4 Phys520.nb 2 Drude theory ~ Chapter in textbook 2.. The relaxation tie approxiation Here we treat electrons as a free ideal gas (classical) 2... Totally ignore interactions/scatterings Under a static electric field (E ), electrons get a constant force. F -e E (2.) The second Law of Newton tell us that F d P d t (2.2) If we have a constant force, P increases linearly as a function of t P (t) P (t 0) + F t Conclusion: If we wait long enough, the velocity of an electron will diverge, which is obviously ipossible and wrong. (2.3) 2..2. The relaxation tie approxiation The reason why we get a wrong answer is because electrons are not really a free gas. There are scatterings/interactions (between electrons, between an electron and an ion, and between an electron and an ipurity). Thought these scatterings, electron gives its oentu and kinetic energy to other particles, e.g. ions or ipurities or another electron, i.e. dissipation Dissipation slows down the flow of electrons, like viscosity in a liquid. To describe this effect, we can introduce a force along the -P direction. F -e E + F dissipation (2.4) Note: this is an approxiation, because the real force on an electron is very coplicated. It varies with tie and it could be pointing at any direction (depending on the locations of other electrons and ions). Here, we take an average value for siplification. We know 2 things about this force. it is along the -P direction (on average) 2. it is zero, if there is no current, i.e. the force is zero if P 0. Here, we introduce an extra assuption: the force is an analytic function of P, i.e., we can do a Taylor expansion for the function F dissipation P. According to the two facts listed above, this Taylor expansion ust take the following for
Phys520.nb 5 F P dissipation - + O P 2 The coefficient of the first order ter / has the unit of /tie, so has the unit of tie. We call it the relaxation tie or the collision tie. Roughly speaking, this is the average tie interval between two collisions. Now the Newton's second law says (2.5) d P d t -e P E - This is the relaxation tie approxiation (Read Chapter for the historical approach that people used when they tried to "derive" this approxiation) (2.6) 2..3. Collision tie The relaxation-tie approxiation doesn t care about the source of the dissipation. It assues that wherever it coes fro, it can be described by only a single paraeter, which is. In reality, we know that it coes fro ainly three sources: 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 i (2.7) At low T, ipurity scattering doinates. At high T e-p scatterings doinates. Think about why we add inverse fro different sources, instead of adding s directions 2..4. Stable solution d P d t -e P E - 0 (2.8) P -e E (2.9) e v - E (2.0) We know that j -e n v where n is the density of electron (2.) e j e n E E (2.2) This is the Oh s Law j σ E where the conductivity Or say σ (2.3) (2.4)
6 Phys520.nb ρ (2.5) We know that + +, so ee ep i ρ + + ee ep i + ee + ep i (2.6) If we define ρ ee and ρ ep ee and ρ i ep i (2.7) we will find that ρ ρ ee + ρ ep + ρ i (2.8) 2..5. Resistivity and teperature-dependence In a real etal ρ is a function of T. Typically, ρ decreases as T goes down. At T 0, ρ goes to a finite value ρ(t 0), which is known as the residue resistivity. At low T, ρ ee and ρ ep decreases to 0 as T is reduced down to 0, while ρ i is T independent. Very typically, ρ ee T 2, ρ ep T 5 and ρ i constant ρ ρ i + A ee T 2 + A ep T 5 (2.9) At low teperature, ρ i >> ρ ee >> ρ ep, so ipurity scattering is the doinate contribution for ρ. At high teperature, ρ is doinated by phonons (scattering between electrons and lattices) and typically it is a linear function of T (Page 524) ρ A T This is because the nuber of phonons is proportional to T at high teperature. And thus the scattering probability is proportional to T. Q: why the nuber of phonons is proportional to T at high teperature? A: The equipartition theore in classical statistical physics: in a classical syste, each degree of freedo contributes k B T /2 to the average total energy. At high teperature, quantu effects becoes negligible, and we always recover the classical liit. Here, the equipartition theore tells us that the total energy in elastic waves ust be T. Particle-wave duality in quantu echanics tells us that waves are particles. Each particle has energy ħ ω. Total nuber of particles is about total energy. Because the total energy is T, the total particle nuber ust also be linear in T. ħω n exp ħ ω k B T - + ħω k B T + O T-2 - k B T ħω (2.20) (2.2) 2.2. The Hall effect If we applying unifor a agnetic field along z, then by applying an electric current along x, an electric field will be generated along y direction
Phys520.nb 7 2.2.. treatent #: balancing the force along y The Lorentz force along y is balanced by the electric field along y e v x B z - e 0 v x B z e n j x -e n v x - B z Resistivity ρ xx E x / j x?? The value of ρ xx is not universal. It is very sensitive to scattering tie (), which depends on teperature, ipurity density, etc. The Hall resistivity ρ xy j x - e n B z The Hall resistivity is a linear function of B. The coefficient is known as the Hall coefficient R H - e n The Hall coefficient directly easures the carrier density (electron density). NOTE: (2.22) (2.23) (2.24) (2.25) (2.26) (2.27) If you use the Gaussian units, there will be an extra factor c in R H The sign of R H is very iportant. If the electric current is carried by particles with a negative charge (electrons), R H < 0. If particles that carries the current has positive charge (holes), R H > 0. Materials with R H < 0 is called electron-like and for R H > 0, they are called holelike. The electron density easured fro this forula is in good agreeent with the total nuber of valence electrons for aterial with low electron density (e.g Z ), but the agreeent is bad for Z2 or 3 (even the sign is wrong). See Table.4 on Page 5. Be careful about Hall resistance and Hall resistivity. 2.2.2. Hall Resistance: In ost experients, the row data is Hall resistance. L y ρ xy R xy V y I x - B z - B z (2.28) j x L y L z L z e (n L z ) e n 2 D Hall resistance easures the 2D carrier density, n 2 D, which is the nuber of particle per 2D area. For a 2D aterial, this is the 2D carrier density. For a 3D aterial, the 3D carrier density is n 2 D divided by the thickness. 2.2.3. treatent #2: Drude theory 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 -e E + v B - v Using the second law of Newton, we find that (2.29) d v d t -e E + v B - v So we have two equations of otion (2.30)
8 Phys520.nb d v x v x + + e v y B -e E x d t d v y v y + - e v x B -e d t The static solution (d v / d t 0) v x v y + e v y B -e E x - e v x B -e which iplies that e E x - ω c v x - + ω 2 c 2 e v y - + ω c E x + ω c 2 2 (2.3) (2.32) (2.33) (2.34) (2.35) (2.36) Here, ω c e B /. If the current is along x v y 0, we find that And -ω c E x (2.37) Thus e E x - ω c e E x + ω 2 c 2 E x e v x - - - + ω 2 c 2 + ω 2 c 2 E x (2.38) j x -e n v x E x We recovered the resistivity ρ xx E x / j x For the Hall resistivity (2.39) ρ xy j x E x E x ω c (-ω c ) - - e B - e n B (2.40) Warning: we used resistivity ρ xx and the Hall resistivity ρ xy. We didn t ention anything about conductivity σ xx or the Hall conductivity σ xy. There is a reason for that. In fact, it is iportant to keep in ind that for the Hall bar geoetry, ρ xx E x / j x but σ xx j x /E x Siilarly, ρ xy j x (2.4) (2.42) (2.43) but σ xy j x (2.44) 2.2.4. Resistivity and conductivity In a solid, the relation between the resistivity and the conductivity could be ore coplicated than a siple nuber inverse. Here are two exaples