Free Electron Fermi Gas, Heat Capacity, and Electrical Conduction

Free Electron Theory: Overview and Context

  • We start from lattice‐vibration discussions; now we examine electrons and how they dominate metallic properties.

  • Central idea: treat valence (bonding) electrons as a gas of non-interacting, free particles subject only to Pauli exclusion.

  • Key simplifications (Free Electron Fermi Gas – FEFG model)- Ionic cores form a uniform, positively charged background (constant potential taken as V=0V=0).

    • Conduction electrons do not interact with each other (electron–electron repulsion neglected).

    • Pauli principle restricts occupancy (2 electrons per spatial orbital owing to spin ms=±12m_s = \pm \tfrac12).

    • Electrons are confined to the crystal volume; they cannot escape (infinite barriers at the surface).

  • Relevance of bonding types- Metals vs semiconductors differ in how impurity addition changes resistivity (↑ for metals, ↓ for semiconductors).

    • Mobile (loosely bound) valence electrons appear when outer-shell electrons are weakly held (e.g.

      Na: 1 s22 s22 p63 s1\text{Na: } 1\,\mathrm s^2 2\,\mathrm s^2 2\,\mathrm p^6 3\,\mathrm s^1).

Classical Conductors vs Quantum Conductors

  • Classical view (Drude, 1900): all electrons drift slowly; current I=V/RI=V/R, resistivity explained via collisions.

  • Quantum view (Fermi gas): only electrons near the Fermi surface contribute; they move at Fermi velocity vFv_F.

  • Both yield same macroscopic σ\sigma, but quantum picture clarifies temperature dependence and mean free path.

One-Dimensional “Particle in a Box”
Schrödinger Equation (time-independent)

  • In 1-D with V(x)=0V(x)=0 inside 0</p><p>0</p><p>\frac{d^2\psin}{dx^2}+\frac{2m\varepsilonn}{\hbar^2}\,\psi_n=0</p></li><li><p>Generalsolution:</p></li><li><p>General solution:\psin=A\sin knxwithwithk_n=\tfrac{n\pi}{L} \;(n=1,2,3,\dots).</p></li></ul><h6id="020b1481−88ac−4ada−9c0e−1481382663e7"data−toc−id="020b1481−88ac−4ada−9c0e−1481382663e7"collapsed="false"seolevelmigrated="true">Energyquantisation</h6><ul><li><p>Standing–wave(boundary)condition.</p></li></ul><h6 id="020b1481-88ac-4ada-9c0e-1481382663e7" data-toc-id="020b1481-88ac-4ada-9c0e-1481382663e7" collapsed="false" seolevelmigrated="true">Energy quantisation</h6><ul><li><p>Standing–wave (boundary) condition\psi(0)=\psi(L)=0→wavelengths\rightarrow wavelengths\lambda_n=\tfrac{2L}{n}.</p></li><li><p>Orbitalenergies:<br></p><p>.</p></li><li><p>Orbital energies:<br></p><p>\varepsilonn=\frac{\hbar^2kn^2}{2m}=\frac{\hbar^2\pi^2n^2}{2mL^2}

  • Each orbital holds two electrons (spin up & down).

Fermi level in 1-D

  • For N(even)electrons:(even) electrons:2nF=N \;\Rightarrow\; nF=\tfrac{N}{2}.</p></li><li><p>Fermienergy:<br></p><p>.</p></li><li><p>Fermi energy:<br></p><p>\varepsilon_F^{1\text D}=\frac{\hbar^2\pi^2}{2mL^2}\Big(\frac{N}{2}\Big)^2

