6 Free Electron Fermi Gas

Transcription

1 6 Free Electron Feri Gas 6.. Electrons in a etal 6... Electrons in one ato One electron in an ato (a hydrogen-lie ato): the nucleon has charge Z e, where Z is the atoic nuber, and there is one electron oving around this nucleon Four quantu nuber: n, l and lz, sz. Energy levels En with n,,... Z e4 En - Π Ε Ñ n where e N He N L» e where e is the ass of an electron and N is the ass of the nucleon. En - H.6 e L Z (6.) (6.) n For each n, the angular oentu quantu nuber [L Ψ lhl L Ψ] can tae the values of l,,,, 4, n -. hese states are nown as the s, p, d, f, g, states For each l, the quantu nuber for Lz can be any integer between -l and l Hlz -l, -l,, l -, ll. For fixed n, l and lz, the spin quantu nuber sz can be / or -/ (up or down). At each n, there are n quantu states. In a real ato In a hydrogen-lie ato, for the sae n, all the n quantu nuber has the sae energy. In a real ato, the energy level splits according to the total angular oentu quantu nuber l. ypically, the state with lower l has lower energy (not always true). Due to the rotational syetry, states with the sae l has the sae energy (if we ignore the spin-orbital coupling, etc.). n4 n n n n n n l l l l l l l 4s d p 6 s p 6 s s

2 4 Many electrons at Electrons are ferions. he Pauli exclusive principle requires that we can have at ost electron per quantu state. At, to iniize the total energy, the electrons want to state at the lowest Ne quantu state. Ne Eleent No. of electrons on each shell H s He s Li s s 4 Be s s 5 B s s p 6 C Ne s s p s s p6 Na s s p s... Mg s s p s... alence electrons Electrons in the low energy states (inner layers) are bonded tightly to the nucleon. (It costs about - e to reove one of these electrons fro an ato. e is 4 K. It is a very high energy cost in solid state physics). Electrons in high energy states (outer layers) are loosely bonded to the nucleon (easy to reove). hese electrons are called the valence electrons and these energy states are called the valence shells. alence electrons are the electrons in a ato which can participate in the foration of cheical bonds. hey are typically electrons in the outerost (or second-outer-ost) shell Electrons in a etal In a etal, an ato ay lose soe or all of its valence electrons and thus turns into an ion (nucleoninner electrons). hese ions for a crystal and their otions are phonons. he valence electrons are no longer bonded to nucleons. hey can ove freely in a crystal. A crystal A lattice of ions valence electrons (6.) Motions in a crystal phonons valence electrons (6.4) At low teperature, the interactions between phonons are typically very wea. we can consider treat the as a quantu gas (a Bose gas). In a etal, because valence electrons can ove around, we can treat the as a quantu fluid (a ferion fluid). ypically, we call this fluid a Feri liquid. It is called a Feri liquid, instead of a Feri gas, because the interactions between electrons (Coulob interactions) are typically pretty strong (coparable to the inetic energy in ost etals). A solid can be considered as the ixture of two type of fluids: A Bose gas of phonons A Feri liquid of electrons (6.5) 6... Free electrons in D at zero teperature Here, we start fro the siplest situation: a free Feri gas he word free here eans two things.. We ignore interactions between electrons. We ignore interactions between electrons and ions (nucleons) Within this approxiation, electrons are free particles. their Hailtonian is (for a D syste) p H â -ä Ñ Ñ â âx One electron in D he Schrodinger equation is â x (6.6)

