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O =u  Coulomb glass$ 0  Computer simulations  Nz{|}~ O6TUV79LxMuW#:v  QRSYdoqbpstefghirZ[]\j_lk^w`amn(yuIyJ./0X6@ACBBKPFP  ` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>>0 d(    6d$ P  TKliknij, aby edytowa styl tytuBu z Wzorca+ +  0$   Kliknij, aby edytowa style tekstu z Wzorca Drugi poziom Trzeci poziom Czwarty poziom Pity poziom,   c  0$% ``  Y*  0% `   [*  0% 0p   ^*/80H  0޽h ? ff3333f *Pusta prezentacja.potR0  @( '   0d P    Y*   0ĸ     [* d  c $ ?    0$  @  Kliknij, aby edytowa style tekstu z Wzorca Drugi poziom Trzeci poziom Czwarty poziom Pity poziom,   c  6 `P   Y*   6 `   [* H  0޽h ? ̙33 tl08(  8 8 H ?P    Y*  8 HD ?    [*  8 N ?`P   Y*  8 N ?`   [* H 8 0޽h ? ̙33 6.P (     C P@<$ 0    C dH_<$0  T  0Ļ Jacek Matulewski, Sergei Baranovski, Peter Thomas Departament of Physics Phillips-Universitat Marburg, Germany Faculty of Physics, Astronomy and Informatics Nicolaus Copernicus University in ToruD, Poland Marburg, V 2007~1%0/         ,  (     B  s *޽h ? ̙33 `%( w   04 ?  Outline 0. Example of disordered system with long-range interactions 1. Simulation procedure for computation of single particle DOS 2. The dynamics of the Coulomb gap 3. The Coulomb glass and the glass transition 4. Phononless AC conductivity in Coulomb glass. 26     ! H  0޽h ? ff3333fM p%9T@](  T T 0t5 ? ;Realisation of disorder: the impurity band in semiconductor2< 2" < d T < ?@  T <5?0`* QConduction band    T <6?` XImpurity band (donors)  |B T T D?00 T B6?p XDE( j T B ?@pB T H D? p pB T H D?`pB T H D?  pB  T H D? pB  T H D? 2 T HT7?`P > 2 T B7? P > 2  T B8?P  > 2 !T Bt8?@ > 2 "T B8?  > d #T < ?p@  $T <9?`? N Valence band   j 'T B ?0 @ pB (T H D?  pB )T H D? ` pB +T H D? 0 l8 @`  6T@` $@ @  3T0 ` rB -T B D?@0 rB .T B D?0 @ rB /T B D? @ rB 0TB B D? @P rB 1T B D?P  rB 2T B D?xB 4T H D?0 0 xB 5T H D?0@00 jB 7T B D?2 8T B9? >  9T <:?9 VR3  U Acceptors   H T 0޽h ? ff3333f G?&&\(  \d \ < ?0 @ jB \ B D?   \ 0t; ? ;Realisation of disorder: the impurity band in semiconductor2< 2" < d \ < ?@ \ <;?0`* QConduction band   \ <<?` XImpurity band (donors)  vB \ N D?00 \ <<?p XDE( d \ < ?@jB  \ B D? p jB  \ B D?`jB  \ B D?  jB  \ B D? jB  \ B D? B \ H D?P@P,$D02 \ BT=?@ > 2 \ B=?P  > 2 \ B>?`   > d \ < ?p@  \ <>?`? N Valence band   jB \ B D? ` jB \ B D? 0 F @`  \ @` 2N @  \ 0 ` rB \ B D?@0 rB \ B D?0 @ rB \ B D? @ rB \B B D? @P rB  \ B D?P  rB !\ B D?xB "\ H D?0 0 xB #\ H D?0@00 jB $\ B D?2 %\ B4?? PP  >  &\ <$?9 VR3  U Acceptors   2 \ B?  > 2 \ BD? > H \ 0޽h ? ff3333f! 9!1!69d (  dd d < ?0 @ jB d B D?   d 0 ? ;Realisation of disorder: the impurity band in semiconductor2< 2" < d d < ?@ d <?0`* QConduction band   d <d?` XImpurity band (donors)  vB d N D?00  d <?p XDE( d  d < ?@jB  d B D? p jB  d B D?`jB  d B D?  jB d B D? jB d B D? pB d H D?P@P2 d B$?`P > 2 d B? P > 2 d B?P  > 2 d BD?`   > d d < ?p@  d <?`? N Valence band   jB d B D? ` jB d B D? 0 F @`  d @` 2N @  d 0 ` rB d B D?@0 rB d B D?0 @ rB d B D? @ rB dB B D? @P rB d B D?P  rB  d B D?xB !d H D?0 0 xB "d H D?0@00 jB #d B D?2 $d Bd? PP  >  %d <$?9 VR3  U Acceptors   2 &d B?  > 2 'd B?@ > 2 (d BD? > 2 )d B?  > 2 *d B? ` 0  > 2 +d Bd? `  > 2 ,d B?P `  >  .d 0? t  z Occupied donor (electron) q = 0,! 2 !  1d 0D?` 4  ~$Empty donor (hole) q = +,% 2 %  2d 0?   -Occupied (electron) q = acceptor@. 2      3d <?   C_  4d <d?0 P  C_  5d <?0   C_  6d 0$?  C_  7d <?0^ C+  8d <? | V C+  9d <D? l C+ H d 0޽h ? ff3333f 08l ( w ld l < ?0 @ jB l B D?   l 03 ? ;Realisation of disorder: the impurity band in semiconductor2< 2" <  l <dS?` XImpurity band (donors)  d  l < ?@jB  l B D? p jB  l B D?`jB  l B D?  jB l B D? jB l B D? pB l H D?P@P2 l BS?`P > 2 l B$T? P > 2 l BT?P  > 2 l BT?`   > jB l B D? ` jB l B D? 0 F @`  l @` 2N @  l 0 ` rB l B D?@0 rB l B D?0 @ rB l B D? @ rB lB B D? @P rB l B D?P  rB  l B D?xB !l H D?0 0 xB "l H D?0@00 jB #l B D?2 $l BDU? PP  >  %l <V?9 VR3  U Acceptors   2 &l BdV?  > 2 'l BV?@ > 2 (l B$W? > 2 )l BW?  > 2 *l BW? ` 0  > 2 +l BDX? `  > 2 ,l BX?P `  >  -l 0dY? t  z Occupied donor (electron) q = 0,! 2 !  .l 0$Z?` 4  ~$Empty donor (hole) q = +,% 2 %  /l 0Z?   -Occupied (electron) q = acceptor@. 2      0l <Z?   C_  1l <D[?0 P  C_  2l <[?0   C_  3l 0\?  C_  6l <d\? l C+  7l <\?0^ C+  8l <$]? | V C+ H l 0޽h ? ff3333f %%t>(  t t 0D^ ? ;Realisation of disorder: the impurity band in semiconductor$< 2# < 2 t B^? @  > X2 t 0?p `d2 t <?` 0 2 t B]?   > 2 t B? p0 > 2 t B?`   > 2  t BD?   > X2  t 0?00 X2  t 0? ` X2  t 0?pX2  t 0? @pd2 t <?  d2 t <?pPd2 t <?@@ d2 t <?0P^ t 6? 0 5 t 0?p The electrostatic potential at every donor site is due to Coulomb interaction with every acceptor (-) and every other empty site (+) in the system.   t 6d?p* RN = 10 K = 0.5 2 2 t BĮ?@ >  t 0?@ ROccupied donor 2  X2 t 0? @ d2 t <? @  t 0D?@  O Empty donor 2    t 0? @`.  UOccupied acceptor 2  B t 0d?p,$D0 PSince the sites positions are random - site potential are random too (disorder) &QPP  t <ı?  C+  t <$?p D C+  t <?