Superconductivity Discovery & Basic Phenomenon • 1911 – H. Kamerlingh-Onnes found that the resistivity of Hg drops abruptly to ρ → 0 \rho \to 0 ρ → 0 T < e m > C ( 4.15 K ) . T<em>C \;(4.15\,\text K). T < e m > C ( 4.15 K ) . T < / e m > C T</em>C T < / e m > C T C T_C T C
Critical Field • External field destroys superconductivity; the threshold is the critical field H < e m > C ( T ) . H<em>C (T). H < e m > C ( T ) . H < / e m > C ( T ) = H < e m > C ( 0 ) [ 1 − ( T / T < / e m > C ) 2 ] . H</em>C (T)=H<em>C(0) \,[1-(T/T</em>C)^2]. H < / e m > C ( T ) = H < e m > C ( 0 ) [ 1 − ( T / T < / e m > C ) 2 ] . H C H_C H C
Meissner–Ochsenfeld Effect (1933) • A bulk super-conductor expels magnetic flux: B = 0 \mathbf B=0 B = 0 C and H C HC H C
Type-I vs Type-II Property
Type I (soft)
Type II (hard)
Typical composition
Pure metals (Pb, Hg, Sn)
Alloys/oxides (NbTi, YBCO)
Fields
Single H C H_C H C
to 100 mT)
Two criticals H C 1 , H C 2 H{C1},H{C2} H C 1 , H C 2 H C 1 C 2 H{C1}{C2} H C 1 C 2
Meissner behaviour
Complete
Partial (flux lines penetrate as vortices)
T C T_C T C
< 10 K
Up to 135 K for cuprates
London Equations ∂ J s ∂ t = n s e 2 m E \dfrac{\partial \mathbf Js}{\partial t}=\dfrac{ns e^2}{m}\;\mathbf E ∂ t ∂ J s = m n s e 2 E
∇ × J s = − n s e 2 m B \nabla\times\mathbf Js=-\dfrac{ns e^2}{m}\;\mathbf B ∇ × J s = − m n s e 2 B London penetration depth \lambdaL=\sqrt{\dfrac{m}{\mu0 ns e^2}},\;\;B(x)=B0 e^{-x/\lambdaL}. • Temperature dependence λ ( T ) = λ ( 0 ) 1 − ( T / T C ) 4 . \lambda (T)=\dfrac{\lambda (0)}{\sqrt{1-(T/TC)^4}}. λ ( T ) = 1 − ( T / T C ) 4 λ ( 0 ) .
Coherence Length • Mean size of a Cooper pair: ξ ≈ ℏ v F π Δ ( 0 ) . \xi\approx \dfrac{\hbar v_F}{\pi \Delta(0)}. ξ ≈ π Δ ( 0 ) ℏ v F . κ = λ / ξ \kappa=\lambda/\xi κ = λ / ξ
BCS Theory (Bardeen-Cooper-Schrieffer, 1957) • Electron–phonon interaction gives effective attraction (<0.1\;\text eV). Cooper pairs of opposite k , ↑ ↓ \mathbf k,\,\uparrow\downarrow k , ↑↓ Δ ( 0 ) = 1.76 k B T C , Δ ( T ) → 0 at T C . \Delta (0)=1.76\,kB TC,\quad \Delta(T)\to 0 \text{ at }TC. Δ ( 0 ) = 1.76 k B T C , Δ ( T ) → 0 at T C . Φ 0 = h / 2 e = 2.07 × 10 − 15 Wb . \Phi0=h/2e=2.07\times10^{-15}\,\text{Wb}. Φ0 = h /2 e = 2.07 × 1 0 − 15 Wb . T C ∝ M − α , α ≈ 0.5. T_C\propto M^{-\alpha},\;\alpha\approx 0.5. T C ∝ M − α , α ≈ 0.5.
Thermodynamic Properties • Entropy S s n Ssn S s n T C TC T C Δ C = 1.43 C n ( T C ) . \Delta C=1.43\,Cn(TC). Δ C = 1.43 C n ( T C ) . T C TC T C
High-T e (Cu-oxide) Superconductors
Compound
T C T_C T C
La${1.85}$Ba${0.15}$CuO$_4$
36
YBa$2$Cu$3$O$_{7-\delta}$
92
Tl$2$Ba$2$Ca$2$Cu$3$O$_{10}$
125
• Highly anisotropic layered perovskites; short ξ \xi ξ H C 2 H_{C2} H C 2
Applications • Zero-loss power cables; MRI & NMR magnets; Maglev transport; SQUID sensors; fast cryotron switches; high-field laboratory magnets (50 T).
