UvA-VU Master Course: Advanced Solid State Physics

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UvA-VU Mste Couse: Advnced Solid Stte Physics

phenomen (week 8, AdV) Supeconductivity (week 9&0,

1 UvA-VU Mste Couse: Advnced Solid Stte Physics Contents in 005: Diffction fom peiodic stuctues (week 6, AdV) Electonic bnd stuctue of solids (week 7, AdV) Motion of electons nd tnspot phenomen (week 8, AdV) Supeconductivity (week 9&0, RW) Mgnetism (week &,JB) Anne de Visse Rinke Wijngden Jügen Buschow

2 Litetue, softwe nd homewok The couse is bsed on the book: H. Ibch nd H. Lüth: Solid Stte Physics 3 d edition (Spinge-Velg, Belin, 003) ISBN X See lso: N.W. Ashcoft nd N.D. Memin: Solid Stte Physics (Sundes College Publ.) ISBN Compute simultions fom n essentil pt of the couse: R.H. Silsbee nd J. Däge: Simultions fo Solid Stte Physics (Cmbidge Univesity Pess, Cmbidge 997) ISBN Softwe (feewe): Homewok execises will be distibuted thoughout the couse Completing the couse gives 6 ECTS ~ 6 x 8 hous

3 Couse : Diffction fom peiodic stuctues K k ' k 0 G S f e ig

4 Couse : Diffction fom peiodic stuctues Question: how do we len the stuctue of peiodic solid? Diffction of wves o pticles with λ lttice constnt Coheent sctteing events give Bgg peks Bgg condition: dsinθ λ sctteing vecto: Kk-k 0 This pictue (nd some othes!) tken fom the Solid Stte Couse by Mk Jel (Cincinnti Univesity).

5 Intemezzo peiodic stuctues Fuit TEM silicon STM gphite???

6 Lttice vectos nd unit cells el o diect lttice n n + n + n33

7 The 4 Bvis lttices

8 ) ( b 3 3 π 3 b l kb hb G + + ij i i b πδ ) ( b 3 3 π ) ( b 3 3 π 3 3 n n n n + + el lttice ecipocl lttice Inside out: the ecipocl lttice length ecipocl lttice vecto π/(length diect vecto)

Pobe bems diffe in: wve length λ h/p sctteing coss section mgnetic moment Pobe bems Fo lttice constnt ~0-0 m ~ λ πh/p the elevnt enegy scle is electons: E e p /m e ~ 300 ev neutons: E n p /m n ~ 0.

9 Pobe bems diffe in: wve length λ h/p sctteing coss section mgnetic moment Pobe bems Fo lttice constnt ~0-0 m ~ λ πh/p the elevnt enegy scle is electons: E e p /m e ~ 300 ev neutons: E n p /m n ~ 0.6 ev photons: E ph pc ~ kev Typicl pobes: electons (0 ev - kev) neutons (0 mev - ev) x-y photons ( kev -00 kev) The de Boglie wvelength (λπ/k) of photons, electons, neutons, helium toms s function of enegy

10 Diffeent pobe bems fo diffeent esech electons typiclly penette only ~ 50 Å nd e used to pobe the sufce (e.g. LEED) Clen, not oxidized, sufces needed! neutons sctte t nuclei nd cy mgnetic moment used fo phonon dispesion, esolving mgnetic stuctues etc. x-ys, wide enegy nge, esolve cystllogphic stuctues of solids, complex molecules etc.

If the smple consists of some tens of ndomly oientted single cystls, the diffcted bems lie on the sufce of sevel cones. A smple of some hundeds of cystls (i.e. powdeed smple) show tht the diffcted bems fom continuous cones.

11 Exmple diffction expeiment: powde diffction Used to detemine the vlue of the lttice pmetes ccutely. If monochomtic x-y bem is diected t single cystl, then only one o two diffcted bems my esult. If the smple consists of some tens of ndomly oientted single cystls, the diffcted bems lie on the sufce of sevel cones. A smple of some hundeds of cystls (i.e. powdeed smple) show tht the diffcted bems fom continuous cones. Ech cone intesects the film giving diffction lines. Fo evey set of cystl plnes, by chnce, one o moe cystls will be in the coect oienttion to give the coect Bgg ngle to stisfy Bgg's eqution. Ech diffction line is mde up of lge numbe of smll spots, ech fom septe cystl.