`4 C+  t <?p `D  C+   t <D?P $ C+  !t <? D t  C_  "t <?Pp $ $ C_  #t <d?  C_  $t <Ĵ?`  C_  %t <$?pD C_ H t 0޽h ? ff3333f  x%(  x x 0D ? ;Realisation of disorder: the impurity band in semiconductor$< 2# < ~ x B A& ? ?pb  x 6?`eZ >System of randomly distributed sites with Coulomb interaction:.? 2%3 ? z Xp  x pX ,$D0 x B A5 ? ?p   x 6?XF  SSite potential: 2   x <ķ?   ^isolated sites are identical    x 6$?@ x4Total energy (classical electrostatic interactions):5 25 5   x <]? &" ]dimensionless units  H x 0޽h ? ff3333fI     q ( w   0 ? lWhat is the Coulomb glass?$ 2    6D? =System of randomly distributed sites with Coulomb interaction> 2> >    0?;,$D0 mIf the system is so sparse that the distances between sites are larger than the localisation length (n < nC)Lnf  j   &   6d?qk,$D0 2disorder => electronic wavefunctions are localised3 23    Z   6?1+,$D0 |the chemical potential is localised in  localised part of DOS? 2? ?   <$?,$D0 y7=> the quantum overlap may be neglected (no tunnelling)88 8   <?,$D0 z8=> classical system (electrons move via incoherent hops)99 9   <? K,$D0 TExamples: compensated lightly doped semiconductors amorphous semiconductors and alloys hopping behaviour of quasicrystals granular films silicon MOSFET s heterostructures electrically conducting polymers and stannic oxides nanowiresLl    4      <??{ ,$D0 n=> disorder isolator,  H  0޽h ? ff3333f ~v(    0$ ?  vpOutline 0. Example of disordered system with long-range interactions 1. Simulation procedure for computation of single particle DOS a) Searching for the pseudo-ground state (T = 0K). Coulomb gap b) Monte Carlo simulations (T > 0K) 2. The dynamics of the Coulomb gap 3. The Coulomb glass and the glass transition 4. Phononless AC conductivity in Coulomb glassq 29;=!6  <   ! H  0޽h ? ff3333f ^V$p(  p p 0 ? Simulation procedure (T = 0K)( 2  2  p H? @  > ^2  p 6?p `j2  p B?` 0 2 p BD?   > 2 p B? p0 > 2 p B?`   > 2 p Bd?   > X2 p 0?00 X2 p 0? ` X2 p 0?pX2 p 0? @pd2 p <?  d2 p <?pPd2 p <?@@ d2 p <?0P^ p 6? 0  p 0?pZ !Metropolis algorithm: the same as used to solve the salesman problem General: Searching for the configuration which minimise some parameter In our case: searching for electron arrangement which minimise total energy   p 6?p* RN = 10 K = 0.5 2 2 p B?@ >  p 0?@ ROccupied donor 2  X2 p 0? @ d2  p <? @  !p 0o?@  O Empty donor 2    "p 0p? @`.  UOccupied acceptor 2   $p <p? *,$D0 The calculating procedure isn t a simulation of the relaxation process. (no transition rates),^? ^ H p 0޽h ? ff3333f  R J  ( n   0tr ? ySimulation procedure (T = 0K) 2  ~  B A& <? ?pb <  6q?