Chapter 6 Free Electron Fermi Gas – Heat Capacity Chapter 6 Free Electron Fermi Gas � Heat Capacity Properties of Free Electron Metals � Last lecture, we defined terms such as the Fermi Energy, Fermi[.]
Chapter 6: Free Electron Fermi Gas – Heat Capacity Properties of Free Electron Metals • • • Last lecture, we defined terms such as the Fermi Energy, Fermi Temperature, and Fermi Velocity (which all depend upon the density of conduction electrons) We will use these concepts for calculating the electronic heat capacity Here are some examples: (note Tf is usually a large number, so for most materials at room temperature, this is a low- T regime) The Density of States in 3D • We also looked at the Density of States in 3D for a Free Electron Fermi Gas: Density of States in 3D Another way of looking at last lecture: No of states ~ surface area of Fermi sphere (4πk2) Heat Capacity of a Free Electron Fermi Gas • • • • We know that the heat capacity goes like T3 at low temperatures from phonons Experimentally, we also know that there is a linear component which goes like T Can we derive this result with our free electron fermi gas theory to show that this component is from the free electrons? You can actually predict this without a rigourous proof (although you will see this proof in graduate level solid state physics courses) C/T Slope = β Intercept = γ T2 C = γT + βT3 Electron component Phonon comp Heat capacity by intuition • • • • • • • What is the average thermal energy of a free particle at temperature T? Energy/particle ~ kT (actually, it is 3/2 kT from the equipartition theorem – ½ kT for the ½ mv2 component of the kinetic energy in the x, y, and z direction) So, U = 3/2 NkT (for N electrons at temperature T) The electronic heat capacity would then be: Cel = dU/dT = 3/2 Nk In the real world, however, if we measure this, we find that most materials have an electronic component which is only ~ % of this value In addition, this does not vary as T, but is a constant at all temperatures What is wrong with this interpretation? • • • • • • • We have made an assumption which is wrong In reality, we know that only a small fraction of the electrons can absorb energy Why? All of the electron energy levels are mostly filled up to the Fermi energy Actually, only a fraction of electrons ~ T/Tf can be excited to higher levels (because we only have about ~ kT of thermal energy at temperature T) Therefore, we have to multiply the no of electrons by this factor to get the total no of electrons which can absorb energy at temp T: U ~ 3/2 NkT (T/Tf) ~ T2 And we get C = dU/dT ~ T (the right result!) D(εF) Heat capacity by intuition εF Only a fraction of electrons can absorb energy – those near the Fermi Energy within ~kT of energy “Ripples in the Fermi Sea” Energy levels fill up until N/2 at T= K, and at T > K, we have some higher levels occupied • T>0 K T=0K ~kT Some electrons from here are excited up to here • • One way of looking at this is like a “sea” of electrons, but only those electrons near the surface (at the highest energies) can absorb energy as heat So, it is the “ripples” in the Fermi sea that determine electronic properties The higher the T, the more ripples we have, and the more electrons we can excite to higher levels electrons down here are “trapped”: only electrons within ~kT of the Fermi Surface can absorb energy Electronic Heat Capacity • • • • • • • • How would you calculate this more rigourously? What we really need to is calculate the number of electron states that can be excited to higher energies, and then take the derivative to find Cel At the graduate level, you will see this done as the Sommerfeld expansion The actual result is: Cel = ½ π2NkT/Tf Compare to our “back of the envelope” calculation: Cel = 3/2 NkT/Tf (within a factor of 3!) This is often written as : Cel = γ T, where γ = ½ π2Nk/Tf This is called the Sommerfeld constant (γ) You can also write this in terms of the density of states at the Fermi surface Electronic Heat Capacity • Remember, from the last time, that: 3/ V 2mE N (ε ) = ⇒ ln N = ln ε + const 3π h dN dε dN N (ε ) ⇒ = ⇒ D(ε ) = = N ε dε ε • So, using this, and that εf = kTf, we have: D(ε f ) = 3N 3N = 2ε f 2kT f T 2 D(ε f ) 2 = π k D(ε f )T ⇒ Cel = π Nk = π Nk T Tf 2 3N Meaning of the Sommerfeld Constant • • • • • • So, the Sommerfeld constant is ~ density of states at the Fermi energy (γ = 1/3 π2k2 D(εf)) This makes sense – it is only these electron states that can absorb energy The Sommerfeld constant is also related to an “effective mass” of the conduction electrons (which should be ~ mass of an electron) Mass of conduction electron How? The Fermi Energy is εf = ħ2kf2/2m = (ħ2/2m)(3π2N/V)2/3 So, rearranging our terms we have: Nk 2 2m N ⇒γ = π = π Nk 3π εf V h • And therefore, γ ~ m −2 “Effective Mass” • • • • • What is meant by an “effective mass”? Well, if the electrons were actually free, meaning they could move without feeling a potential, then: γ ~ melectron However, the electrons are not completely free, so we usually say that γ ~ mth (an effective or thermal mass, which takes care of all other interactions) What sort of interactions can be present? Interaction of conduction electrons with the periodic potential of the lattice (we have ignored the effect of the positive ion cores, for example) Interaction of conduction electrons with phonons (phonons can distort the lattice that the electrons see – electron-phonon scattering is common in most materials) Interaction of the conduction electrons with themselves (these are negative charges, which should repel one another!) All of these effects change the effective mass Sommerfeld Constants • • • Typical Sommerfeld constants for metals (and the corresponding effective masses) are: The observed values are from the linear heat capacity experiments, calculated values are assuming mth = melectron and using the conduction electron density Most materials have a fairly good approximation that mth = melectron (ie The Free Electron Theory works well) γobserved γcalculated mth/m Li 1.63 0.749 2.18 Na 1.38 1.094 1.26 K 2.08 1.668 1.25 Rb 2.41 1.911 1.26 Cs 3.20 2.238 1.43 Cu 0.695 0.505 1.38 Ag 0.646 0.645 1.00 Au 0.729 0.642 1.14 (in mJ/(mol K2)) Heavy Fermion Materials • • • Most materials have mth ~ 1-2 melectron, meaning that the electrons are basically free to move throughout the lattice However, some materials have mth ~ 100 -1000 mel (!) These are called “heavy fermion” materials (electrons are fermions) Usually contain U or Ce (felectron rare earth materials) Typical values for other metals are ~ 1-10 mJ/(mol K2) so these have high mth values Heavy Fermion Materials • • • • • What is happening in these materials? The high “effective” mass means that the electrons are highly correlated with one another – each electron can feel the other electrons and they “drag around” the rest of the electron cloud when they move These are what are known to be “highly correlated” electron systems, and they arise from very complicated f-electron physics These materials often show bizarre behaviour, such as strange magnetic order (URu2Si2), strange superconductivity, and sometimes even the coexistence of magnetic ordering and superconductivity (which is strange!) You will learn about why this is strange in the second part of this course Another way of looking at the Sommerfeld constant • • • • • Some scientists look at the Sommerfeld constant as being a direct measure of the number of states at the Fermi Energy, D(εf) This is using: γ = 1/3 π2D(εf)k2 Why is this useful? Another way of looking at heavy fermion materials it to notice that a high gamma coefficient means a high density of states at the Fermi Energy You can calculate this and make theories about heavy fermion physics (same as mth)