Three-Dimensional Free Electron Gas
Wavefunctions & Boundary Conditions

  • Cubic box of edge L;applyperiodic(Born–vonKarman)orinfinite−wellconditions.</p></li><li><p>Standing−wavesolution:<br></p><p>; apply periodic (Born–von Karman) or infinite-well conditions.</p></li><li><p>Standing-wave solution:<br></p><p>\psi{nxnynz}(\mathbf r)=A\sin\frac{nx\pi x}{L}\,\sin\frac{ny\pi y}{L}\,\sin\frac{nz\pi z}{L}wherewhereki=\tfrac{n_i\pi}{L}(integerscanbepositive/negativeforperiodicBCs).</p></li></ul><h6id="bf7318c3−1ee1−44ce−9137−35a38560c0e3"data−toc−id="bf7318c3−1ee1−44ce−9137−35a38560c0e3"collapsed="false"seolevelmigrated="true">Energyspectrum</h6><p>(integers can be positive/negative for periodic BCs).</p></li></ul><h6 id="bf7318c3-1ee1-44ce-9137-35a38560c0e3" data-toc-id="bf7318c3-1ee1-44ce-9137-35a38560c0e3" collapsed="false" seolevelmigrated="true">Energy spectrum</h6><p>\varepsilonk=\frac{\hbar^2}{2m}(kx^2+ky^2+kz^2)=\frac{\hbar^2k^2}{2m}</p><h6id="58fafeba−bf86−4580−b128−825cb0969bd8"data−toc−id="58fafeba−bf86−4580−b128−825cb0969bd8"collapsed="false"seolevelmigrated="true">Occupationink−space</h6><ul><li><p>Groundstate→allstatesinsideFermisphereofradius</p><h6 id="58fafeba-bf86-4580-b128-825cb0969bd8" data-toc-id="58fafeba-bf86-4580-b128-825cb0969bd8" collapsed="false" seolevelmigrated="true">Occupation in k-space</h6><ul><li><p>Ground state \rightarrow all states inside Fermi sphere of radiusk_Farefilled.</p></li><li><p>Electronconcentrationare filled.</p></li><li><p>Electron concentrationn=N/Vgives<br></p><p>gives<br></p><p>N=2\times \frac{\tfrac43\pi kF^3}{(2\pi/L)^3}=\frac{V}{3\pi^2}kF^3<br></p><p><br></p><p>\Rightarrow\; k_F=(3\pi^2n)^{1/3}</p></li><li><p>Fermienergy(3−D):<br></p><p></p></li><li><p>Fermi energy (3-D):<br></p><p>\varepsilonF^{3\text D}=\frac{\hbar^2kF^2}{2m}=\frac{\hbar^2}{2m}(3\pi^2n)^{2/3}</p></li><li><p>Fermivelocity</p></li><li><p>Fermi velocityvF=\tfrac{\hbar kF}{m},Fermitemperature, Fermi temperatureTF=\varepsilonF/k_B(typically(typically\sim10^4\,\text K).</p></li></ul><divdata−type="horizontalRule"><hr></div><h5id="fab7f796−9fd7−461a−9e45−a094f2cfe7f0"data−toc−id="fab7f796−9fd7−461a−9e45−a094f2cfe7f0"collapsed="false"seolevelmigrated="true">DensityofStates(DoS)</h5><ul><li><p>Definition:).</p></li></ul><div data-type="horizontalRule"><hr></div><h5 id="fab7f796-9fd7-461a-9e45-a094f2cfe7f0" data-toc-id="fab7f796-9fd7-461a-9e45-a094f2cfe7f0" collapsed="false" seolevelmigrated="true">Density of States (DoS)</h5><ul><li><p>Definition:D(\varepsilon)=\dfrac{dN}{d\varepsilon}.</p></li><li><p>Forfree3−Dgas:<br></p><p>.</p></li><li><p>For free 3-D gas:<br></p><p>D(\varepsilon)=\frac{V}{2\pi^2}\Big(\frac{2m}{\hbar^2}\Big)^{3/2}\sqrt{\varepsilon}(perspindirection);multiplyby2fortotal.</p></li><li><p>Totalstatesupto(per spin direction); multiply by 2 for total.</p></li><li><p>Total states up to\varepsilon_FreproducereproduceN.</p></li></ul><divdata−type="horizontalRule"><hr></div><h5id="52daab12−fdc2−4295−9e1a−ea3131961853"data−toc−id="52daab12−fdc2−4295−9e1a−ea3131961853"collapsed="false"seolevelmigrated="true">Fermi–DiracDistribution</h5><ul><li><p>Generalform(finite.