Magnetism Fundamental Quantities • Magnetization M = magnetic moment volume . \mathbf M=\dfrac{\text{magnetic moment}}{\text{volume}}. M = volume magnetic moment . B = μ 0 ( H + M ) = μ H . \mathbf B=\mu0(\mathbf H+\mathbf M)=\mu\mathbf H. B = μ 0 ( H + M ) = μ H .
Magnetic Classes Class
χ m \chi_m χ m
Key features
Diamagnetic
–, small (≈−10⁻⁵)
All paired electrons; field expelled weakly; χ \chi χ
Paramagnetic
+, small (10⁻⁴)
Unpaired moments; random; Curie law χ = C / T . \chi=C/T. χ = C / T .
Ferromagnetic
+, large (10²–10⁵)
Spontaneous alignment in domains; T C T_C T C
Antiferromagnetic
small; + or –
Adjacent spins antiparallel; Néel temp. T_N.$
Ferrimagnetic
moderate +
Unequal antiparallel sub-lattices (e.g., Fe$3$O$4$).
Langevin Diamagnetism \chi = -\dfrac{\mu0 n e^2}{4m}\sum rj^2( t e m p e r a t u r e − i n d e p e n d e n t , u n i v e r s a l ) . < / e m > < / p > < h 5 i d = " 46302 d a 0 − 0852 − 4155 − a 729 − 8 d 00 c 1653433 " d a t a − t o c − i d = " 46302 d a 0 − 0852 − 4155 − a 729 − 8 d 00 c 1653433 " c o l l a p s e d = " f a l s e " s e o l e v e l m i g r a t e d = " t r u e " > < e m > L a n g e v i n P a r a m a g n e t i s m < / e m > < / h 5 > < p > < e m > A v e r a g e m o m e n t (temperature-independent, universal).</em></p><h5 id="46302da0-0852-4155-a729-8d00c1653433" data-toc-id="46302da0-0852-4155-a729-8d00c1653433" collapsed="false" seolevelmigrated="true"><em>Langevin Paramagnetism</em></h5><p><em>Average moment ( t e m p er a t u r e − in d e p e n d e n t , u ni v er s a l ) . < / e m >< / p >< h 5 i d = "46302 d a 0 − 0852 − 4155 − a 729 − 8 d 00 c 1653433" d a t a − t oc − i d = "46302 d a 0 − 0852 − 4155 − a 729 − 8 d 00 c 1653433" co l l a p se d = " f a l se " seo l e v e l mi g r a t e d = " t r u e " >< e m > L an g e v in P a r ama g n e t i s m < / e m >< / h 5 >< p >< e m > A v er a g e m o m e n t < b r > F o r s m a l l x : <br>For small x: < b r > F or s ma l l x :
Weiss Theory & Curie–Weiss Law (Ferromagnets) Internal molecular field BE=\lambda M.A b o v e Above A b o v e : : : D o m a i n s + h y s t e r e s i s : c o e r c i v i t y Domains + hysteresis: coercivity D o main s + h y s t er es i s : coer c i v i t y , r e m a n e n c e , remanence , r e man e n ce
Super-paramagnetism Nanocrystals (< 100 nm): single-domain particles; thermal flipping gives \chi=C/Td e s p i t e l a r g e m o m e n t s . < / e m > < / p > < d i v d a t a − t y p e = " h o r i z o n t a l R u l e " > < h r > < / d i v > < h 4 i d = " 808 a 8432 − 814 c − 4307 − a 5 c 5 − e 7 d 42 b 40287 f " d a t a − t o c − i d = " 808 a 8432 − 814 c − 4307 − a 5 c 5 − e 7 d 42 b 40287 f " c o l l a p s e d = " f a l s e " s e o l e v e l m i g r a t e d = " t r u e " > < e m > D i e l e c t r i c s < / e m > < / h 4 > < h 5 i d = " c e 68 b 1 a f − f d 13 − 4661 − a c b a − c 3 d c 7 b b c 23 f 3 " d a t a − t o c − i d = " c e 68 b 1 a f − f d 13 − 4661 − a c b a − c 3 d c 7 b b c 23 f 3 " c o l l a p s e d = " f a l s e " s e o l e v e l m i g r a t e d = " t r u e " > < e m > K e y R e l a t i o n s < / e m > < / h 5 > < p > < e m > • P o l a r i z a t i o n despite large moments.