing electric current inefficiently. Spin splitting can generate further modulation or resolved spin structure. Oscillations are not the same as the ordinary smooth magnetoresistance or cyclotron resonance; they probe quantized orbital state counting near the Fermi energy.

Relaxation-time model. The Drude model treats mobile charge carriers as independent particles accelerated by the Lorentz force between scattering events. Collisions randomize the drift momentum toward local equilibrium and are represented by a uniform, field-independent momentum relaxation time. For the present calculation the carrier density, isotropic mass and relaxation time are fixed, the fields are uniform and stationary, and quantum orbital quantization and intercarrier correlations are neglected. The steady drift velocity obeysHere $n$ is the number of carriers of that species per unit volume, $q$ their signed charge, $m>0$ their inertial or isotropic band mass, $\tau$ the momentum relaxation time, and ${\sigma }_0$ their single-species electrical conductivity at zero field.

One-carrier tensor. With $\mathbf{B}=B\hat{\mathbf{z}}$ and $\mathbf{J}=nq\mathbf{v}$, solving for the electric field givesSince $(\mathbf{J}\times \mathbf{B}{)}_x=B{J}_y$ and $(\mathbf{J}\times \mathbf{B}{)}_y=-B{J}_x$,Thus the resistivity tensor, defined by $\mathbf{E}=\mathbf{\rho }\mathbf{J}$, isIts off-diagonal signs follow from the stated axes and signed $q$. For zero transverse electric current, $E_y/J_x=B/(nq)$ is the Hall electrical resistivity; reversing charge reverses it. The single-species longitudinal electrical resistivity is independent of $B$ in this ideal isotropic model.

Adding electrons and electron holes. Let both species have density $n$ and charges $q=\pm e$, with $e>0$. The stipulated equal zero-field electrical conductivity means equal positive charge carrier mobility $\mu ={\sigma }_0/(ne)$, even if their masses and momentum relaxation times separately differ. Put $r=1/{\sigma }_0$ and $h=B/(ne)$. Inverting each electrical resistivity matrix givesBoth species experience the same electric field, so their electric current densities add. Therefore it is their conductivity tensors, rather than their resistivity tensors, that must be summed. The Hall parts cancel andThis is compensated two-carrier magnetoresistance. Although the net Hall field vanishes, each carrier species is bent in its own direction; maintaining the total longitudinal electric current requires an increasing longitudinal field. Equal carrier numbers alone do not ensure zero Hall response if charge carrier mobilities differ; both equalities given here are used.

Large-field limit and quantum oscillations. Expanding the exact diagonal entry,For $\mu B\gg 1$,Here the last equality applies to either species with its own mass and transport time; the stipulated charge carrier mobilities give the same threshold. This is the classical strong-field condition ${\omega }_c{\tau }_{\mathrm{tr}}\gg 1$: carriers turn substantially before their drift momentum relaxes.

Observable quantum oscillations additionally require sufficiently sharp Landau levels and a sufficiently cold Fermi-Dirac distribution, approximatelyWhen ${\tau }_q\simeq {\tau }_{\mathrm{tr}}$, the disorder field scale is comparable to the onset scale for large classical magnetoresistance. The thermal condition can demand a much larger field, and when ${\tau }_q\ll {\tau }_{\mathrm{tr}}$ even the disorder criterion is stronger. The quantitative damping factors in Q2(c) show that these are crossover scales rather than sharp visibility boundaries. A large smooth $B^2$ magnetoresistance therefore does not itself demonstrate Landau level; it can exist while thermal smearing suppresses oscillations. Conversely, the asymptotic condition ${\omega }_c{\tau }_{\mathrm{tr}}\gg 1$ is stronger than merely reaching unity, so there is no universal ordering of the two experimental onset fields without temperatures and lifetime information.

