SHANGHAI JIAO TONG UNIVERSITY LECTURE

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1 SHANGHAI JIAO TONG UNIVERSITY LECTURE Athoy J. Leggett Departmet of Physics Uiversity of Illiois at Urbaa-Champaig, USA ad Director, Ceter for Complex Physics Shaghai Jiao Tog Uiversity

2 SJTU 9.1 Lecture 9 Dirty Supercoductors Experimetal fact: Quite strog omagetic disorder (e.g. alloyig) does little harm to supercoductivity, while eve tiy amouts (~ a few ppm) of magetic impurities suppress it completely. Why?

3 A. Nomagetic Disorder sigleelectro H 0 = H = H 0 + V Assume: k f l 1 (but possibly l ξ 0 ) i 2 pƹ i 2m + U r i iter-electro iteractio potetial due to atomic cores spi-idepedet SJTU 9.2 Eigestates of H 0 are of form ψ r, σ = φ r σ, σ, σ (, ) with eergy ε where φ r is very complicated. However, ote that average desity of states d dε 2 σ δ ε ε is much the same (for k f l 1) as i origial (crystallie) case. Crucial poit: sice H 0 is ivariat uder time-reversal (, φ r φ r ത ), the if state, is a eigestate of H 0 with eergy ε, the ത, is also eigestate of H 0 with eergy ε. Note: φ ത r may or may ot be idetical to φ r, i.e. φ r may or may ot be real, (does t matter!)

4 Recall that i free space, BCS groud state was Ψ = k Φ k Φ k state vector i occupatio space of k,, k, SJTU 9.3 So, replace k, Ψ = k by Φ, ത ad geeralize BCS asatz: Assume as i free-space case that at T=0 0,1, 1,0 are irrelevat, the Φ = u 0,0 + v 1,1 u 2 + v 2 = 1 i.e. pair i time-reversed states KE is idetical to free-space case with k : 2 T = 2 ε v For the PE, as i the free-space case, we eed to calculate the matrix elemet ψ f V ψ i with ψ i, ത occupied;, ത empty ψ f, ത empty;, ത occupied For a δ-fuctio iteractio V r i r j = V 0 δ r i r j, this is equal to V 0 u v u v න φ Φ k state vector i occupatio space of,, ത, r φ ത r φ r φ ത r dr But sice φ ത ad v s) r = φ r (etc.), this ca be rewritte (regroupig the u s V 0 u v u v න φ r 2 φ r 2 dr For ormalizatio i uit volume the itegral, thought ot exactly equal to 1, is very close to it, so V V 0 u v u v, V 0 F F u v

5 SJTU 9.4 The subsequet algebra goes through exactly as i the freespace case, ad we ed up with the gap equatio Δ = V 0 2E Δ E ε 2 + Δ Assumig Δ = Δ = cost ad turig the Σ ito dε: ε c ρ ε dε 1 = V 0 න ε c 2 ε 2 + Δ Sice ρ ε is (almost) the same as for the origial free-space case, this is (almost) the origial BCS gap equatio ad has the same solutio Thus, Δ = 2ε c e 1ΤN 0 V N thermodyamics almost uaffected by alloyig (i zero magetic field, for k F l 1) (we have simply shuffled the origial plae-wave states aroud ) d dε ε=εf Similar results at o-zero T, e.g. χ T χ = Y(T) (Yoshida fuctio) (sice, ത still eigestates of spi) However, calculatio of ormal desity does ot go through ( sigle-particle eergy eigestates, σ are ot eigestates of mometum)

6 SJTU 9.5 OK, so which properties are affected by alloyig? (a) Pair radius Recall: i pure metal, with pairs at rest, pair wave fuctio F is idepedet of COM variable R, ad as fuctio of relative coordiates r is give (at T = 0) by F r = F k exp(ik r) F k /2E k E k (ε 2 k + 2 ) 1/2 k The rage of F i ε is ~, hece i k it is /ħv F, so by idetermiacy priciple k r ~1 we have, r~ħv F /π ξ 0 ( pair radius ) (Techically, F r ~ exp r /ξ, ξ ~ξ 0 ) I the dirty system, pair wave fuctio F is give by F r, r = u v φ r φ ത r u v φ R + r/2 φ ത (R r /2) so is techically a fuctio also of COM variable R. So let's defie F r F R + r/2, R r /2 where average is over R What is depedece of F r o relative coordiate r? Rewrite

7 SJTU 9.6 F r = ( /2E )φ R + r/2 φ (R r /2) sice φ ത φ Ituitive (semiclassical) argumet: F r will drop below its r = 0 value as soo as differece i phase of the product φ R + r/2 φ (R r /2) for differet becomes ~2π. Semiclassically, a wave packet with spread i eergy ε will be dephased (idetermiacy!) i a time t~ħ/ ε. I our case ε~, so dephasig time is t~ħ/ How far does the packet travel i t? I pure metal, r~v F t so r~ħv F / leadig to r p ~ħv F / as above. But i a dirty metal (l ξ 0 ) motio is diffusive, ad we have r 2 ~D t D~ 1 3 v Fl so puttig t~ħ/ r p ~(ħv F l/ ) 1/2 or sice ξ 0 ~ħv F / r p ~(ξ 0 l) 1/2 (dirty limit) i.e. pair radius decreases by factor (l/ξ 0 ) 1/2 (which ca be 1). (Also i limit T T c, i.e. ρ s dirty (T)~(l/ξ 0 ) 1/2 ρ s clea (T)).