</em></p><div data-type="horizontalRule"><hr></div><h4 id="808a8432-814c-4307-a5c5-e7d42b40287f" data-toc-id="808a8432-814c-4307-a5c5-e7d42b40287f" collapsed="false" seolevelmigrated="true"><em>Dielectrics</em></h4><h5 id="ce68b1af-fd13-4661-acba-c3dc7bbc23f3" data-toc-id="ce68b1af-fd13-4661-acba-c3dc7bbc23f3" collapsed="false" seolevelmigrated="true"><em>Key Relations</em></h5><p><em>• Polarization d es p i t e l a r g e m o m e n t s . < / e m >< / p >< d i v d a t a − t y p e = " h or i z o n t a l R u l e " >< h r >< / d i v >< h 4 i d = "808 a 8432 − 814 c − 4307 − a 5 c 5 − e 7 d 42 b 40287 f " d a t a − t oc − i d = "808 a 8432 − 814 c − 4307 − a 5 c 5 − e 7 d 42 b 40287 f " co l l a p se d = " f a l se " seo l e v e l mi g r a t e d = " t r u e " >< e m > D i e l ec t r i cs < / e m >< / h 4 >< h 5 i d = " ce 68 b 1 a f − f d 13 − 4661 − a c ba − c 3 d c 7 bb c 23 f 3" d a t a − t oc − i d = " ce 68 b 1 a f − f d 13 − 4661 − a c ba − c 3 d c 7 bb c 23 f 3" co l l a p se d = " f a l se " seo l e v e l mi g r a t e d = " t r u e " >< e m > K ey R e l a t i o n s < / e m >< / h 5 >< p >< e m > • P o l a r i z a t i o n • D i s p l a c e m e n t • Displacement • D i s pl a ce m e n t w i t h with w i t h < / e m > < / p > < h 5 i d = " 39 b 7 b 861 − 2 b d 5 − 4175 − b b 28 − 1711 f 2736559 " d a t a − t o c − i d = " 39 b 7 b 861 − 2 b d 5 − 4175 − b b 28 − 1711 f 2736559 " c o l l a p s e d = " f a l s e " s e o l e v e l m i g r a t e d = " t r u e " > < e m > P o l a r i z a t i o n M e c h a n i s m s < / e m > < / h 5 > < o l > < l i > < p > < s t r o n g > < e m > E l e c t r o n i c < / e m > < / s t r o n g > < e m > </em></p><h5 id="39b7b861-2bd5-4175-bb28-1711f2736559" data-toc-id="39b7b861-2bd5-4175-bb28-1711f2736559" collapsed="false" seolevelmigrated="true"><em>Polarization Mechanisms</em></h5><ol><li><p><strong><em>Electronic</em></strong><em> < / e m >< / p >< h 5 i d = "39 b 7 b 861 − 2 b d 5 − 4175 − bb 28 − 1711 f 2736559" d a t a − t oc − i d = "39 b 7 b 861 − 2 b d 5 − 4175 − bb 28 − 1711 f 2736559" co l l a p se d = " f a l se " seo l e v e l mi g r a t e d = " t r u e " >< e m > P o l a r i z a t i o n M ec hani s m s < / e m >< / h 5 >< o l >< l i >< p >< s t r o n g >< e m > E l ec t r o ni c < / e m >< / s t r o n g >< e m > ( ≈ 10 − 40 F ⋅ m 2 ; 10 − 15 s ) . < / e m > < / p > < / l i > < l i > < p > < s t r o n g > < e m > I o n i c < / e m > < / s t r o n g > < e m > ( v i b r a t i o n a l ) (≈10⁻⁴⁰ F·m²; 10⁻¹⁵ s).</em></p></li><li><p><strong><em>Ionic</em></strong><em> (vibrational) ( ≈ 1 0 − 40 F ⋅ m 2 ; 1 0 − 15 s ) . < / e m >< / p >< / l i >< l i >< p >< s t r o n g >< e m > I o ni c < / e m >< / s t r o n g >< e m > ( v ib r a t i o na l ) ( 10 − 38 ; 10 − 13 s ) . < / e m > < / p > < / l i > < l i > < p > < s t r o n g > < e m > O r i e n t a t i o n a l < / e m > < / s t r o n g > < e m > (~10⁻³⁸; 10⁻¹³ s).