`eZ >System of randomly distributed sites with Coulomb interaction:.? 2%3 ? z Xp   pX ,$D0  H A5 =? ?p  =  <4s?XF  SSite potential: 2    Bs?   ^isolated sites are identical     6Tt?@ x4Total energy (classical electrostatic interactions):5 25 5 l @p m  p @m ,$D0  6u?@p j  ]Single electron transfer: 2  h   c $AD U?? m  U  <u? &" ]dimensionless units  H  0޽h ? ff3333fV 18r( w  ) c $A$ M??  8 M$D0  0v ?,$D0 5Total energy change during single electron transition$6 25 6   64v?` 0  _A (all acceptors)  2    0Tw?,$D 0 ^Di. 2    0w? P W jDj. 2 z pu  @pu ,$D0h  c $A C??` M Ch  c $A D??su Dh   c $A E??p E.F p s  p sh  c $A F??p  Fh  c $A G??s G  <tx?p  Y Site energies  l   0 ,$D07    #  ,$D0N 0    0@ xB   HD?p ~B   NDԔ?0` xB  HD?0N      2  <x?  > l2  <? Pl2  <?P p   64y? p  M+ 2 n  s *A" I??p  In  s *A! J?? J/   # @ ,$D0N @    2  By?@p > `2   0?@0l2 ! <?0   " 6y? M+ 2 N 0p # 0 ~B $ NDԔ?0 0p~B % NDg ?pxB & HD?`p 1z  p  6 p  ,$D0f 1 6? p   Bz?p  ]Total energy of the system:  n  s *AE H??  Hn ' s *A K??  Kn ( s *A L??p   L?z C I 8 C I,$D0? + B{?C   In order to make the calculation possible we need to express the energy difference using sites energy values before the transition  n , s *AA N?? I NxB - HD3?xB . HD3?00 / 0t{?  O_  2  H  0޽h ? ff3333f  Z R   ( su   0d ? ySimulation procedure (T = 0K) 2  ~  B A& O? ?pb O~  B A) P? ?p  P  6?`Z >System of randomly distributed sites with Coulomb interaction:.? 2%3 ?   6?p !j  ]Single electron transfer: 2    6?XF  SSite potential: 2    <T?   ^isolated sites are identical     6?@ x4Total energy (classical electrostatic interactions):5 25 5    s *A# Q?? U8 Q$D0b  6?,$D0 HSalesman says: transitions for which DH < 0 leads to pseudo-ground stateRI 2%  I   s *AD ^?? p~m 8 ^$0l  `   ` ,$D0  <? `z  [hole-electron interaction  xB  H D?yp y H  0޽h ? ff3333f JB09x (  x x 0T ? Simulation procedure (T = 0K)( 2   x 0?p DMetropolis algorithm for searching the pseudo-ground state of systemED E 0 8 @  6x`xB x HD?@@xB x HD?xB x HD?  xB x HD?  xB  x HD?` ` xB !x HD?  xB "x HD?PPxB #x HD?  xB $x HD?  xB %x HD?pp2 )x H?P` > 2 *x H?@P > 2 +x Ht? P@  > 2 ,x H? P0  > 2 -x H4?P P  >   .x # 0e0e    BgC`DE F( Ԕ 8c8c     ?A)BCD|E||5gN `@p`  x <?J 9Step 0 1. Place N randomly distributed donors in the box 2. Add KN randomly distributed acceptors (all occupied) 3. Distribute KN electrons over donors Step 1 (m -sub) 3. Calculate site energies of donors 4. Move electron from the highest occupied site to the lowest empty one 5. Repeat points 3 and 4 until there will be no occupied empty sites below any occupied (Fermi level appears)b 2{Y    L   5x#  @g rB 2x BD?  rB 3x BD? xB 4x HD?H x 0޽h ? ff3333f @$P%( tf3   <?  