3 Ñ â â x Ψn HxL Ε Ψn HxL 5 (6.7) he solutions of this equation are plane waves Ψn HxL A exphä n xl (6.8) he eigen-energy Εn is Ñ n (6.9) Εn For a D syste with length L and periodic boundary conditions, Ψn HxL Ψn Hx LL, we now that the wavevector (oentu) is quantized n L n Π (6.) with n being an integer, n, -, -,,,, Or say, Π n n (6.) L he eigen-energy here is also quantized Ñ n Εn Ñ n Π L (6.) For a large syste L, the energy and oentu becoe continuous variables. Many electrons in D at At, the electrons want to stay in the lowest energy states to iniize the energy. But electrons are ferions, which eans that they cannot all just go to the lowest energy state Ε. Due to the Pauli exclusive principle, one quantu state can host at ost one electron. hese eans that for each n, we can have at ost two electrons (one with spin up and one with spin down). If we have N electrons, at, the electron occupies the lowest N states. he energy of the highest filled state is nown as the Feri energy. he oentu of this state is nown as the Feri oentu PF. he wavevector of this state is nown as the Feri wavevector F. Obviously, PF PF Ñ F and Ñ F (6.) For a large syste (L ),? he nuber of quantu state between -PF to PF is (the factor coes fro the spin degrees of freedo): PF âp ΠÑ L - PF L ΠÑ L PF âp - PF L PF ΠÑ PF ΠÑ (6.4) Nuber of electrons in these states L Ne PF (6.5) ΠÑ Π Ñ Ne PF Π Ñ Ne L (6.6) L Here Ne L is the density of electrons (nuber of electrons per length). PF Π Ñ Ne 8 L he botto line: PF, and F are deterined by the density of electrons. (6.7)

4 Free electrons in D at finite teperature Distribution function he distribution function f HΕL easures the average nuber of electrons on a quantu state with energy Ε. At zero teperature, the nuber of electrons on a quantu state is if Ε < and if Ε >. In other words, this function is a step function f HΕL : Ε < Ε > (6.8) At finite, the electron nuber on a quantu state is NO fixed. eties this state is occupied ( electron) and soeties it is epty ( electron). the average electron nuber is soe nuber between and. < f HΕL < at finite (6.9) Feri distribution: For non-interacting ferions, at finite teperature, the distribution function taes this for f HΕL expj Ε- N (6.) where is nown as the Feri-Dirac distribution. Let s copare it with the Planc distribution (for phonons) we learned in the previous chapter. fphonon HΕL expj Ε N- (6.) We found two differences: () - turns into and () we have an extra paraeter for electrons. he ± is deterined by the nature of the particles. For bosons (photons, phonons, etc.), it is -. For ferions (electrons, protons), it is always. he paraeter is called the cheical potential. It is a very iportant paraeter in therodynaics and statistical physics (it is as iportant as the concept of teperature). It controls the density of particles in a syste. Here, for a Feri gas, we can consider it as a generalization of at finite. In fact, we will show below that when,. How to deterine? he cheical potential can be deterined by counting the total nuber of electrons Ne âp ΠÑ L - f HΕ L L ΠÑ âp - exp (6.) - Divided by L on both sides: Ne L ΠÑ âp - exp (6.) - If we now the density of electrons Ne L, we can solve this equation to find. In general, depends on and the electron density. As,. e properties of f HΕL (a) f HΕ L turns into a step function at. At low, Ε- becoes very large. Depending on the sign of Ε - Ε- Li herefore, : - Ε < Ε > (6.4)

5 Li f HΕL Li expj Ε- N : exph -L exph L 7 Ε< (6.5) Ε> his result recovers the zero teperature liit we studied above. And it is easy to notice that the cheical potential plays the role of here. In other words, at. (b) At any, f H L f HL expj - N exphl (6.6) (c) At < B, f HΕ L is close to a step function. Only the part with Ε~ shows strong deviation fro the step function f HΕ L PlotB: ExpA Ε- ExpA Ε- E E..,.., ExpA Ε- E ExpA Ε- E.,.., ExpA Ε- E. >, 8Ε,, <, PlotStyle hic, AxesLabel 8"Ε ", "fhεl"<, LabelStyle Mediu, PlotLegends :"", " ", " ", " ", " ">F B B B B fhεl B B B. B Ε Free electrons in D at zero teperature For free electrons, the Hailtonian in D is px p y pz p H One electron in D B -ä Ñ -ä Ñ x -ä Ñ y z F- Ñ x y z (6.7) he Schrodinger equation is Ñ - x y z Ψ Hx, y, zl Ε Ψ Hx, y, zl (6.8) he solutions for this equation are D plane waves Ψ Hx, y, zl A expaä Ix x y y z zme (6.9)