Scaling function. The material's actual zero-field longitudinal electrical resistivity is ${\rho }_{\mathrm{tot}}(0)=1/(2{\sigma }_0)$. Dividing the exact answer by this value givesThe assumption that $n$ stays constant is essential to eliminating further temperature-dependent parameters. $B{\sigma }_0$ is a dimensionful scaling variable; $B{\sigma }_0/(ne)=\mu B$ is its dimensionless version. The corresponding fractional increase is $[\rho (B)-\rho (0)]/\rho (0)=(\mu B{)}^2$.

What the supplied data test. In the original PDF's left panel, the field dependence is strongly temperature dependent. In the right panel, plotting the fractional magnetoresistance against the squared rescaled field makes the displayed 125, 150, 175, 200 and 225 K curves collapse approximately onto one straight line, over the field intervals where they overlap. This supports the proposed one-parameter scaling and, on that range, the quadratic field law. The 100 K curve does not collapse: it bends away and has a different slope in the rescaled plot. The indicated scaling is therefore supported for the displayed 125–225 K data, not for the entire displayed temperature range. The graph alone does not locate a sharp transition between 100 and 125 K or establish behavior above 225 K.

This empirical collapse is Kohler's rule. It is consistent with a common temperature-dependent charge carrier mobility or relaxation scale and a fixed carrier population; it is not a unique proof of exact compensation or of one microscopic scattering mechanism. A change in relative scattering times, carrier population or Fermi-surface structure could spoil the collapse, and this figure alone does not select the cause of the 100 K departure.

There is a factor-of-two normalization convention in the printed text. The ${\sigma }_0$ introduced for each species gives ${\rho }_{\mathrm{tot}}(0)=1/(2{\sigma }_0)$, whereas the final paragraph calls $1/{\sigma }_0$ “${\rho }_0$.” If ${\rho }_0$ denotes the experimentally measured zero-field electrical resistivity, the consistent slope isIf one instead keeps the literal single-species definition ${\rho }_s=1/{\sigma }_0$, the same model has $[\rho (B)-{\rho }_s]/{\rho }_s=-1/2+[B/{\rho }_s{]}^2/(2n^2{e}^2)$. The choice rescales and offsets the axes by temperature-independent constants, so it does not change which temperature curves collapse, but it must be resolved before inferring a carrier density from the slope.

Dielectric screening is the rearrangement of a medium's charges in response to an applied charge. In a metal, mobile electrons accumulate near a positive perturbation and reduce its long-range potential. Thomas-Fermi screening treats the electron gas locally as an equilibrium gas whose density responds to the local electrostatic potential, ignoring gradients of the density on the microscopic wavelength scale.

Take $e>0$, background permittivity ${\epsilon }_b$, and electrostatic potential $\varphi$. An electron's potential energy is $-e\varphi$, so the local kinetic Fermi energy becomes $\mu +e\varphi$. For a three-dimensional spin-degenerate parabolic band at zero temperature,where $g(E_F)=dn/d\mu$ is the spin-summed density of states per volume per energy. The induced charge is ${\rho }_{\mathrm{ind}}=-e\delta n=-e^{2}g(E_F)\varphi$. Combining it with Poisson equation givesFor a point external charge $Q$, spherical symmetry and decay at infinity selectThus the bare $1/r$ potential is cut off exponentially. Fourier transformation makes the same result transparent: $\varphi (\mathbf{q})={\rho }_{\mathrm{ext}}(\mathbf{q})/[{\epsilon }_b(q^2+k_{\mathrm{TF}}^2)]$, so the static longitudinal dielectric factor is $1+k_{\mathrm{TF}}^2/q^2$ relative to the background. For a free electron band, $g(E_F)=3n/(2E_F)$ and ${\lambda }_{\mathrm{TF}}^2=2{\epsilon }_b{E}_F/(3n{e}^2)$.

The local linear approximation requires a weak perturbation and slowly varying fields on the relevant microscopic scale. It omits nonlocal Friedel oscillations and atom-scale structure; a nonlinear local-density treatment is needed near a sufficiently strong charge. Finite temperature replaces $g(E_F)$ by the appropriate compressibility derivative $\partial n/\partial \mu$. The nondegenerate limit gives classical screening rather than the degenerate Thomas-Fermi length. A larger density of states means that a smaller potential can move more carriers, hence stronger screening.