</p></li></ul><div data-type="horizontalRule"><hr></div><h5 id="52daab12-fdc2-4295-9e1a-ea3131961853" data-toc-id="52daab12-fdc2-4295-9e1a-ea3131961853" collapsed="false" seolevelmigrated="true">Fermi–Dirac Distribution</h5><ul><li><p>General form (finiteT):<br></p><p>):<br></p><p>f(\varepsilon,T)=\frac{1}{\exp[(\varepsilon-\mu)/k_BT]+1}</p></li><li><p>At</p></li><li><p>AtT=0,,\mu=\varepsilonFandandf(\varepsilon,0)=\begin{cases}1,&\varepsilon<\varepsilonF\0,&\varepsilon>\varepsilon_F\end{cases}</p></li><li><p>Classical(Boltzmann)limitfor</p></li><li><p>Classical (Boltzmann) limit for\varepsilon-\mu\gg kB T::f \approx \exp[-(\varepsilon-\mu)/kBT].</p></li></ul><divdata−type="horizontalRule"><hr></div><h5id="188850e2−9686−4897−a969−1afd522a621d"data−toc−id="188850e2−9686−4897−a969−1afd522a621d"collapsed="false"seolevelmigrated="true">ElectronicHeatCapacity</h5><ul><li><p>Classicalequipartitionwouldpredict.</p></li></ul><div data-type="horizontalRule"><hr></div><h5 id="188850e2-9686-4897-a969-1afd522a621d" data-toc-id="188850e2-9686-4897-a969-1afd522a621d" collapsed="false" seolevelmigrated="true">Electronic Heat Capacity</h5><ul><li><p>Classical equipartition would predictC{el}=\tfrac32NkB(ruledoutexperimentally).</p></li><li><p>Quantumresult(low(ruled out experimentally).</p></li><li><p>Quantum result (lowT \ll TF):):C{el}=\gamma T\,,\quad \gamma=\frac{\pi^2}{2}\,NkB\,/TF<br></p><p>i.e.<br></p><p><br></p><p>i.e.<br></p><p>C{el}=\frac{\pi^2}{2}NkB\,\frac{T}{T_F}</p></li><li><p>Lattice(phonon)contributionatlow</p></li><li><p>Lattice (phonon) contribution at lowT::C_{ph}=AT^3(Debyelaw).</p></li><li><p>Total:(Debye law).</p></li><li><p>Total:C=\gamma T+AT^3.</p></li><li><p>Experiment:.</p></li><li><p>Experiment:\gamma{exp}slightlyexceedsslightly exceeds\gamma{FEFG};ratiogiveseffectivemass; ratio gives effective massm^*/m.</p></li></ul><divdata−type="horizontalRule"><hr></div><h5id="5ff0f163−b036−4c47−94b6−310a7b722210"data−toc−id="5ff0f163−b036−4c47−94b6−310a7b722210"collapsed="false"seolevelmigrated="true">ElectricalConductivity–Drude(Classical)Model</h5><h6id="6e6e1096−d88d−4f74−bb17−61b0864d111d"data−toc−id="6e6e1096−d88d−4f74−bb17−61b0864d111d"collapsed="false"seolevelmigrated="true">Coreassumptions</h6><ul><li><p>Electronsareclassicalfreeparticlesbetweeninstantaneouscollisions.</p></li><li><p>Collision(relaxation)time.</p></li></ul><div data-type="horizontalRule"><hr></div><h5 id="5ff0f163-b036-4c47-94b6-310a7b722210" data-toc-id="5ff0f163-b036-4c47-94b6-310a7b722210" collapsed="false" seolevelmigrated="true">Electrical Conductivity – Drude (Classical) Model</h5><h6 id="6e6e1096-d88d-4f74-bb17-61b0864d111d" data-toc-id="6e6e1096-d88d-4f74-bb17-61b0864d111d" collapsed="false" seolevelmigrated="true">Core assumptions</h6><ul><li><p>Electrons are classical free particles between instantaneous collisions.</p></li><li><p>Collision (relaxation) time\tau:probabilityperunittime: probability per unit time1/\tau.

