Fermi Gas_ Free electrons in a solid

8/7/2019 Fermi Gas_ Free electrons in a solid

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Chapter 6

Free electrons in a solid: The Fermi

gas

Valence electrons of the atoms

usually delocalized over entire crystal

move in periodic crystal potential

Pauli principle: 2 electrons cannot occupy the same quantum state

most simple model: Free electron gas (=Fermi gas)

consider ions in a crystal as homogeneously distributed background of (pos-

itive) electric charge crystal = cube, lenght L, choose periodic boundary conditions

description of electrons as independently moving quantum objects in a cubevia Schrodinger equation:

2

2m

2

x2+

2

y2+

2

z2

k(r) = kk(r)

(note: does not include a term from the crystal potential)

With: k = energy of an electron with wave vector k

is a very good approximation for many solids (!)

Ansatz: k = eikr (i.e. planar wave)

Periodic boundary conditions: k(x + L) = k(x) analog for y, z directionkx = 0,2p/L,4p/L,...; ky, kz analog

due to finite crystal volume the possible values of k become discrete

Insert k = eikr in Schrodinger equation

k =2k2

2m

2(k2x + k2y + k

2z )

2m

p2

2m

each energy state can be occupied with 2 e (one spin up; one spin down)

113


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114 Chapter 6 Free electrons in a solid: The Fermi gas

ground state: electrons occupy states with lowest possible (total) energy(i.e. with as small |k| as possible in the case of a Fermi gas)

results (approximately)in a sphere in k-space (Fermi sphere)

radius kF (=Fermi wave vector) of the Fermi sphere is determined by

N = 2V

(2)24

3k3F

total number N of electrons = 2 (density of states) (volume of sphere);factor 2 due to spin

kF =

32N

V

13

n1

3

Energy of electrons at surface (=Fermi surface) of the Fermi sphere:

F =2k2F2m

=2

2m(32n)

2

3 (=Fermi energy)

Figure 6.1: In the ground state of a system of N free electrons the occupied orbitals ofthe system fill a sphere of radius kF, where F =

2k2F/2m is the energy of an electronhaving a wave vector k

F. [from Kittel, Einfuhrung in die Festkorperphysik (1999);

Abb.6.4].

velocity of the electrons on the Fermi surface:

vF =kFm

=

m(32n)

1

3 (= Fermi velocity)

frequently used: Fermi temperature TF =FkB


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115

Typical values:

electron density n: 1023/cm3 . . . 1 022/cm3

kF: 1.5 108/cm .. .0.8 108/cm (i.e. same order of magnitude as the reciprocal

lattice constant)

vF: 1.8 108 cm/s . . . 0.8 108 cm/s (i.e. 1% of the velocity of light in vacuum!)F: 10 eV . . . 1.6 eV

TF: 105 K . . . 2 104 K

Density of states in a Fermi gas:

D() =dN

d=

V

22

2m

2

32

1

2

Figure 6.2: Free electron gas: Density of states, Fermi distribution and distribution ofoccupied states (in red)

Remarks:

The Fermi surface plays a very important role for the electronic propertiesof metals.

real Fermi surfaces: almost spheric surface, if all e inside the 1. Brillouin-Zone; if not, the Fermi surface can become very complex


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Heat capacity of a Fermi gas

probability for the energy state being occupied with an electron:

f() =1

e()/kBT + 1(Fermi-Dirac distribution)

here, the electrochemical potential has to determined in such a way, that thetotal number N of electrons is the same at any temperature:

N =

0

d D()f()

at T = 0: = F

for T TF: F

Note: in most cases TF at R.T.; then F ; the quantities a nd F are

frequently used synonymical

Figure 6.3: Fermi-Dirac distribution function at various temperatures, for TF

F/kB = 50000 K. The results apply to a gas in three dimensions. The total numberof particles is constant, independent of temperature. The chemical potential at eachtemperature may be read off the graph as the energy at which f = 0.5. [from Kittel,Einfuhrung in die Festkorperphysik(1999); Abb.6.3].