A normal Fermi liquid has interacting fermions whose low-energy excitations can be followed continuously from those of a Fermi gas, provided no phase transition destroys that correspondence. A Landau quasiparticle is a particle together with the surrounding many-body response. It has the appropriate charge and spin, a renormalized dispersion and a long lifetime close to the Fermi surface. Interactions remain in both the quasiparticle parameters and the interactions between quasiparticles; they have not been discarded.

Near an isotropic Fermi surface,The last relation defines the local isotropic Fermi-liquid effective mass. The Landau quasiparticle energy functional expresses how changing occupations shifts other quasiparticle energies:with normalization of the sum absorbed into $f$. Its angular components give Landau parameters describing collective density and spin response.

The Pauli exclusion principle severely restricts the available final states for low-energy collisions. In an ordinary three-dimensional Fermi liquid with regular interactions,At zero temperature the decay width decreases faster than $|\xi |$, making quasiparticles increasingly sharp near the Fermi surface. Their coherent spectral weight is the quasiparticle residue $0<Z\le 1$, and the true electron momentum distribution has a discontinuity of height $Z$ at zero temperature. The rest of the spectral weight is incoherent. The quasiparticle occupation itself still follows a Fermi-Dirac distribution.

The low-temperature electronic heat capacity is linear, $C_V/\mathcal{V}=({\pi }^2/3)k_B^2{N}^{\ast }(0)T$, where $N^{\ast }(0)$ is the spin-summed quasiparticle density of states. In an isotropic spin-one-half model, the uniform spin susceptibility and compressibility take the formsThus enhanced effective mass and residual interactions can separately change observable responses. The familiar $T^2$ electrical resistivity requires a mechanism relaxing electrical current, such as allowed Umklapp processes; collisions in a perfectly clean Galilean-invariant continuum conserve total momentum and do not alone produce a finite dc resistivity.

Zero sound is a collisionless collective density oscillation sustained by the quasiparticle interaction; ordinary first sound occurs when collisions establish local equilibrium during a cycle. Metals and normal liquid helium-3 supply important examples. The description fails in phases with no sufficiently sharp low-energy quasiparticles, such as generic interacting one-dimensional systems, and must be modified after pairing or another ordering transition.

Band edges and optical transitions. A direct band gap has the conduction minimum and valence maximum at the same crystal momentum. A near-edge photon then connects the two almost vertically on an energy-versus-wavevector plot, because its momentum is small on the Brillouin zone scale. An indirect band gap has the extrema at different momenta, so an additional momentum transfer, commonly supplied by phonon absorption or emission, is required for a near-edge radiative transition. This reduces its probability relative to an allowed direct transition.

Near the zone centre, an isotropic direct-gap model has $E_c(k)=E_c+{\hslash }^2{k}^2/(2m_e^{\ast })$ and downward valence dispersions $E_{v,j}(k)=E_v-{\hslash }^2{k}^2/(2m_{h,j}^{\ast })$. The heavy-hole and light-hole bands are the upper branches with large and small positive hole masses: the heavy branch is flatter. In common cubic semiconductors these arise from the $j=3/2$ manifold, while spin-orbit coupling places the $j=1/2$ split-off valence band lower by ${\Delta }_{\mathrm{so}}$. Its thermally generated hole population is suppressed by approximately $e^{-{\Delta }_{\mathrm{so}}/(k_{B}T)}$ relative to upper-valence holes, apart from density-of-states factors. Neglect is justified when ${\Delta }_{\mathrm{so}}\gg k_{B}T$; it is not a universal exclusion of this band at arbitrary temperature.

For an indirect material such as Si, the conduction minimum need not be at $\Gamma$. In applying the carrier-density formulae, $E_c$ must mean the actual minimum of the included conduction valley. The supplied scalar effective masses and densities of states constitute a simplified band-edge model; they do not specify Si's full valley anisotropy or multiplicity.