6 8 Considering periodic boundary conditions Ψ Hx Lx, y, zl Ψ Hx, y, zl (6.) Ψ Ix L y, y, zm Ψ Hx, y, zl (6.) Ψ Hx Lz, y, zl Ψ Hx, y, zl (6.) We find that the oenta are quantized lπ Π x and y nπ and z Lx Ly (6.) Lz where l,, and n are integers. he eigen-energy is also quantized: Εn Ñ x y z Ñ Ñ 4 Π Lx l 4 Π Ly 4 Π Lz n (6.4) For a very large syste HL L, the discrete energy and oenta turn into continuous variables. Many electrons in D at At, the electrons occupy the lowest N quantu states to save energy. In other words, the quantu states with Ε are occupied, while states with Ε > are epty. Since Ε Ñ, this eans that states with oentu F are occupied and states with > F are epty. Here F Ñ. In other words, in the -space, the occupied states for a sphere with radius F. his sphere is nown as the Feri sphere (or the Feri sea). Q:? HF? L A: It is deterined by the density of electrons. he total volue of the Feri sphere is 4Π the total nuber of quantu states is 4Π F F. Each quantu state occupies the volue H ΠL, which coes fro the uncertainty relation. F H ΠL (6.5) 6 Π here are two electrons per state (because we have electrons with spin up and spin down), so the total nuber of electrons is F N F (6.6) 6 Π Π F Π N (6.7) F is deterined by the electron density N. he Feri energy Ñ F Ñ Π N (6.8) he Feri velocity vf Ñ â Ñ F Here, the forula v group velocity. Ñ F Ñ â Π N (6.9) is nothing but the definition of the group velocity. Notice that Ε Ñ Ω and v â Ω â is precisely the definition of the

7 9 Another equivalent definition of the Feri velocity is pf vf (6.4) If the energy Ε is a quadratic function of HΕ Ñ L, these two definitions are identical. If Ε is NO a quadratic function of (which could happen as will be discussed in the next chapter), we use the first definition vf Ñ â F he botto line: the first definition is ore generic. he second one H pf L only wors for Ε Ñ Conductivity Under a static electric field (E), electrons feel a constant force. (6.4) F -e E he second Law of Newton tell us that âp (6.4) F ât If we have a constant force, P increases linearly as a function of t PHtL PHt L F t (6.4) In quantu echanics, P Ñ, so we have HtL Ht L F Ñ t Ht L - e E (6.44) t Ñ Here, if we turn on the E field at t, at t, the Feri sea is a sphere centered at. At t >, the Feri sphere is centered at HtL - e E t Ñ, because the wavevector of every electron is shifted by this aount at tie t. For free electrons in a perfect crystal, HtL as t. However, in a reals solid, HtL will NO diverge, because there are collisions between electrons, between electrons and phonons, and between electrons and ipurities. hese scatterings will reduce the oentu towards. hey will introduce viscosity to the Feri liquid, which reduces the velocity and oentu of the liquid. Eventually, the scatterings and the Efield will balance each other and will turn to a fixed value he total force can be written as F -e E - p (6.45) v -e E - Here the first ter is the electric force and the second ter is a viscosity ter, which is proportional to -v, reducing the speed. he coefficients and are the electron ass and the collision tie (average tie between two collisions). Using the second law of Newton: âp p - p (6.46) -e E ât where p is the oentu before we turn on the electric field and we assue the initial condition pht L p. It is easy to find that the solution to this equation is p p - e E I - ã-t M (6.47) his eans that the change of oentu p p - p -e E I - ã-t M In the static liit Ht L, (6.48)

8 4 (6.49) p -e E he change in wavevector ee - (6.5) Ñ his approxiation is nown as the Drude approxiation (the Drude odel). he classical theory (ignore the fact that electrons are ferions) he total electric current is e n Ñ j -e n v -e n (6.5) E his is the Oh s Law (6.5) j ΣE where the conductivity e n (6.5) Σ he quantu theory (taen into account the Pauli exclusive principle) Assuing that the E field is along the z direction. he total velocity along z is e j -Xe n v\ -[ e N v_ F vf -e Π H ΠL â Θ cos Θ -e Π Π â Φ â Θ F cos Θ vf cos Θ Π F vf F vf -e 4 Π Because ee 4 Π e Ñ F F vf 4 Π Ñ Ee vf Ñ 4 Π (6.54) e F E 4 Π E N F Π (6.55) e Π e F j 4 Π E N 4 Π en E E 4 (6.56) en (6.57) Σ 4 ypically, we absorb the extra /4 factor into the definition of, so the conductivity turns bac to its classical for en (6.58) Σ which is identical to the classical forula shown above. eties, we use the ean free path l vf in the forula, instead of. he physical eaning of the ean free path is the average distance that an electron travels between two collisions. Using l, we have enl Σ (6.59) vf he resistivity Ρ is