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117

Figure 6.4: Variations with tem-

perature of the chemical poten-tial , for free electron Fermigases in one and three di-mensions. In common metals /F 0.01 at room tem-perature, so that is closelyequal to F. These curveswere calculated from series ex-pansions of the integral for thenumber of particles in the sys-tem. [from Kittel, Einfuhrung

in die Festkorperphysik (1999);Abb.6.8].

Energy of electrons at temperature T, referred to F:

U =

0

d ( F)D()f()

heat capacity

Cel =

dU

dT =

0 d ( F)

df

dTf()

for T TF:dfdT

only significant close to F D() D(F) and F. Then:

df

d(kBT)=

F(kBT)2

e(F)/kBT

[e(F)/kBT + 1]2

Cel = k2BT D(F)

F/kBT

dx x2ex

(ex + 1)2

with /kBT 1 ( )

Cel k2BT D()

dx x2ex

(ex + 1)2=

1

32D(F)k

2BT

mit D(F) =32

NkB

TF

Cel =1

22NkB

T

TF=: T

I.e. the heat capacity increases linearly with temperature.


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For comparison: classical treatment of free electrons

classical thermodynamics: each e has energy 32

kBT

Cel,class =32

N kB

Cel is reduced by factorT

T F(< 102) as compared to Cel,class. !

basically only a fraction TT F

of all e contributes to the heat capacity

electrons deep inside the Fermi sphere are not thermally excited (Pauli prin-ciple!)

Total heat capacity (electrons + lattice): C = T + AT2 (at low temperature)

Figure 6.5: Experimental heat capacity values for potassium, plotted as C/T vs. T

2

.(After W. H. Lien and N. E. Phillips.) [from Kittel, Einfuhrung in die Festkorperphysik(1999); Abb.6.9].


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119

Electric conductivity and Ohms law

classical: Force acting on electron: F = mv = e E in electric field E

semi-classical: p k

k = e E

without scattering processes: Fermi sphere is displaced from origin:

k(t) = k(0)e Et

flow of netto current

Figure 6.6: (a) The Fermi sphere encloses the the occupied electron orbitals in k-spacein the ground state of the electron gas. The net momentum is zero because for everyorbital k there is an occupied orbital at k. (b) Under the influence of a constant forceF acting in a time interval t every orbital has its k vector increased by k = F t/. Thisis equivalent to a displacement of the whole Fermi sphere by k. The total momentumis Nk, if there are N electrons present. The application of the force increases theenergy of the system by N(k)2/2m. [from Kittel, Einfuhrung in die Festkorperphysik(1999); Abb.6.10].

include consideration of scattering (average collision time ) of electrons byimpurities, phonons, etc.: Displacement k = e E / in the stationary state

- current density

j = nqv = ne2k

mne2

E

m E

with = ne2

m(conductivity) and = 1 = m

ne2(resitivity)


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Remark:

same result follows from classical consideration of free electrons

Difference between classical und quantum mechanical treatment: Nature ofscattering processes

classical: each e

can be scattered anywherequantum mechanical: - no scattering on ideal, periodic lattice

- no scattering into states which are already occupied(Pauli principle)

Figure 6.7: Relaxation of the displaced Fermi sphere

Figure 6.8: Temperature dependence of the resistivity (schematically)

Typical values:300 K: a few cm, mean free path approx. 106 cmT 0: 0 < 10

9 cm possible for high purity metals (if not superconducting);then: mean free path of the order of mm


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121

contribution to thermal conductivity

as for phonons:Thermal conductivity coefficient (Warmeleitzahl) K = 1

3Cv

mean free path determined by e on Fermi surface: = vF

with Cel = 132D(F)k2BT, D(F) = 23

nF

and F = 12mv2F

Kel =2

3

nk2BT

mv2FvF =

2nk2B

3mT

Remark:Due to different T-dependencies, we get at low temperature Kel > Kphonon

Wiedemann-Franz law

ratio

Kel

=

2k2B

T n

mne2

m

=2

3

kBe

2

T

Lorenz number:

L =KelT

=2

3

kBe2

= 2.45 108W/K2

independent of material properties , m, n!