j8Step 2 (Coulomb term) 6. Searching the pairs checking for occupied site i and empty j If there is such a pair then move electron from i to j and call m -sub (step 1) and go back to 6. Effect: the pseudo-ground state (the state with the lowest energy in the pair approximation) " Energy can be further lowered by moving two and more electrons at the same step (few percent) 2L      f  s *AH X ??P` X   0T ? Simulation procedure (T = 0K)( 2    0t ?p DMetropolis algorithm for searching the pseudo-ground state of systemED E pB  HD?P pP D F ` $ `xB  HD?``xB  HD?  xB   HD?0 0 xB   HD?xB   HD?  xB   HD?@@xB   HD?ppxB  HD?  xB  HD?  xB  HD?  2  H ?pP   > 2  H4 ?p @  > 2  H ?p `  > 2  H ?p P  > 2  HT ?pp > ~B  ND?  ~B  ND?0  ~B  ND? ~B  ND?@H  0޽h ? ff3333f$ j$b$P? #(  , F ` 6 `xB  HD?``xB  HD?  xB  HD?0 0 xB  HD?xB   HD?  xB ! HD?@@xB " HD?ppxB # HD?  xB $ HD?  xB % HD?  2 & H?pP   > 2 ' H ?p @  > 2 ( HTq?p `  > 2 ) Ho?p P  > 2 * H$p?pp > z   ,$02 8 B? P  ,$D02 9 Bp?`  > r2 : B? ` 0r2 ; B? 0 2 < Bp?@p > 2 = HDq?`0   > 2 > Bq?  > r2 ? B?  l2 @ <?` 0 f2 A 6?`Pr2 B B? @r2 C B?``  r2 D B?@@r2 E B? 0 r2 F B? l p@  p@ ,$D0  6r?p w-The new hole appears in the neighbourhood ....- . 2  Bdr? P   > 2  Hr?p @  >   0$s ? -The origin of Coulomb gap in the ground state(. 2- . ^  6? 0 B I HD1? 0P ,$D0z p0  0p,$D0xB K HD?p xB L HD?xB N HD? 0 xB O HD?  0  l p p,$D0pT P # f2 0 6? P P 2  Hs?P`  > f2  6?  0r2  B?0 ` 2  Hs? p > 2  BDt?0 0  > 2   Ht? P  > f2   6?@r2   B? 00 f2   6?Pf2   6?P @r2  B?` 0 r2  B?@r2  B? p r2  B?`  4 6u?p \Other holes don t like it - they move away ...// / H  0޽h ? ff3333f% 4%,%`=kL0 $( \iK` LF ` L `rB L BD?``rB L BD?  rB L BD?0 0 rB L BD?rB L BD?  rB L BD?@@rB  L BD?pprB  L BD?  rB  L BD?  rB  L BD?  2  L B$v?pP   > 2 L Bv?p @  > 2 L Bu?p `  > 2 L Bv?p P  > 2 L BDw?pp >  #L 6w?p w-The new hole appears in the neighbourhood ....- . 2 %L Hx? p@  >  &L 0dx ? -The origin of Coulomb gap in the ground state(. 2- .  ?L 6x?p \Other holes don t like it - they move away ...// /  AL B$y?ph JDistances between sites with the same (different) occupancy raise (lessen)KJ K z @  CL @P ,$D0 DL B AC >? ?`   >z EL <y?@ Holes escape increase the distance between them and therefore lessen the total energy:YXX #z 0 O* kL 0 O*,$D0~B GL ND?0  ~B HL ND?  ~B IL ND?0 JL H AB ?? ?z "* ? KL By? O  m+Occupied sites are closer to new empty site,, , B LL T Do?pp-z z  jL z ,$D0 NL H AG @? ?` P z  @ OL BDz?` Z  k)Empty sites are farther from the new hole** * B PL T Do?p  ~B QL ND?p~B RL ND? ~B SL ND?`~B TL ND?  ~B UL ND?   z p VL 0,$D0l WL < ?pxB XL HD?P pP rB YL BD?``rB ZL BD?  rB [L BD?  rB \L BD?rB ]L BD?  rB ^L BD?rB _L BD?rB `L BD?  rB aL BD?  rB bL BD?2 cL Bz?p  > 2 dL B{?p   > 2 eL Bd{?p`   > 2 fL B4?pp   > 2 gL B?pp > xB hL HD?0  xB iL HD? @ H L 0޽h ? ff3333fm- -,pKT,( w   0D ? }+Coulomb gap in density of states for T = 0K$, 2+ ,   6?  <Coulomb gap created due to Coulomb interaction in the system= 2= = D 8 `p p@xB  HD?P pP rB  BD?``rB  BD?  rB  BD?  rB  BD?rB  BD?  rB   BD?rB   BD?rB   BD?  rB   BD?  rB   BD?2  B?p  > 2  B?p   > 2  Bd?p`   > 2  B?pp   > 2  B$?pp > xB  HD?0  xB  HD? @ `  c $A- 4??0 k 4  B?  USingle-particle DOS  .F - T -  <D? J  TSi:P 2BB  3 F=_>BB  B 3 =>BB " 3 F _ BB #B 3   BB % 3 F _ BB &B 3   BB ( 3 Fu _v BB )B 3 u v BB + 3 F3 _4 BB ,B 3 3 4 BB . 3 F_BB /B 3 BB 1 3 F_BB 2B 3 BB 4 3 Fj_kBB 5B 3 jkBB 7 3 F(_)BB 8B 3 ()T Y O#  ! 6  V 0&  $ 6 Yv  Y 0.05&  ' 6dL <3  X 0.1&  * 6C Y  Y 0.15&  - 6$L <  X 0.2&  0 6Yk Y 0.25&  3 6L{<( X 0.3&  6 6D9Y Y 0.35&  9 6L< X 0.4& BB : 3  % =BB ; 3  ( @ < <d  - V-4& BB = 3 W %X =BB > 3 W (X @ ? <d  - V-2& BB @ 3  %=BB A 3  (@ B <J- V 0& BB C 3 %=BB D 3 (@ E <$- V 2& BB F 3 z%{=BB G 3 z({@ H <?- V 4& H I C F(= L BPJCBDE F(B&BBBFBBAfAA%AAAEAAAeAA%AAADzAhAvAdmAkA$ DA 0A AD A A Ac @ @# @ @ {@C c@ V@3@b@?"?z?Y?B4?>>b_>>!==0=A<O<;a;:!998@77P6`4i3 10;/?-*(_&$x"4?N^[  ~>J] !}!!~="" "Q ]#d#$|$2$*!<%&%*%0[&54&&8'|;{'>'7A:(NB(B(rBZ)A)?*(=y* :*68+L3+.+*X,,',E$- x-B-8. ..W/K/0fw0071u1d1W2m23v33644,4V5 5!6I#u6$6a%57&7'7(U8y)8O*9r+u9e,9q-4::.:/:/T;e0;W185>6?6s?Q7?[73@7@/8@8RA9A*9B9rB":B:2C:C;Cf;RD;D;<E-H>0Ij>I>I>PJ>@F=2 M BCDEPFX`<i ?_ !8W?}^),@  H N C F(=~B P N D?HH R 6D?0p kN=500 1000 real., PBC( 2  H  0޽h ? ff3333f    Z ( w   0 ? |* Dimensionless units$ 2  ` N c $A q??pw q` O c $A7 r??  r P <d?pzj m+The temperature is measured in energy units,, ,  Q <` ?4 z QThe length unit   R <` ?` Z hThe unit of energy ` S c $A% t?? O  tl *7   YP N ,$D0h M c $A p??7   p T <$a ?*   FThus   U <a ?   j(=> the Coulomb interaction energy reads)) ) h V c $A( u??@   u W 0a ? ,$D0 For example: nD=69% of nC, nC = 3.521018 cm-3 => t[d.u.] = 1 <=> T[K] H" 200KP       t            Z <Db ?,$D0  nD= 8% of nC, nC = 3.521018 cm-3 => t[d.u.] = 1 <=> T[K] H" 100KZ       t         H  0޽h ? ff3333fv) ))C\@(( w @ @ 0d  ? ~ Shape of Coulomb gap for T = 0K $! 