  • Collisions randomise velocity; only during free flight does electric force act.

Drift & Ohm’s law derivation

  • Equation of motion (with friction):

    m\frac{d\mathbf v}{dt}+\frac{m}{\tau}\mathbf v = -e\mathbf E</p></li><li><p>Steady−statedriftvelocity:</p></li><li><p>Steady-state drift velocity:\mathbf v_d=-\tfrac{e\tau}{m}\mathbf E.</p></li><li><p>Currentdensity:.</p></li><li><p>Current density:\mathbf j=-ne\mathbf v_d=\frac{ne^2\tau}{m}\mathbf E\equiv\sigma\mathbf E.</p></li><li><p>Conductivity:<br></p><p>.</p></li><li><p>Conductivity:<br></p><p>\sigma=\frac{ne^2\tau}{m};resistivity; resistivity\rho=1/\sigma.</p></li></ul><h6id="4593dff4−52fd−4710−b15a−4a8ce86a44b3"data−toc−id="4593dff4−52fd−4710−b15a−4a8ce86a44b3"collapsed="false"seolevelmigrated="true">Temperatureeffects(Matthiessenrule)</h6><ul><li><p>Totalscatteringrate.</p></li></ul><h6 id="4593dff4-52fd-4710-b15a-4a8ce86a44b3" data-toc-id="4593dff4-52fd-4710-b15a-4a8ce86a44b3" collapsed="false" seolevelmigrated="true">Temperature effects (Matthiessen rule)</h6><ul><li><p>Total scattering rate1/\tau = 1/\tau{ph}+1/\taui.</p></li><li><p>Hence.</p></li><li><p>Hence\rho(T)=\rho{ph}(T)+\rhoiwherewhere\rhoiisresidual(impurities)andis residual (impurities) and\rho{ph}\propto T(high−Tlinearrange).</p></li></ul><divdata−type="horizontalRule"><hr></div><h5id="a4965bf0−03d4−4825−8ded−90ac85914ff7"data−toc−id="a4965bf0−03d4−4825−8ded−90ac85914ff7"collapsed="false"seolevelmigrated="true">QuantumViewofConductivity</h5><ul><li><p>ElectricfielddisplacesFermisphereby(high-T linear range).</p></li></ul><div data-type="horizontalRule"><hr></div><h5 id="a4965bf0-03d4-4825-8ded-90ac85914ff7" data-toc-id="a4965bf0-03d4-4825-8ded-90ac85914ff7" collapsed="false" seolevelmigrated="true">Quantum View of Conductivity</h5><ul><li><p>Electric field displaces Fermi sphere by\Delta k = -\tfrac{e\tau}{\hbar}\mathbf E.</p></li><li><p>Onlyelectronsinthinshellnearsurface(thickness.</p></li><li><p>Only electrons in thin shell near surface (thickness\sim \Delta k)carrycurrent.</p></li><li><p>Meanfreepath:) carry current.</p></li><li><p>Mean free path:\ell = v_F\tau;room−temperaturevalues; room-temperature values\sim100\,\text{Å},uptocmatlowTinpuremetals.</p></li><li><p>Perfectperiodiclatticewouldyieldinfinite, up to cm at low T in pure metals.</p></li><li><p>Perfect periodic lattice would yield infinite\tau; real scattering arises from phonons & defects.