Examples for metals (at T=0

C)Metal L 108W/K2

Ag 2.31Au 2.35Cd 2.42Cu 2.23

Very good agreement with theory!


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Energy bands

We consider now explicitly the periodic potential of the lattice. New results ascompared to the free electron gas will be:

Bragg reflection of the electrons, if k = G gaps in the energy spectrum E(k) of the electrons und hence the formation

of energy bands

new wave functions (Bloch functions) k(r)) = uk(r)eikr,

with uk(r) = uk(r +T)

(i.e. amplitude-modulated planar waves instead of simple planar waves)

crystal lattice is described by periodic potential U(r), i.e. displacement by trans-

lation vector T results in the same potential: U(r + T) = U(r)

Schrodinger equation:

2

2m

2

x2+

2

y2+

2

z2

k(r) + U(r) = kk(r)

Fourier expansion for U(r): U(r) =

G u Gei Gr

Ansatz for k(r): k(r) =

k Ckeikr

(i.e. general superposition of planar waves; k hast to be found consistently)

insert into Schrodinger equation

2m

k

k2Ckeikr +

G

k

UGVkei(k+G)r = k

k

Ckeikr

equation has to be valid for each Fourier component separately

2k2

2m k

Ck +

G

CkGUG = 0 (Hauptgleichung)

The Hauptgleichung is a system of algebraic equation, yieldsCk

s.

From the Hauptgleichung obviously follows:

If there exists a k which is a solution to the Hauptgleichung, then also the wavevector k G is a solution

built the solution k(r) in the following way:

k(r) =

G

CkGei(kG)r =

G

CkGei Gr

eikr = uk(r)eikr

with uk(r)eikr =

G CkGei Gr


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123

uk(r) obeys:

uk(r +T) =

G

CkGei G(r+T) =

G

CkGei Grei

G T = uk(r)

since ei G T

= 1

Hence:

k(r) = uk(r)eikr , mit uk(r +

T) = uk(r) (Bloch waves)

Remarks:

k is called crystal momentum or Quasiimpuls of an electron; for scat-tering events (e.g. electron-phonon): k + q = k + q + G

note: there is no scattering of electrons with an ideal lattice; Bloch waves fitto the lattice.

for crystal translations r k + T: k(r) k(r +T) = k(r) e

ik T (i.e.

generation of a phase factor eik T due to translation)

if U = 0 (corresponds to free electron gas):

Hauptgleichung:2k2

2m

Ck = 0

solution with CkG = 0 forG = 0

uk = const.

allowed values for k: again consider cube, length L, with periodic boundaryconditions

k = 2L

(nx, ny, nz), with integer nx, ny, nz as for the free electron gas

i.e. values for k are now discrete and only definied modulo reciprocal latticevector G

solution of the Hauptgleichung close to a Brillouin zone boundary, k G2

:

assumption: Fourier component of the potential UG shall be small as compared

to the kinetic energy of the electron

direction of motion shall be along x; furthermore, one assumes U(x) = U(x)(i.e. mirror symmetry)

U(x) = 2

G

UG cos Gx 2UG cos Gx


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Note: Wave vectors k = G have identical energy (follows from Hauptgleichung) consider only coeffizients Ck with k =

G2

and k = G2

in the Hauptgleichung.

1. Hauptgleichung for k = G2

With =2

2mG22

:

( )CG2

+ UGCG2

= 0

( )CG2

+ UGCG2

= 0

(i.e. Hauptgleichung is reduced to a set of two equations)

Solution exists for

det

UGUG = 0

( )2 = U2G

= UG =2

2m

G

2

2 UG

Energy gap 2U at zone boundary

For coefficients CG2

follows:

CG2

CG2

=

UG

= 1

(x) = eiGx/2 eiGx/2

(x) + = 2cos(Gx/2) for+

(x) = 2i sin(Gx/2) for

+ and are standing waves (as expected at the zone boundary).