2    :B @ 3 +,:B @@ 3 +, @ <c  a V 0& :B @ 3 0 1 :B @@ 3 0 1  @ <d    X 0.1& :B @ 3 5 6 :B @@ 3 5 6  @ <De    X 0.2& :B  @ 3 ;<:B !@@ 3 ;< "@ <e  X 0.3& :B #@ 3 @A:B $@@ 3 @A %@ <f  X 0.4& :B &@ 3 EF:B '@@ 3 EF (@ <df   X 0.5& :B )@ 3 JK:B *@@ 3 JK +@ <f  X 0.6& :B ,@ 3 +:B -@ 3 Jg .@ <$g z=5' X-0.2& :B /@ 3 WX+:B 0@ 3 JWXg 1@ <g z' V 0& :B 2@ 3  +:B 3@ 3 J g 4@ <g z ' X 0.2& :B 5@ 3  +:B 6@ 3 J g 7@ <Dh z6 & ' X 0.4& :B 8@ 3 MN+:B 9@ 3 JMNg :@ <h z' X 0.6& :B ;@ 3 +:B <@ 3 Jg =@ <i zu' X 0.8& :B >@ 3 +:B ?@ 3 Jg @@ <di zY' V 1& @ A@ C J+p D@ HB"QCbDEpFxbZw : bS#@]yt"!%9(+ / %2B5^8C{; >"ADGG  K(NV"QI9<@ * G@ B"QC,DEFeeZ'Z''x(u(G~))c**+`j+2+ + + ,y ,K , , ,,,d ,6+++j+}+O*!c*)~)(hx(9' Z'&(&%S$%G$# "!Q"l"!># $S $%&PV''((N)*o+PA,--X./ Z0g,11!23s4>E566e780^90:;f; <; w= H> ? ? @b A bBG 4CD0DE{FLG HHI|JfKv7L MtMNu~OQP{"Q@+* J@ B3CCADEPFX3TTAA@@u@Gw@:@??\?`?2> I> = y=y =J < < ;;:d :6z98Q87}7Ow6!5'5z43g38^2 10%0`/R.#-, .,!X+k"*=#)$($'%'&&V'4%'(H$(Z#)g"*s!o+{ A,--./|Y0t,1i1\2M3:s4$E5 6678]9q0:J ; ; < w=H>d?.?@AbBA3C@Qm** M@ B3CCt$DEPFX3TTt$t$t$t$ur$Go$d$J$ $#`#27# " K" !y /!J   Id?6)w}O!ijg8! DR# _!k"G=#$:$ %7 & V'= '( (K ) *b o+ A, - -.A/Y0t,112M3s4E57667.8]90:3;;?b?@AbBD3C@m+@ N@ C J+8 v@ Q@@ xB @ HD?v`` @ 6$j ?@ ]numerical simulation result  8 @p j  Z@p@ j HB L@ C @0 `0  @ 6j ?p j  }fitting of ax2 (soft gap)6     vB O@@ N D?9 XY)#F @ Y R@ P@ S@ 0k ?_Y {#fitting of (Efros)$$    h T@ c $A; a??0  B arB U@ BD?@`F @ P V@ @PrB W@ BD?@` X@ 0?P h&fitting of (BSE)'' ' h Y@ c $A: b??0   b [@ 64 ?  9Single-particle-DOS  ` \@ c $A9 c??0 pi cH @ 0޽h ? ff3333f# v#n#B[P"( w   0  ? ~ Shape of Coulomb gap for T = 0K $! 2      6t ?  9Single-particle-DOS  vB  NDo?     <4 ?/ ?)  Jhard gap   8 @t [@txB  HD?@```   6 ?t ]numerical simulation result  '8 @ Y ZP@   6 ?_Y {#fitting of (Efros)$$    n  s *A; i??0  B ixB  HD?@`8 @ P Y@PxB   HD?@`  6t ?P h&fitting of (BSE)'' ' n  s *A: j??0   j`  c $A9 k??0 pi k4B  # !"4B @ # !~"  6    V 0& 4B  #   4B @ #  ~ 4B  # } ~ 4B  @ # } ~~  ! 64 C P  Y 0.04& 4B " # + , 4B #@ # + ~, 4B % # 4B &@ # ~ ' 6 PL  Y 0.08& 4B ( # 4B )@ # ~4B + # 674B ,@ # 6~7 - 6 P Y 0.12& 4B . # 4B /@ # ~4B 1 # 4B 2@ # ~ 3 6T XP Y 0.16& 4B 4 # @A4B 5@ # @~A4B 7 # !4B 8 # @] 9 6 p] V 0& 4B : # !4B ; # @] < 6 paQ X 0.1& 4B = #   !4B > # @  ] ? 6t p  X 0.2& 4B @ # 89!4B A # @89] B 6 p  X 0.3& 4B C # jk!4B D # @jk] E 64 p X 0.4& 4B F # !4B G # @] H 6 p) X 0.5& : I 3 @! L BOCDDE4F< DjD CZ>8#1j*+*1o"9y@G9.OO@ O |BOCEDEFddEEEElE:E EE|EuhEDNE-EE D~ DL TD D C YCBUB#BAA@^@-u?