</em></p></li><li><p><strong><em>Orientational</em></strong><em> ( 1 0 − 38 ; 1 0 − 13 s ) . < / e m >< / p >< / l i >< l i >< p >< s t r o n g >< e m > O r i e n t a t i o na l < / e m >< / s t r o n g >< e m > ( d i p o l a r ; m s r a n g e ) . < / e m > < / p > < / l i > < l i > < p > < s t r o n g > < e m > S p a c e − c h a r g e / i n t e r f a c i a l < / e m > < / s t r o n g > < e m > ( s l o w , k H z ) . < / e m > < / p > < / l i > < l i > < p > < s t r o n g > < e m > S p o n t a n e o u s ( f e r r o e l e c t r i c ) < / e m > < / s t r o n g > < e m > ; d o m a i n s ; C u r i e t e m p e r a t u r e . < / e m > < / p > < / l i > < / o l > < h 5 i d = " 800 b f 83 a − d 329 − 4326 − 8201 − 3 f 5 c b 9 c 6 a 510 " d a t a − t o c − i d = " 800 b f 83 a − d 329 − 4326 − 8201 − 3 f 5 c b 9 c 6 a 510 " c o l l a p s e d = " f a l s e " s e o l e v e l m i g r a t e d = " t r u e " > < e m > C l a u s i u s – M o s s o t t i < / e m > < / h 5 > < p > < e m > (dipolar; ms range).</em></p></li><li><p><strong><em>Space-charge / interfacial</em></strong><em> (slow, kHz).</em></p></li><li><p><strong><em>Spontaneous (ferroelectric)</em></strong><em>; domains; Curie temperature.</em></p></li></ol><h5 id="800bf83a-d329-4326-8201-3f5cb9c6a510" data-toc-id="800bf83a-d329-4326-8201-3f5cb9c6a510" collapsed="false" seolevelmigrated="true"><em>Clausius–Mossotti</em></h5><p><em> ( d i p o l a r ; m sr an g e ) . < / e m >< / p >< / l i >< l i >< p >< s t r o n g >< e m > S p a ce − c ha r g e / in t er f a c ia l < / e m >< / s t r o n g >< e m > ( s l o w , k H z ) . < / e m >< / p >< / l i >< l i >< p >< s t r o n g >< e m > S p o n t an eo u s ( f er r oe l ec t r i c ) < / e m >< / s t r o n g >< e m > ; d o main s ; C u r i e t e m p er a t u r e . < / e m >< / p >< / l i >< / o l >< h 5 i d = "800 b f 83 a − d 329 − 4326 − 8201 − 3 f 5 c b 9 c 6 a 510" d a t a − t oc − i d = "800 b f 83 a − d 329 − 4326 − 8201 − 3 f 5 c b 9 c 6 a 510" co l l a p se d = " f a l se " seo l e v e l mi g r a t e d = " t r u e " >< e m > C l a u s i u s – M osso tt i < / e m >< / h 5 >< p >< e m > < / e m > < / p > < d i v d a t a − t y p e = " h o r i z o n t a l R u l e " > < h r > < / d i v > < h 4 i d = " 8 d a 8 e d b 5 − d a d d − 4 c 40 − 98 e e − 35075 d 9 f 3189 " d a t a − t o c − i d = " 8 d a 8 e d b 5 − d a d d − 4 c 40 − 98 e e − 35075 d 9 f 3189 " c o l l a p s e d = " f a l s e " s e o l e v e l m i g r a t e d = " t r u e " > < e m > K e y F o r m u l a e S h e e t < / e m > < / h 4 > < p > < e m > • C r i t i c a l f i e l d : </em></p><div data-type="horizontalRule"><hr></div><h4 id="8da8edb5-dadd-4c40-98ee-35075d9f3189" data-toc-id="8da8edb5-dadd-4c40-98ee-35075d9f3189" collapsed="false" seolevelmigrated="true"><em>Key Formulae Sheet</em></h4><p><em>• Critical field: < / e m >< / p >< d i v d a t a − t y p e = " h or i z o n t a l R u l e " >< h r >< / d i v >< h 4 i d = "8 d a 8 e d b 5 − d a dd − 4 c 40 − 98 ee − 35075 d 9 f 3189" d a t a − t oc − i d = "8 d a 8 e d b 5 − d a dd − 4 c 40 − 98 ee − 35075 d 9 f 3189" co l l a p se d = " f a l se " seo l e v e l mi g r a t e d = " t r u e " >< e m > K ey F or m u l a e S h ee t < / e m >< / h 4 >< p >< e m > • C r i t i c a l f i e l d : • L o n d o n d e p t h • London depth • L o n d o n d e pt h < b r > • F l u x q u a n t u m <br>• Flux quantum < b r > • F l ux q u an t u m • B C S g a p • BCS gap • B C S g a p • C u r i e c o n s t a n t • Curie constant • C u r i eco n s t an t • C l a u s i u s – M o s o t t i • Clausius–Mosotti • C l a u s i u s – M oso tt i