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125

Figure 6.9: (a)Variation of potential energy of a conduction electron in the field of the

ion cores of a linear lattice. (b) Distribution of probability density in the lattice for|()|2 sin2 x/a; |(+)|2 cos2 x/a; and for a traveling wave. The wavefunction(+) piles up electron charge on the cores of the positives ions, thereby lowering thepotential energy in comparison with the average potential energy seen by a travelingwave. The wavefunction () piles up charge in the region between the ions, therebyraising the potential energy in comparison with that seen by a traveling wave. Thisfigure is the key to understanding the origin of the energy gap. [from Kittel, Einfuhrungin die Festkorperphysik(1999); Abb.7.3].

2. Hauptgleichung for k G2

Set Ckeikx + CkGei(kG)x(i.e. again only 2 coefficients Ck have to be taken into accound)

With = 2k2m

:

(k )Ck + UGCkG = 0

(kG )CkG + UGC = 0

2 (kG + k) + kGk U2G = 0

=1

2 (kG + k)1

4 (kG k)2 + U2G


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Figure 6.10: Solutions of (7.50)in the periodic zone scheme, inthe region of near a boundaryof the first Brillouin zone. Theunits are such that U = 0.45;G = 2 a nd 2/m = 1. Thefree electron curve is drawn forcomparison. The energy gapat the zone boundary is 0.90.The value of U has deliberatelybeen chosen large for this illus-tration, too large for the two-term approximation to be accu-rate. [from Kittel, Einfuhrung

in die Festkorperphysik (1999);Abb.7.9].

Figure 6.11: Ratio of the coeffi-cients in (x) = C(k) exp(ikx) +C(k G)exp

i(k G)x

as cal-

culated near the boundary ofthe first Brillouin zone. Onecomponent dominates as we

move away from the bound-ary. [from Kittel, Einfuhrungin die Festkorperphysik (1999);Abb.7.10].

Analogous splitting of k occurs at all zone boundaries

many energy bands, separated by an energy gap


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127

number of energy levels in one band:

1. Brillouin zone: kx, ky, kz = 0,2p/L,4p/L,.. . ,Np/L = p/a 2N electrons in one band if there are N primitivecells in a crystal (factor 2 due to spin)

higher Brillouin zones: analog

different representations of the energy bands:

1. (k) for < k < : extended zone scheme

2. (k) = (k + G) higher bands are transformed onto 1. Brillouin zone by subtraction of asuitable G.

= reduced zone scheme

3. periodic repetition of (2.) = periodic zone scheme

representation of the above mentioned schemes for the example of a linear lattice:

Figure 6.12: Three energy bands of a linear lattice plotted in the extended (Brillouin),reduced, and periodic zone scheme. [from Kittel, Einfuhrung in die Festkorperphysik(1999); Abb.9.4].


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Insulators, Semiconductors and Metals

in many solids: delocalized valence electrons

energy levels of an atom energy bands in a solid

energy gaps (band gaps) due to Bragg reflection at zone bondaries

size of the energy gap at highest band which is occupied by electrons, plus occu-pancy of this band determines whether a material is an isulator, semiconductor,semimetal or metal

highest fully occupied band: valence band (VB)

lowest, empty or partially filled band: conduction band (CB)

insulators and semiconductors:

at T = 0: all bands up to the valence band are fully occupied; conduction bandis empty

at T > 0: thermal excitation of electrons into the conduction band is possible

Figure 6.13: Schematic band structure of a semiconductor.

density of thermally excited electrons: n eEg/kBT (Eg: energy gap)

leaves holes in the valence band (behave like positiv electric charge)

insulators: Eg kBT ( 25meV at 300K)example diamond: Eg = 5.4 eV negligible amount of free charge carriers resistivity > 1014cm


