Metals: Free Electron Model Physics 355 Free Electron Model Schematic model of metallic crystal, such as Na, Li, K, etc. The equilibrium positions of the atomic cores are positioned on the crystal lattice and surrounded by a sea of conduction electrons. For Na, the conduction electrons are from the 3s valence electrons of the free atoms. The atomic cores contain 10 electrons in the configuration: 1s22s2p6. + + + + + + + + + + + + + + + + + + + + + + + + + Paul Drude (1863-1906) вЂў resistivity ranges from 10пЂ8 пЃ—п‚ m (Ag) to 1020 пЃ—п‚ m (polystyrene) вЂў Drude (circa 1900) was asking why? He was working prior to the development of quantum mechanics, so he began with a classical model: вЂў positive ion cores within an electron gas that follows Maxwell-Boltzmann statistics вЂў following the kinetic theory of gasesthe electrons in the gas move in straight lines and make collisions only with the ion cores вЂ“ no electronelectron interactions. Paul Drude вЂў He envisioned instantaneous collisions in which electrons lose any energy gained from the electric field. вЂў The mean free path was approximately the inter-ionic core spacing. (1863-1906) вЂў Model successfully determined the form of OhmвЂ™s law in terms of free electrons and a relation between electrical and thermal conduction, but failed to explain electron heat capacity and the magnetic susceptibility of conduction electrons. OhmвЂ™s Law Experimental observation: V п‚µ I E OhmвЂ™s Law: Free Electron Model пЂЅ number e пЂЅ ne volume j пЂЅ nev d Conventional current OhmвЂ™s Law: Free Electron Model 12 Predicted behavior 10 Resistivity 8 B High T: Resistivity limited by lattice thermal motion. 6 4 2 0 0 20 40 60 80 100 120 Temperature The mean free path is actually many times the lattice spacing вЂ“ due to the wave properties of electrons. Low T: Resistivity limited by lattice defects. Wiedemann-Franz Law (1853) (Ludwig) Lorenz Number (derived via quantum mechanical treatment) LпЂЅ пЃ« 1 пЃі T пЃ° kB 2 пЂЅ 3e 2 2 пЂЅ 2 . 45 п‚ґ 10 пЂ8 W пѓ—пЃ— K 2 Free Electron Parameters пЃҐF Li N/V Г—1022 /cm3 4.70 kF Г—108 /cm 1.11 vF Г—108 cm/s 1.29 eV 4.72 TF Г—104 K 5.48 Na 2.65 0.92 1.07 3.23 3.75 Cu 8.45 1.36 1.57 7.00 8.12 Au 5.90 1.20 1.39 5.51 6.39 Be 24.20 1.93 2.23 14.14 16.41 Al 18.06 1.75 2.02 11.63 13.49 Pb 13.20 1.57 1.82 9.37 10.87 Electron Heat Capacity NaГЇve Thermodynamic Approach: вЂў You could start out by considering the average thermal energy of a free electron at some temperature T. E пЂЅ 3 2 k BT вЂў Then, U пЂЅ 3 2 Nk B T вЂў And the electronic heat capacity would then be: C el пЂЅ dU dT пЂЅ 3 2 Nk B вЂў However, when we go out and measure, we find the electronic contribution is only around one percent of this. Electron Heat Capacity D(пЃҐF) вЂў The electron energy levels are mostly filled up to the Fermi energy. вЂў So, only a small fraction of electrons, approximately T/TF, can be excited to higher levels вЂ“ because there is only about kBT of thermal energy available. пѓ¦T пѓ¶ 3 вЂў Therefore, U пЂЅ Nk B T пѓ§пѓ§ пѓ·пѓ· 2 пѓЁ TF пѓё which goes as T2. вЂў вЂ¦and the heat capacity, Cel = dU/dT goes as T, which is the correct result. Sommerfeld Constant вЂў The Sommerfeld constant is proportional to the density of states at the Fermi energy, since 1 2 2 пЃ§ пЂЅ пЃ° k B g (пЃҐ F ) 3 вЂў Now, we look at this and say, вЂњObviously!вЂќ вЂ“ because only the electrons very close to the Fermi energy can absorb energy. The Sommerfeld constant is also related to another, rather important, concept in Solid State Physics вЂ“ effective mass. 2 2 2 пЃЁ kF пЃЁ пЃҐF пЂЅ пЂЅ 2m 2m пѓ¦ 2 N пѓ¶ пѓ§ 3пЃ° пѓ· V пѓё пѓЁ 2/3 Effective Mass ~m + + + + + + + + + + + + + + + + + + + + + + + + + Electrons interact with вЂў periodic lattice potential вЂў phonons вЂў other electrons пЃ§ ~ m effective Sommerfeld Constant пЃ§ observed пЂЁ mJ/ mol пѓ— K 2 пЂ© пЃ§ calculated пЂЁ mJ/ mol пѓ— K 2 пЂ© me / m Observed values come from the linear heat capacity measurements. Calculated values are determined using the conduction electron density and from assuming me = m, Heavy