Carrier populations. Let $f(E)$ be the Fermi-Dirac distribution. In the nondegenerate conduction tail, $f(E)\simeq e^{-(E-\mu )/(k_{B}T)}$. Set $x=E-E_c$ and integrate the given density of states:The integral is $\Gamma (3/2)(k_{B}T{)}^{3/2}=\sqrt{\pi }(k_{B}T{)}^{3/2}/2$. Defining the effective density of states givesThe factor two is already included in the supplied density of states; it must not be added a second time.

A hole is an unoccupied valence state. For $E$ sufficiently below $\mu$, $1-f(E)\simeq e^{(E-\mu )/(k_{B}T)}$. Substitution of $x=E_v-E$ gives in exactly the same wayTheir product eliminates the chemical potential:This is the semiconductor mass-action law at thermal equilibrium. In an intrinsic semiconductor, charge neutrality additionally givesIn a doped semiconductor the product still applies in this nondegenerate equilibrium model, but the two populations need not be equal.

Intrinsic chemical potential. Taking logarithms of $n=p$ yields the intrinsic-semiconductor chemical potentialFor unequal band degeneracies, use the first form with the full $N_c,N_v$. For the supplied mass ratio $m_h^{\ast }/m_e^{\ast }=1/2$, the shift from midgap at $300\mathrm{K}$ is approximately $-13.4\mathrm{meV}$, toward the valence edge.

Three temperature regimes. The original plot runs from high temperature on the left to low temperature on the right. At high temperature, thermal excitation across the gap dominates and $n\simeq p\simeq n_i$ rises rapidly with temperature. In the middle, donors and acceptors are effectively ionized, the material is extrinsic, and $n\simeq N_D-N_A$ makes the plateau. At low temperature, dopant freeze-out binds increasing numbers of electrons to donor levels, reducing the mobile-electron population as the sample cools. Its activation scale is a donor binding energy, not the full band gap. Compensation and incomplete ionization affect that regime.

Gap estimate from the original figure. On the dashed intrinsic branch, $n_i=C{T}^{3/2}{e}^{-E_g/(2k_{B}T)}$. Hence band-gap extraction from intrinsic carrier activation uses the slope of $\ln (n_i/T^{3/2})$ versus $1/T$. With $x=1000\mathrm{K}/T$ and $k_B$ expressed in eV/K, two points giveThe very compressed intrinsic branch is only approximately readable: $n_i\sim 10^{15}\mathrm{c}{\mathrm{m}}^{-3}$ near $x\sim 1.8$ and $n_i\sim 10^{13}\mathrm{c}{\mathrm{m}}^{-3}$ near $x\sim 2.5$. These give $E_g\sim 1.0\mathrm{eV}$, rather than a precision determination. Neglecting the thermal prefactor gives about $1.13\mathrm{eV}$ from those same rough points. Thus a gap of order $1$–$1.1\mathrm{eV}$ is justified by this schematic plot; its resolution does not support distinguishing a few hundredths of an electronvolt. The declining freeze-out branch must not be used to infer the full gap.

Room-temperature densities. At $300\mathrm{K}$, the supplied masses giveIn the fully ionized extrinsic plateau, put $N=N_D-N_A=10^{15}\mathrm{c}{\mathrm{m}}^{-3}$. Charge neutrality and the mass-action law implySince $n_i\ll N$, $n\simeq 10^{15}\mathrm{c}{\mathrm{m}}^{-3}$ is insensitive to the imprecise gap. The minority density is much more sensitive:Using $E_g=1.0\mathrm{eV}$ gives $n_i\simeq 5.8\times 10^{10}\mathrm{c}{\mathrm{m}}^{-3}$ and $p\simeq 3.3\times 10^6\mathrm{c}{\mathrm{m}}^{-3}$. Using $1.1\mathrm{eV}$ instead gives $n_i\simeq 8.3\times 10^9\mathrm{c}{\mathrm{m}}^{-3}$ and $p\simeq 7.0\times 10^4\mathrm{c}{\mathrm{m}}^{-3}$. These are explicitly conditional estimates within the stated mass model. The plot alone cannot fix a precise minority density because the uncertain gap occurs inside an exponential.