9 Ρ (6.6) en Σ 4 Collision tie he collision tie coes fro collisions between electrons ee collisions between electrons and phonons: ep collisions between electrons and ipurities: i If we assue that different scatterings are independent, the total satisfies ee ep (6.6) i Notice that Ρ (6.6) e n Ρ en ee ep e n ee i e n ep e n i (6.6) If we define Ρee e n ee and Ρep e n ep and Ρi e n i (6.64) so we found Ρ Ρee Ρep Ρi (6.65) Resistivity in a real etal at finite teperature In a real etal at finite teperature, the approxiation used above are still valid, but the collision tie will show teperature-dependence. As a result, Ρ is a function of. ypically, Ρ decreases as goes down. At, Ρ goes to a finite value ΡH L, which is nown as the residue resistivity. At low, Ρee and Ρep decreases to as is reduced down to, while Ρi is independent. ery typically, Ρee µ, Ρep µ 5 and Ρi µ constant Ρ Ρi Aee Aep 5 (6.66) At low teperature, Ρi >> Ρee >> Ρep, so ipurity scattering is the doinate contribution for Ρ. At high teperature, Ρ is typically a linear function of Ρ A (6.67) his is because the nuber of phonons is proportional to at high teperature. n expj Ε N-» J Ε OH - LN - Ε (6.68) the Hall effect (part I) Consider a thin saple (electrons oving in D). We apply a agnetic field perpendicular to the plane (in the z direction) F -eke v BO (6.69) he second ter is the Lorentz force, which is perpendicular to the velocity. If we pass a current along the x direction. he electrons will feel a Lorentz force along the y direction and thus positive (negative) charge will accuulate along the top and botto edge of the saple. his charge accuulation will induce a electric field along y, perpendicular to the current. If we have a static current, this electric field will balance the :Lorentz force.

10 4 he second ter is the Lorentz force, which is perpendicular to the velocity. If we pass a current along the x direction. he electrons will feel a Lorentz force along the y direction and thus positive (negative) charge will accuulate along the top and botto edge of the saple. his charge accuulation will induce a electric field along y, perpendicular to the current. If we have a static current, this electric field will balance the :Lorentz force. ` ` -e E y e y e v B -e vx B e y (6.7) E y vx B (6.7) he current along x is jx -e n vx (6.7) the ratio between E y and j X, which is called the Hall resistance, is Ey Rxy vx B jx -e n vx B (6.7) en he ratio between Rxy and B is nown as the Hall coefficient Rxy RH (6.74) B en RH is proportional to n (inverse of the electron density), this easureent is the standard technique to deterine the density of electrons in a aterial. n- (6.75) e RH Notice that the density n here is D electron density (nuber of electron per area) Ne (6.76) n n D A For a D aterial, the D electron density is Ne n n D Ne Ad n D d (6.77) where d is the thicness of the saple the Hall effect (part II with scatterings) In the calculations above, we didn t consider the electron collisions. If we treat the collisions using the Drude approxiation (assuing that the collisions are described by a single paraeter, the collision tie), the force on an electron is F -eke v BO v (6.78) Using the second law of Newton, we find that âv ât -eke v BO (6.79) v we have two equations of otion â vx vx ât â vy (6.8) - e vx B -e E y (6.8) vy ât e v y B -e Ex Using the initial condition vx v y at t, the solution of these differential equations is vx - e IEx - B e E y M B e - -ã t e AI-Ex B e E y M cos Bet IE y B e Ex M sin B e Bet E (6.8)

