MIT OpenCourseWarehttp://ocw.mit.edu 8.512 Theory of Solids II Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.� � � � 18.512 Theory of Solids II Problem Set 7 Due April 6, 2009 1. Opticalconductivityofdisorderedsuperconductors.Followingourdiscussionofdisorderedmetals,theopticalconductivityofadisorderedsuperconductorisgivenbytheKuboformula(whichiseasilyderivedbyconsideringtherateofabsorptionofelectromagneticradiation):π 1� σ(q → 0, ω) =ω Ω|�0| drjP (r)|n�|2σ(ω − (En − E0)) (1)x n where Ω is the volume. The paramagnetic current operator is written in the exacteigenstaterepresentationas:drjxP (r) =e vα,β c†, (2)β,σcα,σ α,β,σ andvα,β =1drφβα �x φα (3)m i isthevelocitymatrixelementsbetweenexacteigenstatesoftheHamiltonianH1,whichdescribesfreefermionswithadisorderedpotentialH1φα =εαφβ . (4)In Eq. (1), |0� and |n� are the ground and excited states of the BCS mean fieldHamiltonianinthepresenceofdisorder.(a) UsingtheBogolinbovtransformation,showthatσ(q → 0, ω) =eω 2 Ωπ � (uαvβ − vαuβ )2|vαβ |2σ(ω − Eα − Eβ ) (5)αβ where� � � � � � � � � � � 2u 2 α =121 +Eξαα (6)vα 2 =121− Eξαα (7)Eα = ξα 2 + ∆2 , ξα =εα − µ . (8)Bydefining1� f(ξ, ξ�) =Ω|vαβ|2σ(ξ − ξα)σ(ξ� − ξβ ) , (9)αβ showthatEq. (5)canbewrittenase2 �� ∞ � ∞ σ(q → 0, ω) =ω dξ dξ�(uv� − vu�)2f(ξ, ξ�)δ(ω − E − E�) (10)−∞ −∞ wheretherelationbetweenu,v,E andξ isgivenbyEqs. (6–8).(b) Showthat(uv� − vu�)2 =1 ξξ� ∆2 . (11)21− EE� − EE� (c) Notethatf(ξ, ξ�)dependsonlyonthenormalstateproperties.Indeeditappearedinourtreatmentofdisorderedmetals. Byfactorizingtheimpurityaverage,arguethatf(ξ, ξ�)can be approximatedbya constantforsmall |ξ| and |ξ�| , i.e. forenergiesneartheFermilevel. (Moreaccurately,|ξ − ξ�| � 1 . Amuseyourselfbyτ tryingtopointoutatwhatstepintheargumentwasthiscondition imposed.)Bytakingthelimit∆ 0,showhowtheexpressionwederivedforthenormal→ state conductivity σN can be recovered. (Note how the spin sum is magicallyincluded.)(d) ShowthatEq. (10)simplifiesto1∞ ∞ ∆2 σ(q → 0, ω) =σN ω 0 dξ 0 dξ� 1− EE� σ(ω − E − E�) . (12)ThisisknownastheMattis-Bardeenformula. Bychangingtheintegrationvariablefromξ toE,Eq. (12)reducestoaone-dimensionalintegralwhichcanbe3donenumericallyorbymathematics. Sketchσ/σN andcommentonthekeyfeatures. WeemphasizethattheMattis-Bardeenformulaisvalidonlyfordisorderedsuperconductorsandfor∆τ �
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