12 Indexing powde ptten λ.54 Å, W 80 mm, cubic stuctue Diffction ngle θ is: π S π S θ o θ W W Bgg s lw: n λ d sinθ Intepln spcing d, lttice pmete d h + k + sin l θ λ 4 ( h + k + l ) Stuctue fcto clcultion: fce centeed cubic h,k,l ll even o odd 4.0 Å

13 Diffction in kinetic ppoximtion

14 Genel theoy of diffction in kinetic ppoximtion - single sctteing, emission of spheicl wves - coheent sctteing, phse eltion fixed - souce Q, sctteing cente P, obseve B - ppoximte spheicl wve by plne wve t lge distnce fom souce - sctteing mteil with sctteing density ρ() emits spheicl wves mplitude t P, time t: A ik0 ( R+ ) iω0t Ap A0e mplitude t B, time t: B A p ik R' e (, t) ρ( ) R' totl sctteing mplitude sttic lttice: A B ( t ) e iω 0 t ρ( )e i( k 0 k ) d with k long R ' identicl fo ll P elstic sctteing ωω 0

In diffction expeiment intensity is mesued I( K) A B ρ( ) e ik d with sctteing vecto K k ' k 0 - intensity is bsolute sque of the Fouie tnsfom of the sctteing density ρ() - if one could mesue the

15 In diffction expeiment intensity is mesued I( K) A B ρ( ) e ik d with sctteing vecto K k ' k 0 - intensity is bsolute sque of the Fouie tnsfom of the sctteing density ρ() - if one could mesue the mplitude(time) (phse), stuctue could be detemined by invese Fouie tnsfom, this is NOT possible Pocedue: choose possible cystl stuctue (symmety) clculte diffction ptten compe with mesued ptten Fowd sctteing Lue imge of hexgonl cystl

16 Fo disodeed systems nlysis fcilitted by Ptteson function: P() the utocoeltion function of the sctteing density P( ' ) ρ( ) ρ( ' + )d ik ' I( K) P( ') e d ' If mteil consists of single type of toms sctteing density t position i ρ( ) ρ ( i t i ) Decompose Ptteson function in - coeltion of tom with itself nd - coeltion of tom with ll othe toms P ( ' intensity is Fouie tnsfom of Ptteson function ) Nf + N ρ( ) ρ( ' + )d δ 0,' f is tom fcto : mesue of the mgnitude of the sctteing mplitude of n tom if lttice vecto P( ) peks ρ( ) j i ρt ( j )

17 stuctue fcto S(K) nd pi coeltion function g() I( K) N V f S( K) g( ') + N V g() Fouie tnsfom of S(K)- ρ( ) ρ ( g( ) e ik + ') d d N numbe of toms V volume Pi coeltion function fo mophous silicon nd liquid ion (833 K) mophous silicon dioxide

18 Sctteing fom peiodic stuctues one dimensionl exmple: peiodic y of toms, tnsltionl invince Fouie seies ρ( x e i( nπ / ) x ) ρn n geneliztion to thee dimensions el lttice n n + n + n33 tnsltionl invince implies ρ( x) ρ( x + n) n 0, ±, ±,... ρ ( ) G n πm G ρ e ig G m intege G is ecipocl lttice vecto G hg + kg + lg3 g i i πδ ij g π 3 ( 3 ) nd cyclic pemuttions

19 ) ( 0 ' ) ( d e R A K I K G i G G ρ othewise nd K G if V d e K G i ~ 0 ) ( Inset Fouie seies of peiodic y ρ() in expession fo intensity Lue sctteing condition: constuctive intefeence will occu when the chnge in wve vecto is vecto of the ecipocl lttice 0 ' ) ( V R A G K I G ρ significnt contibution only when GK, thus: G k k K 0 ' (since width of intensity distibution ~V - totl intensity ~V) with intensity diffction peks obseved t I ρ 3 g l kg hg G + +

20 Mille indices - distnce between plnes Mille indices (h, k, l): integeed invesed diect spce coodintes n m + n + o3 h' / m, k' / n, l' / o p( h',k',l' ) ( h,k,l ) Z p is smllest int ege Exmple: (m,n,o) (0.5,,) (h,k,l) (,,) The ecipocl lttice vecto o G is diected pependicul to the plne (h,k,l) its length eltes to the distnce between the plnes G π / d G hg + kg + lg3

21 Lue condition is equivlent to Bgg condition Bgg condition λ d sinθ Lue condition K k k 0 G K K k k 0 G / d sinθ / λ 4π K k 0 sinθ sinθ G π / d λ

22 The Ewld constuction to detemine if the conditions e coect fo obtining Bgg pek select point in k spce s oigin. dw the incident wve vecto k 0 to the oigin. fom the bse of k 0 wve vecto k in ll possible diections to fom sphee (elstic sctteing k k 0 ). t ech point whee this sphee intesects lttice point in k spce, thee will be Bgg pek with G k - k 0. Fo instnce 8 peks in the exmple below. ecipocl lttice oigin single cystl ligned with espect to k 0 fo smll chnges in k 0 no Bgg peks!