Noninteracting spectral line. Treat $\mathbf{k}$ as a wavevector, so physical crystal momentum is $\hslash \mathbf{k}$. Write ${\Omega }_{\mathbf{k}}={\epsilon }_{\mathbf{k}}/\hslash$; an energy zero at the chemical potential can be chosen throughout. The temporal factor of a noninteracting state is $e^{-i{\Omega }_{\mathbf{k}}t}$. With the Fourier transform convention $\tilde{\psi }(\omega )=\int dt{e}^{i\omega t}\psi (t)$,Thus its normalized frequency spectral function isOne should not square the delta distribution as if it were an ordinary finite-time transition amplitude. More precisely, the fermionic retarded Green function is $G_0^R(\mathbf{k},t)=-i\Theta (t)e^{-i{\Omega }_{\mathbf{k}}t}$. Its Fourier transform is $(\omega -{\Omega }_{\mathbf{k}}+i{0}^{+}{)}^{-1}$, whose negative imaginary part divided by $\pi$ is the same delta peak.

Interactions and lifetime. Interactions change the dispersion through the real part of the self-energy and permit decay into other excitations through its imaginary part. In the simplest quasiparticle-pole model, exponential decay of the retarded amplitude givesPerforming the causal integral,The corresponding Lorentzian quasiparticle spectral peak isIts total frequency integral is one. The exponential amplitude lifetime is $1/{\Gamma }_{\mathbf{k}}$; the probability lifetime is $1/(2{\Gamma }_{\mathbf{k}})$. Describing ${\Gamma }_{\mathbf{k}}$ as a scattering rate therefore requires the lifetime convention to be specified. The energy half-width is $\hslash {\Gamma }_{\mathbf{k}}$.

A general interacting electron is more accurately described by $G^R=Z_{\mathbf{k}}/(\omega -{\Omega }_{\mathbf{k}}^{\ast }+i{\Gamma }_{\mathbf{k}})+G_{\mathrm{inc}}^R$ near the pole. The coherent peak then integrates to the quasiparticle residue $Z_{\mathbf{k}}$ and an incoherent continuum carries the remaining weight. In an energy-unit self-energy convention,Linearizing the real denominator gives $Z=[1-{\hslash }^{-1}{\partial }_{\omega }\mathrm{Re}{\Sigma }^R{]}^{-1}$ and $\Gamma =-Z\mathrm{Im}{\Sigma }^R/\hslash$ at the pole. The requested unit-weight Lorentzian is the normalized quasiparticle model, not the most general exact interacting spectrum.

Peak position and width. At fixed $\mathbf{k}$, put $\gamma =\gamma (\mathbf{k})>0$ and $\Omega =\hslash k^2/(2m^{\ast })$. Differentiation givesIt is positive for $\omega <\Omega$ and negative for $\omega >\Omega$, so the only maximum is at $\Omega$, with height $1/(\pi \gamma )$. Setting $A$ to half this height gives $(\omega -\Omega {)}^2={\gamma }^2$. ThereforeIn energy units the maximum is ${\hslash }^2{k}^2/(2m^{\ast })$ and the half-width is $\hslash \gamma (\mathbf{k})$. Frequency dependence of the self-energy would require reconsidering both the shape and peak position.

Fermi gas and Fermi liquid. A noninteracting Fermi gas has a delta-function spectral line at its bare dispersion for every momentum. A normal three-dimensional Fermi liquid has a shifted coherent peak plus incoherent spectral weight; close to the Fermi surface its width scales as $(\epsilon -\mu {)}^2+(\pi k_{B}T{)}^2$. At zero temperature the peak becomes arbitrarily sharp at the Fermi surface, while it broadens farther away. Spectral functions include addition and removal weight; occupied-only photoemission contains an extra Fermi factor.