Fermions Heavy fermions are intermetallic compounds containing noncomplete 4f- or 5f-electronic shells. The orbital overlap with ligand atoms in the lattice leads to strong correlation effects in the system of delocalized electrons. As a result, the effective mass of the electrons can increase by orders of magnitude as пЃ§ observed compared to the free electron mass. пЂЁ mJ/ mol пѓ— K Heavy fermion materials exhibit very interesting ground states - such as unconventional superconductivity, smallmoment band magnetism and non Fermi liquid behavior. CeAl 3 CePb 3 1700 UBe 13 CeCu 2 Si 2 1100 CeCu 6 U 6 Fe 1000 1200 1000 24 2 пЂ© Heavy Fermions A large value of the Sommerfeld parameter indicates that heavy fermion materials have a high density of states at the Fermi Energy. g пЂЁпЃҐ пЂ© Electrical Conduction Electrical Conduction Hall Effect In 1879, while working on his doctoral thesis, Hall was pursuing the question first posed by Maxwell as to whether the resistance of a coil excited by a current was affected by the presence of a magnet. Does the force act on the conductor or the current? Hall argued that if the current was affected by the magnetic field then there should be "a state of stress... the electricity passing toward one side of the wire." Hall Effect Initially, v пЂЅ v x xЛ† пЂ« v y yЛ† пЂ« v z zЛ† E пЂЅ E x xЛ† B пЂЅ B z zЛ† пЃІ пЃІ пЃІ пЃІ 1 пѓ¶пЃІ пѓ¦ d F пЂЅ mпѓ§ пЂ« пѓ· v пЂЅ пЂ e (E пЂ« v п‚ґ B ) пѓЁ dt пЃґ пѓё net force in пЂx direction net force in пЂy direction 1пѓ¶ пѓ¦ d Fx пЂЅ m пѓ§ пЂ« пѓ·vx пЂЅ пЂ e( E x пЂ« v y B ) пѓЁ dt пЃґ пѓё Fy 1пѓ¶ пѓ¦ d пЂЅ mпѓ§ пЂ« пѓ·v y пЂЅ e(vx B ) пѓЁ dt пЃґ пѓё Hall Effect As a result, electrons move in the пЂy direction and an electric field component appears in the y direction, Ey. This will continue until the Lorentz force is equal and opposite to the electric force due to the buildup of electrons вЂ“ that is, a steady condition arises. B Hall Effect mv x пЂЅ пЂ e( E x пЂ« v y B ) пЃґ mv y пЃґ пЃ·C пЂЅ eB m пЂЅ пЂ e( E y пЂ« vx B ) vx пЂЅ пЂ vy пЂЅ пЂ e пЃґE x m e пЃґE y m пЂ пЃ· C v yпЃґ пЂ« пЃ· C v xпЃґ Hall Effect vy пЂЅ пЂ e пЃґE y m пЂ« пЃ· C v xпЃґ пЂЅ 0 пѓћ Ey пЂЅ m пЃ·Cvx e E y пЂЅ пЂпЃ·CпЃґ E x пЂЅ пЂ vx пЂЅ пЂ eпЃґ m Ex пѓћ Ex пЂЅ пЂm vx eпЃґ eB пЃґ m Ex Hall Effect The Hall coefficient is defined as: eB пЃґ RH пЂЅ Ey jx B пЂЅ пЂ m Ex 2 ne пЃґ m For copper: n = 8.47 Г— 1028 electrons/m3. пЂЅ пЂ 1 ne ExB Hall Effect Hall Effect: Electrons & Holes вЂў The Hall Effect experiment suggests that a carrier can have a positive charge. вЂў These carriers are вЂњholesвЂќ in the electron sea - the absence of an electron acts as a net positive charge. These were first explained by Heisenberg. вЂў We canвЂ™t explain why this would happen with our free electron theory. вЂў Note: the conditions we derived for the steady state can be invalid for several conditions (for example, when there is a distribution of collision times). But in general, it is a very powerful tool for looking at properties of materials. Hall Effect: Applications For a 100-пЃm thick Cu film, in a 1.0 T magnetic field and through which I = 0.5 A is passing, the Hall voltage is 0.737 пЃV. Hall Effect: Applications Hall-Effect Position Sensors Hall-Effect position sensors have replaced ignition points in many distributors and are used to directly detect crank and/or cam position on distributorless ignition systems (DIS), telling the computer when to fire the coils. Hall-Effect sensors produce a voltage proportional to the strength of a magnetic field passing through them, which can come from a permanent magnet or an electric current. Since magnetic field strength is proportional to an electric current, HallEffect sensors can measure current. They convert the magnetic field into millivolts that can be read by a DMM. Recap: Free Electron Model Some successes: 1. electrical conductivity 2. heat capacity 3. thermal conductivity Some failures: 1. physical differences between conductors, insulators, semiconductors, semi-metals 2. positive Hall coefficients вЂ“ positive charge carriers ??