23 Stuctue fcto nd tomic fom fcto position of Bgg peks Lue condition (ecipocl lttice) intensity of Bgg peks stuctue fcto nd tomic fom fcto stuctue fcto intefeence of wves sctteing fom diffeent toms tomic fom fcto intefeence of wves sctteing fom diffeent pts of the tom

24 Intensity is popotionl to Fouie coefficients of sctteing density I ρ integte ove bsis cell nd sum ove N cells ρ V V n ρ( )e,n cell + n,n 3 cell n ig ρ( )e n d ig V ( + cells cell n ) d with N/V/V c with V c volume of cell ρ( )e + n + n33 N V cell ig d ρ( )e ig G n πm d Next: clculte sctteing density due to diffeent toms in cell

25 With diffeent tomic density of diffeent elements in the cell: ρ ( ) ig ig ' ρ e ρ ( ' )e d' Vc tomic sctteing fcto stuctue fcto S fo lttices with one tom pe unit cell Sf f ( ' )e i G ' ρ d' e ig f ρ V c S + schemtic epesenttion of 3d nd 4f obitls diffeent occuption diffeent tomic sctteing fcto

26 Evlute f using spheicl coodintes f ϑ ρ ( ' )e i G ' ig' cos d' ρ ( ' )e ' d' d(cosϑ ) dϕ θ is pol ngle between G nd, integting ove θ nd ϕ nd f sin G' 4π ρ ( ' )' d' G' with G K k 0 sinθ nd k 0 π/λ f sin[ 4π' (sinθ / λ )] 4π ρ ( ' )' d' 4π' (sinθ / λ ) tomic sctteing function is function f(sinθ/λ) Fo θ0 integl of sctteing density f 4 ( ') ' π ρ d' ove the tomic volume Z numbe electons tom numbe (fo x-ys)

F( k) j0 ( k) + j ( k) g J j 0 nd j descibe dil distibution spin nd cuent g J + J ( J + ) + S( S + ) L( L + )

27 Exmple: tomic fom fcto chomium In 3d metls the obitl moment is mostly quenched (L0) due to the cystlline electic field. The pesence of n obitl component cn be mesued by the tomic fom fcto. F( k) j0 ( k) + j ( k) g J j 0 nd j descibe dil distibution spin nd cuent g J + J ( J + ) + S( S + ) L( L + ) J ( J + ) Mgnetic fom-fcto mesued t the h0l. Bgg eflections of C O 3. The smooth cuve is the spin-only fee ion fomfcto fo C 3+ nomlised to.5 µ B.

28 The stuctue fcto descibes the intefeence fom wves sctteed fom diffeent toms in the unit cell S ig Exmple fo centeed lttice since is in the unit cell u,v,w < S f Fo body-centeed cubic lttice nd f f f S f f e e u + v + w 3 ( + e Exmple: stuctue fcto cubic lttice iπ ( h+ k + l ) πi( hu + kv + lw ) (, 0, 0 ) nd ( 0. 5, 0. 5, 0. 5 ) 0 ) 0 fo h + k + l odd f fo h + k + l even systemtic extinctions

29 Fo ou exmple of the powde diffction ptten: fo fce centeed cubic lttice nd f f f ( 0,0,0), (0,0.5,0.5), 3 (0.5,0,0.5), 4 (0.5,0.5,0) S f 4 ( + e iπ ( k + l) + e iπ ( h+ l) + e f fo h, k, l even o odd iπ ( h+ k ) ) 0 othewise With help of the stuctue fcto one cn unvel complicted cystllogphic nd mgnetic stuctues like these: ntifeoqudupol stte of NpO mgnetic phse of DyFe 4 Al 8

General Physics II. number of field lines/area. for whole surface: for continuous surface is a whole surface

General Physics II. number of field lines/area. for whole surface: for continuous surface is a whole surface Genel Physics II Chpte 3: Guss w We now wnt to quickly discuss one of the moe useful tools fo clculting the electic field, nmely Guss lw. In ode to undestnd Guss s lw, it seems we need to know the concept