Hall Effect

  • Setup: current jxalongalong+x,magneticfield, magnetic fieldBzalongalong+z.</p></li><li><p>Lorentzforcedeflectselectrons.</p></li><li><p>Lorentz force deflects electrons(-e)(\mathbf v\times\mathbf B)→chargeseparation→transverseelectricfield\rightarrow charge separation \rightarrow transverse electric fieldE_y.</p></li><li><p>Steadystate:.</p></li><li><p>Steady state:Ey = vx B_z(inCGS,divideby(in CGS, divide byc).</p></li><li><p>Driftvelocity).</p></li><li><p>Drift velocityvx = -\tfrac{e\tau}{m}Ex→<br></p><p>\rightarrow<br></p><p>Ey = -\frac{e\tau}{m}Ex B_z.</p></li><li><p>Hallcoefficient:<br></p><p>.</p></li><li><p>Hall coefficient:<br></p><p>RH=\frac{Ey}{jx Bz} = -\frac{1}{ne}(simplefree−electronprediction,signshowscarriertype).</p></li><li><p>Measurementof(simple free-electron prediction, sign shows carrier type).</p></li><li><p>Measurement ofRHyieldscarrierconcentrationyields carrier concentrationnand,combinedwithand, combined with\sigma,carriermobility, carrier mobility\mu=e\tau/m=\sigma RH.</p></li></ul><h6id="550e62f5−b704−4aef−87ec−ccdd22ddf937"data−toc−id="550e62f5−b704−4aef−87ec−ccdd22ddf937"collapsed="false"seolevelmigrated="true">Practicalapplications</h6><ul><li><p>Magnetic−fieldsensors(Hallprobes).</p></li><li><p>Identificationofn−orp−typeconductioninsemiconductors.</p></li><li><p>Determiningmobilityanddensityinmaterialsresearch.</p></li></ul><divdata−type="horizontalRule"><hr></div><h5id="39c40ffd−c981−43d2−aa04−0636862b3e66"data−toc−id="39c40ffd−c981−43d2−aa04−0636862b3e66"collapsed="false"seolevelmigrated="true">WorkedExampleHints</h5><ol><li><p><strong>ConductivityfromHalldata(Alat77K)</strong>−Given.</p></li></ul><h6 id="550e62f5-b704-4aef-87ec-ccdd22ddf937" data-toc-id="550e62f5-b704-4aef-87ec-ccdd22ddf937" collapsed="false" seolevelmigrated="true">Practical applications</h6><ul><li><p>Magnetic-field sensors (Hall probes).</p></li><li><p>Identification of n- or p-type conduction in semiconductors.</p></li><li><p>Determining mobility and density in materials research.</p></li></ul><div data-type="horizontalRule"><hr></div><h5 id="39c40ffd-c981-43d2-aa04-0636862b3e66" data-toc-id="39c40ffd-c981-43d2-aa04-0636862b3e66" collapsed="false" seolevelmigrated="true">Worked Example Hints</h5><ol><li><p><strong>Conductivity from Hall data (Al at 77 K)</strong>- GivenRH=-3.9\times10^{-11}\,\text{m}^3/\text C→\rightarrown= -1/(eRH).</p><ul><li><p>Use.</p><ul><li><p>Use\sigma = ne^2\tau/mwithwith\tau=6.5\times10^{-14}\,\text s,,m=m_e.</p></li></ul></li><li><p><strong>Typicalexamderivations</strong>−Derive.</p></li></ul></li><li><p><strong>Typical exam derivations</strong>- Derive\varepsilonnfor1−Dbox,thenfor 1-D box, then\varepsilonF.

    • State Drude assumptions & both forms of Ohm’s law: V=IRandand\mathbf j=\sigma\mathbf E$$.

    • Summarise Hall effect principle & uses.

    • Conceptual & Ethical Connections

    • Free-electron idealisation underpins band theory; refinements (electron–phonon, electron–electron) explain deviations.

    • Technological relevance: heat management (specific heat), resistivity trends for wiring, Hall sensors in cars & smartphones.

    • Philosophical note: simplification (ignoring interactions) enables insight but must be checked against experiment (effective mass, residual resistivity).