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129

semiconductors: Eg kBTe.g. Si: 1.17 eV, Ge: 0.75 eV, GaAs: 1.52 eV

considerable amount of free charge carriers is present(e + holes)

resistivity: 102 . . . 109cm

charge carrier concentration n: 1013 . . . 1022/cm3

electric properties depend on precise shape E(k) of the va-lence band and the conduction band (e.g. effective mass ofthe electrons/holes, is in general = mel)

charge carrier concentration can be manipulated by substitu-tion with impurity atoms (doping)

if energy difference ECB (kCB ) EV B (kV B) at identical k isminimum, i.e. kCB = kV B : direct bandgapotherwise: indirect bandgap

example: Germanium: see next page

specific property of semiconductor interfaces or surfaces:

bending of the energy bands

accumulation (Anreicherung) or depletion (Verarmung) of charge carriers at in-terface

is used for the realization of (i) semiconductor devices (e.g. transistor)

or (ii) exotic states, e.g.:

2-dimensional electron gas

1-dimensional quantum wires

0-dimensional quantum dots (artificial atoms)

detailed discussion: lecture Exp. Phys. V + special lectures


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Figure 6.14: Calculated band structure of germanium, after C. Y. Fong. The generalfeatures are in good agreement with experiments. The four valence bands are shownin gray. The fine structure of the valence band edge is caused by spin-orbit splitting.The energy gap is indirect; the conduction band edge is at the point (2/a)( 1

2, 12

, 12

).The constant energy surface around this point are ellipsoidal. [from Kittel, Einfuhrungin die Festkorperphysik(1999); Abb.8.14].


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131

Metals and semimetals:

conduction band(s) only partially filled

charge carrier concentration n: 1021 . . . 1023 /cm3

resistivity: 1..100 cm

Figure 6.15: Carrier concentrationsfor metals, semimetals and semicon-ductors. The semiconductor rangemay be extended upward by increas-ing impurity concentration, and therange can be extended downward tomerge eventually with the insulator range[adapted from Kittel, Einfuhrung in dieFestkorperphysik(1999); Abb.8.1].

more detailed properties of the electrons: determined by shape of the Fermisurface


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construction of Fermi surfaces

1. free electrons + consideration of Bragg reflections

Figure 6.16: Brillouin zone of a square lattice in two dimensions.The circle shown is a surface of constant energy for free elec-trons; it will be the Fermi surface for some particular value ofthe electron concentration. The total area of the filled region ink space depends only on the electron concentration and is inde-pendent of the interaction of the electrons with the lattice. Theshape of the Fermi surface depend on the lattice interaction,and the shape will not be an exact circle in an actual lattice.The label within the section of the second and third zone referto Fig. 6.17 [from Kittel, Einfuhrung in die Festkorperphysik

(1999); Abb.9.6].

Figure 6.17: Mapping of the first, second and third Brillouin zones in the reduced zonescheme. The sections of the second zone in Fig. 6.16 are put together into a square bytranslation through an appropriate reciprocal lattice vector. A different G is needed oreach piece of a zone. [from Kittel, Einfuhrung in die Festkorperphysik(1999); Abb.9.7].

Figure 6.18: The free electron Fermi surface of Fig. 6.16, as viewed in the reduced zonescheme. The shaded areas represent the occupied electron states. Parts of the Fermisurface fall in the second, third and fourth zones. The fourth zone is not shown. Thefirst zone is entirely occupied. [from Kittel, Einfuhrung in die Festkorperphysik(1999);Abb.9.8].


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133

Figure 6.19: Harrison construction of free electron Fermi

surfaces on the second, third and fourth zones of a squarelattice. The Fermi surfaces enclose the entire first zone,which therefore is filled with electrons. [from Kittel,Einfuhrung in die Festkorperphysik(1999); Abb.9.11].

2. Quasi-free electrons (i.e. including weak periodic potential)

qualitative changes when Fermi surfaces are constructed:

energy gaps at zone boundaries

Fermi surface intersects zone boundary almost always perpendicularly

crystal potential causes rounding of sharp corners on the Fermi surface

volume enclosed by Fermi surface: depends on electron density n, but noton details of lattice the interaction (i.e. volume enclosed by Fermi surfaceshould remain unchanged under the above mentioned deformations)

Figure 6.20: Qualitative impression of the effect of a weak periodic crystal potential ofthe Fermi surface of Fig. 6.18 At one point on each Fermi surface we have shown thevector gradk. In the second zone the energy increases towards the interior of the figureand in the third zone the energy increases towards the exterior. The shaded regionsare filled with electrons and are lower in energy than the unshaded regions. We shallsee, that a Fermi surface like that of the third zone is electron-like, whereas one likethat of the second zone is hole like. It is said that the electrons sink, and holes float.[from Kittel, Einfuhrung in die Festkorperphysik(1999); Abb.9.10].