11 vy - e IE y B e Ex M - ã e AIE y B e Ex M cos t B e e Ex - Ωc E y e I-Ex Ωc E y M cos HΩc tl IE y Ωc Ex M sinhωc tl t - -ã Ωc - ã t Ωc Bet E (6.8) (6.84) Ωc e E y Ωc Ex vy - IEx - B e E y M sin B e We can define the cyclotron frequency Ωc e B, so that vx - Bet 4 e IE y Ωc Ex M coshωc tl IEx - Ωc E y M sinhωc tl (6.85) Ωc he static solution at t is e Ex - Ωc E y vx - (6.86) Ωc e E y Ωc Ex vy - (6.87) Ωc If the current is along x Iv y M, we find that E y -Ωc Ex (6.88) e Ex Ωc Ex e Ex - Ωc E y vx - Ωc e - Ex (6.89) Ωc herefore, Ey Rxy Ey jx -Ωc Ex -n e vx en e Ωc - Ex eb - en e - en e B en (6.9) And thus Rxy RH (6.9) B en Free electrons in D at finite teperature Nuber of electrons on a quantu state with energy Ε (the Feri-Dirac distribution function) f HΕL expj Ε- N (6.9) Notice that f HΕL, which agrees with the Pauli exclusive principle. Density of states (DOS) How any states do we have in the window of Ε Ε â Ε? DHΕL â Ε? (6.9) o answer this question, let s consider the total nuber of electrons N DHΕL â Ε otal nuber of electrons can also be written as (6.94)

12 44 â N á á H ΠL F We now that Ñ â â F 4 Π â 8Π Π F â Π â (6.95) Ñ â (6.96) N Π â Π Because Ε Ñ, N Π Ñ Π Ñ Π Ñ Ñ Ε Ñ DHΕL â Ε Π Ñ (6.97) Ε Ñ â Ε H L Π Ñ Ε Π Ñ Ε DHΕL â Ε (6.98) Ε (6.99) Another way to get DHΕL is ân (6.) DHΕL Because H ΕL N Π (6.) Π Ñ H L ân DHΕL Π Ñ Π Ñ Ε Here, we can also prove that DHΕL N Ε, ân DHΕL H L Ε H ΕL Ε Π Ñ (6.) Ε Π Ñ N (6.) Ε otal energy UHL Ε f HΕL DHΕL â Ε (6.4) Heat capacity U C Ε First we prove that C Ù HΕ - L It is easy to prove that â f HΕL â (6.5) DHΕL â Ε â â DHΕL â Ε â â DHΕL â Ε f HΕL DHΕL â Ε (because the nuber of electrons are fixed, so herefore, ân â ân â ) (6.6)

13 U Ε C v Ε â â - â, â expj Ε- N We now that DHΕL â Ε â DHΕL â Ε - HL Ε â B expj Ε- -B expj â Ε - â Ε expj Ε- N 45 â F exp Ε- N DHΕL â Ε HΕ - L (6.7) DHΕL â Ε â Ε- Ε- (6.8) F exp f HΕL â N â DHΕL â Ε - Ε- (6.9) (6.) As a result, C - HΕ - L At << B, Ε - DHΕL â Ε - HΕ - L DHΕL (6.) has a very high pea at Ε». we only need to consider the integral near Ε. herefore, we can set DHΕL» DH L. his approxiation is very good for ost etals, where B» or larger. we have C -DH L HΕ - L â Ε -DH L HΕ - L âf DH L HΕ - L â f (6.)

14 46 SechB -Ε t PlotB: 4t SechB -Ε t 4t F F. t.5,. t., SechB SechB -Ε t 4t -Ε t 4t F F. t.,. t >, 8Ε,, <, PlotRange 8, 6<, PlotStyle hic, AxesLabel :"Ε ", "- PlotLegends :" ", " B ", " B ">, LabelStyle Mediu, ">, PlotPoints F B B B B Ε Notice that lni f - - M Ε - C DH L DH L (6.) HΕ - L â f - - A lni f - ME â f DH L AlnI f - ME â f DH L C is a linear function of at << B. We now that DH L N, so Π C DH L B Π N B Define Feri teperature F B Π C Π N B Π Π Π DH L B (6.4) N B (6.5) N B (6.6) F Heat capacity in a real etal In a real etal, the heat capacity taes this for at low C Γ A (6.7) Here, the first ter coes for electrons and the second one coes fro acoustic phonons. At very low teperature, Γ >> A, so the electron contribution doinates the heat capacity.

