Lecture 10: Condensed matter systems

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1 Lecture 10: Condensed matter systems Ideal quantum, condensed system of fermions Aims: Degenerate systems: non-interacting quantum particles at high density ermions: ermi energy and chemical potential µ(). Electrons in metals: Heat capacity Pauli paramagnetism Degeneracy pressure (introduction) May 05 Lecture 10 1

2 Degenerate systems Summary of previous results: Particles in a box (cube of side a) Single-particle energies ε k m ( ) l + m + n Density of states in 3-dimensions to this we need to add any spin-degeneracy. or example, electrons have spin 1/ and hence a degeneracy of two. g(ε) from 1/ now on Occupation numbers for D (+ sign) and BE (- sign) distributions: p ε 1 exp ε k ± π m a 3/ 3 a m g ( ε )dε ε 1/ o dε 4π g electron ( ε )dε Aε dε ( ) [ (( ( )) ) 1] i i i µ Aε 1/ dε Spin up and spin down Classical limit when p<<1 (high, low density) Quantum limit when p>~1 (low, high density) hen it is called a degenerate system May 05 Lecture 10

3 Degenerate ermi gas At 0. Low temperature limit of the ermi-dirac distribution is a step function: States fill up to energy µ starting at ε 0. All higher energy states are empty. he value of µ() at 0K is known as the ermi energy, ε. All states within a sphere of radius k, the ermi wavevector, are filled, where ε m k ermi energy May 05 Lecture 10 3

4 ermi energy and µ(0) 0 continued Determine ε from the requirement that we have N electrons (i.e. N filled states) N ε 0 ε ( ε ) g( ε ) dε g( ε ) p m ( 3π n) / 3 ermi function for >0K. p ( ε ) 0 ( ε µ ( )) dε 1 exp( k ) + 1 i Density of particles n N / a 3 N / V May 05 Lecture 10 4

5 Chemical potential, µ(), for >0 Occupied states in 3-D 3 D ermi gas Chemical potential µ() for >0K. Result follows using the previous approach: ( ε ) g N dε 0 exp ε µ k + 1 [( ) ] an implicit equation for µ() Numerical solution: Key point µ()ε at low (true for most practical situations) µ ~ constant May 05 Lecture 10 5

6 Electrons in metals ree electron metals Valence electrons move freely and independently. Best example: Alkali metals (Li, Na, K, ); also noble metals (Cu, Ag, Au) A surprisingly good approximation for subtle reasons to do with e-e interactions ypical data ε /k v κ /m Metal n/(m -3 ) ε /(ev) /(K) v / (m/s( m/s) Na.65x x x10 6 Cu 8.45x x x10 6 Metals (at room temperature) are degenerate ermi systems he original free-electron model (Drude model) assumed the electrons were a classical perfect gas. It largely fails to account for metallic properties. Eg: Heat capacity ~3Nk/ was far too large. May 05 Lecture 10 6

7 hermal properties of metals hermal capacity Electronic contribution to the heat capacity follows from differentiating the electronic C el U el ε g( ε ) [( ε µ ) k ] π Nk energy w.r.t.. U el dε 0 exp + 1 Result from a straightforward, if if lengthy, manipulation. See Ashcroft and Mermin p4-7 A similar result follows from a qualitative argument emphasising the essential physics Compare occupied states at 0K with those at 0K May 05 Lecture 10 7

8 Electronic contribution to the heat capacity of a metal A B cont.. only electrons within ~k are active thermally. A direct consequence of degeneracy. here are n g( ε )k such electrons If we treat these excited electrons like classical electrons. hat is, having kinetic energy 3k/ per electron. U C el el Recall g ( ε ) n ex U ex 3k Combining A and B gives el V 4π g g ( ε ) ( ε ) 3k m 3π N ε m V g 3k 1 ( ε ) 3N ε 3N k 3 3 ε N Cel 3 3k 4. 5Nk k p. p. 4 Note dependence May 05 Lecture 10 8

9 Electronic heat capacity cont.. Notes: Absolute magnitude is much less than the classical result (3Nk/). / ~10 - at room temperature for typical metals. Observed values are in good agreement. E.g. Sodium (Na) a classic, free-electron metal: (C el ) meas 15. J mol -1 K -1 (C el ) f.e. 11. J mol -1 K -1 he result is one of the most important consequences of ermi-dirac statistics. dependence arises because, as the ermiedge broadens, more electrons get excited. N.B. the electronic contribution to the heat capacity of a metal is masked by a much larger contribution from vibration (see future lectures) at all but the lowest temperatures. May 05 Lecture 10 9

10 Pauli paramagnetism Paramagnetism in metals Metals are weakly paramagnetic but, unlike molecular-paramagnets, do not obey Curie s Law (χ 1/) An external field, B, shifts spin-up and spindown electrons in energy by ±µ B B g(ε)dε/ g(ε )µ B B/ Net spin x no. switching-spin x spin-moment M g ε µ B µ V g ε µ µ H V χ P ( ( ) B ) B ( ) g( ε ) µ µ V B 0 Small and independent of temperature. Again, only the electrons near ε are involved. he result is accurate but needs reducing by 1/3 due to Landau diamagnetism (Part II Quantum). May 05 Lecture B 0

11 Degeneracy pressure Pressure due to an electron gas: Consider the energy levels in a box under compression. It takes energy to compress the box, since the single-particles energies rise. Hence there must be an outwards pressure. n n 3 1 a l a l/ Astrophysical examples, typical star: Gravity tends to compress the star. Nuclear energy keeps the star hot and inflated. Mainly composed of ionised hydrogen M ~ 3x10 30 Kg, R ~ 3x10 7 m ~ 10 7 K Electrons form a degenerate gas. ~ 3x10 8 K < What happens when the nuclear reaction of hydrogen stops? Next lecture ε k m π m a ( ) l + m + n May 05 Lecture 10 11

7. FREE ELECTRON THEORY.

7. FREE ELECTRON THEORY. Aim: To introduce the free electron model for the physical properties of metals. It is the simplest theory for these materials, but still gives a very good description of many