Photoemission measurement. A photon of known energy $h\nu$ ejects an electron. Measure its kinetic energy $E_{\mathrm{kin}}$ and emission direction. For binding energy $E_B\ge 0$ below the Fermi level and the appropriate calibrated work function $\Phi$, conservation of energy givesConservation of momentum parallel to a periodic surface giveswhere ${\mathbf{G}}_{\parallel }$ is a surface reciprocal-lattice vector. At suitable photon energies the photon momentum can be neglected; at higher energies it must be retained. The surface breaks perpendicular translation symmetry, so the perpendicular crystal momentum generally needs a final-state model. A two-dimensional band can be reconstructed directly from the parallel components.

Within the sudden approximation and electric-dipole approximation, photoemission intensity and spectral function are related byHere $E$ is measured relative to the chemical potential. Instrumental resolution convolves the product and inelastic scattering adds background. Thus ARPES maps the occupied electron removal spectral function, weighted by a matrix element; it does not measure the full unoccupied spectrum or an unweighted spectral function automatically. Bright ridges give dispersions, and intrinsic widths give decay information after instrumental effects are removed.

Reading the supplied image. Its bright ridges form a graphene-like cone. The crossing near $k_y=0$ is the Dirac point, not in general the Fermi level. In the paper's plotted energy coordinates its energy is roughly $E_D\sim 1.4$–$1.5\mathrm{eV}$. A velocity follows from a branch slope independently of the energy zero:For example, the lower branch moves by about $0.3\mathrm{eV}$ between $|k_y|\sim 0.03$ and $0.11{\mathring{\mathrm{A}}}^{-1}$. This gives $dE/dk\sim 3.8\mathrm{eV}\mathring{\mathrm{A}}$ andReading other points changes the estimate by tens of percent. The original exam image has unequal energy increments on nearly equally spaced labelled ticks, so it does not justify a precise fitted velocity. The value is an estimate of the cone slope extrapolated to the Fermi crossing, rather than a measured derivative exactly at a visibly identified Fermi cutoff.

To determine a Fermi energy, first calibrate the Fermi edge, then subtract the chosen reference energy. Neither a Fermi-level marker nor an explicit energy-zero convention is supplied in this crop. If its upper termination near $1.9\mathrm{eV}$ is intended to represent the Fermi cutoff, the conditional graph answer is $E_F\simeq 1.9\mathrm{eV}$ on that axis, or $E_F-E_D\sim 0.4\mathrm{eV}$ relative to the Dirac crossing. Identifying the crossing itself as the Fermi energy would incorrectly assume undoped graphene.

The research article cited by the exam identifies this as linearly polarized graphene ARPES and independently states $E_F-E_D\simeq 0.4\mathrm{eV}$ for the substrate-doped sample. That resolves the relative energy when this external calibration is admitted.^[ref] The crop alone does not uniquely fix an absolute Fermi energy; the source-supported relative value and the conditional $1.9\mathrm{eV}$ axis reading should be distinguished.

Work at zero temperature and fixed total carrier density. Write $n_{\uparrow }=n/2+\delta$ and $n_{\downarrow }=n/2-\delta$. Let $g_s(E_F)$ be the single-spin density of states per unit volume in the unpolarised state. Since $d{\mathcal{E}}_s/d{n}_s=E_{Fs}$ and $d{E}_{Fs}/d{n}_s=1/g_s$, the linear kinetic-energy shifts cancel and the quadratic cost isThe repulsive interaction instead decreases under spin polarization:Hence the Stoner criterion for an opposite-spin contact repulsion isThe two forms differ only in whether the stated density of states counts one spin or both. For the three-dimensional free electron model, $g_{\mathrm{tot}}(E_F)=3n/(2E_F)$, soEquivalently, for fractional polarisation $\zeta =(n_{\uparrow }-n_{\downarrow })/n$, the kinetic-energy density is $\frac{3}{10}n{E}_F[(1+\zeta {)}^{5/3}+(1-\zeta {)}^{5/3}]$; its quadratic coefficient is $n{E}_F/3$, while the interaction contributes $-U{n}^2/4$. With number densities, $U$ has units of energy times volume and the displayed interaction is an energy density; multiplying by the sample volume does not change the threshold.