8 (b) Superfluid desity The superfluid desity ρ s ca be defied i two apparetly differet ways: (1) as the coefficiet of the depedece of the GL free eergy o bedig of the GL order parameter, for A = 0, SJTU 9.7 F = 1 2 ρ sv s 2 with v s ħ 2m ( φ) Ψ~ Ψ expiφ for A = 0 (2) as the (diamagetic) respose of the curret to a weak trasverse EM vector potetial, J = ρ s ( e m )2 A (I pure case at T = 0, ρ s = m so recover Lodo equatio) To see that the two defiitios are equivalet, cosider a thi supercoductig rig i a weak circumferetial (i.e. trasverse) vector potetial A: the must geeralize defiitio of v s to v s = ħ 2eA ( φ 2m ħ ) If A is weak, SVBC eforces φ = cost. so v s = e m A F = 1 2 ρ s( e m )2 A 2 but quite geerally, J = ( F)/ A J = ρ s ( e m )2 A i accordace with secod defiitio. J A

9 SJTU 9.8 To estimate effects of disorder o ρ s (at T = 0), proceed as follows: Although we have up to ow assumed that whe A is a fuctio of r the curret J r is related to A r by the Lodo relatio e 2 J r = ρ s A(r) m this is actually ot quite right. I fact, the more correct formula is J r = න Κ r, r A r dr For the pure case the rage of Κ r, r = Κ r r is ~ the pair radius ξ 0. I fact the exact formula i BCS theory is close to Pippard's origial guess, Κ r r ~ 3e2 4πmξ 0 1 r r 2 exp r r /ξ 0 If A(r) is slowly varyig over distaces ~ξ 0, this gives back the Lodo relatio J r A(r) න Κ r r dr = e2 m A(r) Now, if l ξ 0, expect ituitively that iduced curret falls off as e r r /l, i.e. Κ r r ~ (prefactor x)exp r r ( 1 ξ l ) (Pippard)

10 SJTU 9.9 Hece, itegral is reduced by factor 1/ξ 0 1/ξ 0 + 1/l ξ 0 /l ~l/ξ 0, for l ξ 0. Hece i dirty limit, ρ s dirty ~(l/ξ 0 )ρ s clea ρ s clea (also i limit T T c ).

11 Effects of disorder i laguage of GL theory (T Tc): F{Ψ(r): T} = (α 0 (T T c ) Ψ β 0 Ψ 4 + γ 0 Ψ 2 ) SJTU 9.10 I dirty limit (l ξ 0 but still k F l 1) but, Recall: Thus, α 0 dirty α0 clea β 0 dirty β0 clea γ 0 dirty λ ξ 0 so, Ψ dirty T Ψ clea (T) γ 0 clea γ0 clea ξ(t) (γ 0 /α 0 (T T c )) 1/2 λ(t) (γ 0 Ψ(T) 2 ) 1/2 ξ dirty (T) (l/ξ 0 ) 1/2 ξ clea (T) λ dirty (T) (ξ 0 /l) 1/2 λ clea (T) κ dirty (λ/ξ) dirty (ξ 0 /l)κ clea κ clea alloyig makes system much more type-ii. i particular, dirty H c1 λdirty 2 dirty H c2 ξdirty 2 Hclea c1 Hclea c2

12 B. Magetic disorder SJTU 9.11 Now we have H^ ^ = H 0 + V^ but ow ^ H 0 = (p^i/2m) + U(r i : σ i ) i so ow TRI (Time Reversal Ivariace) is broke, ad state, σ (whe φ ത (r) φ (r)) is o loger degeerate with, σ, ideed is i geeral ot eve a eergy eigestate. Two obvious proposals for GS: sigle-electro Iter-electro iteractio (a) Pair i exact eigefuctios of sigle-particle Hamiltoia, i.e. if exact eigestates of H^0 for σ = are deoted φ m, pair off with some m ( ). The KE is much the same as i pure (BCS) case. However, V V 0 drφ (r)φ m (r)φ m (r)φ (r) mm ad sice we o loger have φ ത (r) = φ (r), (etc.) the itegral is oscillatig ad hece very small. This scheme is usually very eergetically disadvatageous.

13 SJTU 9.12 (b) Cotiue to pair i time-reversed states, eve though these are o loger eigestates of H 0. How much extra eergy does this cost? Suppose lifetime for differet scatterig of ad is τ s ħγ s 1 the by idetermiacy priciple extra eergy ecessary to keep state of the time-reverse of that of is Γ s extra eergy required Γ s o. of perturbed states Γ s (Γ s d/dε) Γ s 2 d/dε. Γ s O the other had, this scheme keeps the whole of the pure-state codesatio eergy, which is ε (pure) E 2 Δ0 ( d ) cod dε (Δ 0 gap of pure system) ε F Hece we expect that this scheme will give a eergy lower tha (pure) the N state provided E > Γ 2 d/dε, i.e. coditio for cod s magetic impurities to suppress supercoductivity completely is Γ s Δ 0 which is equivalet to l s ξ 0. (i.e. mea free path agaist spidepedet scatterig (pure metal) pair radius). Actually, exact calculatio (Abrikosov-Gor'kov) shows that at T = 0 coditio is i fact simply Γ s > Δ 0 ).