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Motion of electrons in a magnetic field: electron orbits, hole orbits andopen orbits

Figure 6.21: Motion in a magnetic field of the wave vector of an electron on the Fermisurface, in (a) and (b) for Fermi surfaces topologically equivalent to those of Fig. 6.20.In (a) the wave vector moves around the orbit in a clockwise direction; in (b) the wave

vector moves around the orbit in a counter-clockwise direction. The direction in (b) iswhat we expect for a free electron of charge e: the smaller k values have the lowerenergy, so that the filled electron states lie inside the Fermi surface. We call the orbitin (b) electronlike. The sense of the motion in a magnetic field is opposite in (a) to thatin (b), so that we refer to the orbit in (a) as holelike. A hole moves like a particle ofpositive charge e. In (c) for a rectangular zone we show the motion on an open orbit inthe periodic zone scheme. This is topologically intermediate between a hole orbit andan electron orbit. [from Kittel, Einfuhrung in die Festkorperphysik(1999); Abb.9.12].

Figure 6.22: (a) Vacant state at the corners of an almost-filled band, drawn intoa reduced zone scheme. (b) In the periodic zone scheme the various parts of theFermi surface are connected. Each circle formes a holelike orbit. The different circlesare entirely equivalent to each other, and the density state is that of a single circle.(The orbit needs not to be true circles: for the lattice shown it is only required thatthe orbits have fourfold symmetry.) [from Kittel, Einfuhrung in die Festkorperphysik(1999); Abb.9.13].


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135

Figure 6.23: Constant energy surface in the Brillouin zone of a simple cubic lattice,

for the assumed energy band k = 2(cos kxa + cos kya + cos kza). (a) Constantenergy surface = . The filled volume contains only one electron per primitive cell.(b) The same surface exhibited in the periodic zone scheme. The connectivity of theorbits is clearly shown. can you find electron, hole, and open orbits for motion in amagnetic field Bz? (A. Sommerfeld and H. A. Bethe.) [from Kittel, Einfuhrung in dieFestkorperphysik(1999); Abb.9.15].

Figure 6.24: Dogs bone orbit of an elec-tron on the fermi surface of copper orgold in a magnetic field. This orbitis classified as holelike because the en-ergy increases towards the interior of theorbit. [from Kittel, Einfuhrung in dieFestkorperphysik(1999); Abb.9.30].


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Real Fermi surfaces of different metals

Figure 6.25: siehe: http://www.phys.ufl.edu/fermisurface [from Kleiner, Lecture Notes(2000); Abb.108.01].


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137

Notes on superconductivity

Cooper pairing in superconductors

T > Tc: uncorrelated electrons (or holes)

Figure 6.26: [from Kleiner, Lecture Notes (2000); Abb.109.01].

T < Tc: 2 electrons with wavevector k and spin and k, form a spin-singlet Cooperpair

correlation of all pairs common center of mass momentum



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Makroscopic wave function

Figure 6.28: Single electron [from Kleiner,Lecture Notes (2000); Abb.110.01].

Figure 6.29: Two electrons [from Kleiner,Lecture Notes (2000); Abb.110.02].

Figure 6.30: Uncorrelated electrons[from Kleiner, Lecture Notes (2000);Abb.110.03].

Figure 6.31: Cooper pairs [from Kleiner,Lecture Notes (2000); Abb.110.04].


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139

The resistance of normal metals: attenuation of a circulating current in thewave picture


R = 0 in superconductors in the picture of the Fermi sphere:

T > Tc single electrons scatter independently Fermi sphere relaxes fast


T < Tc paired electrons scatter only about their common center Fermi sphere does not relax persistend current, supercon-ductivity



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Fermi Gas_ Free electrons in a solid

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