15 In experients, one can plot the curve C vs. his curve is a linear function yγ Ax 47 (6.8) We can deterine the slope A and the intercept Γ by fitting the experiental data. heral effective ass According to the theory of free electron gases, the coefficient Γ is Π N B Γ (6.9) Notice that Ñ F Ñ Π N (6.) Π N B Γ N I Π M (6.) Ñ If we easure Γ in an experient, we can use it to deterine the ass of an electron - Γ Π N B Π N Ñ (6.) In real etals, this ass is typically different fro the real electron ass (by a factor of.-). o distinguish the ass deterined using this forula fro the true electron ass, we call this ass the theral effective ass Hth L. he reason that this ass is different fro the real electron ass is because the electrons in a real etal are not free electrons. Instead, we have A lattice bacground Interactions between electrons Interactions between electrons and phonons Although we ignored all these effects, the free electron theory discussed above turns out to be valid in any solids (ost etals and seiconductors we have in our daily life). It turns out that indeed we can treat the electrons in a etal as a bunch of free ferions. hese free ferionic particles are NO electrons. hey are called quasi-particles. Landau Feri liquid theory he electrons in a etal can be considered as a gas fored by non-interacting quasi-particles. he total nuber of these quasi-particles is identical to the total nuber of electrons. (Known as the Luttinger theore). hese quasi-particles ay carry electric charge ± e and they have spin-/ (they are charged ferions). he ass of these quasi-particles (the effective ass) differs fro the electron ass. he value of the effective ass depends on aterials, electron density, teperature, etc. (Not going to be discussed in this course) he quasi-particles have a Feri residue which is saller than but larger than. Feri residue is defined as Z f H - L - f H LD (6.) where f HΕL is the distribution function of the particles. For free ferions, Z, because f H - L and f H L. But for interacting ferions, this value will be reduced by the interaction effect. However, for a Landau Feri liquid, the value of Z shall always be positive H < Z < L. Quantu liquids with Z are NO Landau Feri liquids. hey are one of the central focuses in current research (starting fro the 8s). his free ferion theory wors surprisingly well in etals and seiconductors. Heavy ferion aterials In typically etallic aterials, the effective ass is not very different fro the electron ass (differs by a factor or.-). But there are soe aterials, which are nown as heavy-ferion copounds. In those aterials, the effective ass is about ties larger than the electron ass.

16 48 Non-Feri liquids It is also worthwhile to notice that there are soe other aterials, in which these Landau Feri liquid theory totally fails. his liquids are nown as the non-feri liquids. Landau Feri liquid is very powerful, but it also enforce certain liitations to our aterials. For exaple, as will be discussed in latter parts, soe aterials can turn into a superconductor (a conductor with zero resistivity) at low teperature < C. For Landau s Feri liquids, the value of c cannot exceeds ~ - K Habout - 6 degreel, which iplies that we cannot use these superconductors on an industrial scale, because we cannot run our achines at such a low teperature. In the 8s, people realized that for non-feri liquids, such a constrain will no longer exist, so we can have superconductors at a uch higher teperature. hese aterials are nown as high teperature superconductors (high c ). Currently, the highest c is about K (about -7 degree). Increase the transition teperature fro a few K to K is very iportant. he reason is because to cool down a syste down to a few K, one need to use very expensive ethods (e.g. liquid Heliu). But to reach the teperature of -K is extreely easy. One just need to dip the saple into liquid Nitrogen. Nitrogen is very easy to obtain (78% of our air is ade by Nitrogen). he price for liquid Nitrogen is coparable with beer. he boiling point of Nitrogen is about 77K. Liquid Nitrogen can help us reach 77K without any difficulty at low cost. raising C above 77K is a very iportant revolution, which is achieved in the late 8s, using non-feri-liquid aterials heral conductivity In the last chapter, we learned that the theral conductivity for a gas is: C K (6.4) vl We used this forula to copute the theral conductivity of phonons. Here, we use the sae forula to copute the theral conductivity of electrons. he heat conductivity of electrons is Π Cel N B (6.5) and we also nown that the ean free path l vf. Π C Kel vf l N B vf HvF L Π N B vf Π n B (6.6) In a clean etal Kelectron >> Kphonon. In dirty etals, Kelectron is sall because the collision tie is sall. In dirty etals, Kelectron» Kphonon. In a etal, Ketal Kelectron Kphonon >> Kphonon. In an insulator, Kelectron, so Kinsulator Kphonon. we have Ketal >> Kinsulator. his is why etal is cold to touch in the winter. he Wiedeann Franz law. he ratio between the heat conductivity and the electric conductivity is Π n B K en Σ Π B (6.7) e We can define the Lorenz nuber: Π B L e.44-8 W W K (6.8) and the ratio between K and Σ is just K L Σ his relation is nown as the Wiedeann Franz law. In real aterials (etals), the value of L is typically very close to.44-8 W W K. (6.9)