>=>=g<5B<;::pb9>8 77H6x{5G 4!3!3"+2#R1O$v0%/%.&-',X(,&)2+)L**f)+~(`,'/-&-%.$/#j0#81"2)!2< 3Or4bA5s667{8J9::;<S=%">8>I?\@o\A*BB C D dE 3F G G'H=nIS=g=5d<;::pA9>k8 765x4G 4!3!-2">1#M0O$[/%g.%s-&|,'+X(*&)))(*'+&`,%/-$-#."/!j0 81223r4,A5<6M6_7s{8J9::;<S="><>[?}@\A *B B C5 D_ dE 3FGGHDnIw 0K)( 2    0b ?p aMonte-Carlo simulations    <c ?0fj wStep 3 (Coulomb term) 7. Searching the pairs checking for occupied site j and empty i If there is such a pair Then move electron from i to j for sure Else move the electron from i to j with prob. Call m -sub (step 1). Repeat step 3 thousands times (Monte Carlo) Repeat steps 0-3 several thousand times (averaging) Step 2 may be omittedNx 2L  *H6{  X   f  s *A* Z ??Pp` Z f  s *A+ [ ??50  [ pB + HD? p  8 @  @0xB - HD?@@xB . HD?xB / HD?  xB 0 HD?  xB 1 HD?` ` xB 2 HD?  xB 3 HD?PPxB 4 HD?  xB 5 HD?  xB 6 HD?pp2 7 Hdc ?0 P  > 2 8 H8? P0  > 2 9 HT9? P@  > 2 : H9?P > 2 ; H:?P P  > H  0޽h ? ff3333f!G FFEDIF( w D D 0; ? x&Smearing of the Coulomb gap for T > 0K$' 2& ' ` BD c $A- d??`  d CD BT<? (  kSingle-particle DOS 8 Cp  D` qD 6<Cp0 ]T = 0.0( BB rD 3   uD 6=P`  W0.1& BB vD 3 f  yD 6t= P X 0.2& BB zD 3   }D 6= @ X 0.3& BB ~D 3   D 64> 0 X 0.4& BB D 3 f D 0>  4 Y 1& BB D 3  <8  Dp<B FD # <B GDB #  HD 6>  V 0& <B ID #   <B JDB #   <B LD #   <B MDB #    ND 6T?   X 0.1& <B OD #   <B PDB #   <B RD # "#<B SDB # "# TD 6? X 0.2& <B UD # <B VDB # <B XD # )*<B YDB # )* ZD 6@ X 0.3& <B [D # <B \DB # <B ^D # 12<B _DB # 12 `D 6t@ X 0.4& <B aD # { |<B bD # {1|N cD 6@3a V-4& <B dD #  <B eD # 1N fD 64Aea V-2& <B gD #  <B hD #  1 N iD 6A a  V 0& <B jD #  <B kD # 1N lD 6AaD V 2& <B mD # C D<B nD # C1DN oD 6TBav V 4& B pD 3 1 sD BHC,CDE F(B%BBB?BBBZBBBtBB0BuBhBKZBNBLBfEB+B!B BB<B BAV A A Ap A A- A vA bAG 6A A Ab @ @@|x@<@8@??S?E?>m>_>)#>=I=D<|<<_;}:9x9Y8576J5O3H2 ;1j/.&J+L)'@5%"$[zu 1|  LVg"  =!!! W"m "#q##z.${!$&$(+G%0%4&8b&<&T?'A|'B',C8(C(sB(@S)=):*F7m*3*n/)+`++'+$D,2!,-^-K-. x..*5///O00 1<j11&22H2@33y3E [4 4""5#u5$5%16'6Q(6!)L7)7*8+g8,8-"9.9/920=:0:1:2W;z3;37>7?o8c?8?9@9}@9@k:9A:AL;Ah;TB;B;Cb_FI>FI>G>yG>G?6H?HY?Hr?@y, tD BCDEPFX^<iy4O #j;Z&@),@yr   wD BNC:DE F(f::}::H:::y::D:::ty:f:@l:l: h:pQ:B:;9:::l9:6 9 9 9g 9 93 9 9 e9c /9 9. 8 8 8_x8Q8*877[876%6m6"6V55"k43P3R210/.N-,[+f*G(J'$C#z E7avEA&  q/=|nm8  !Mi!!4""' "e##0$$$l!_%w$%'+&s*&-&.\'/'A0&(b0(u0(/W).),"*^+*{(*%R+#+ ,,O,N-e-._..eJ//0z0R0F1f1~2w2G2A3K38 4r44,=5H576Un66a 97!7"8e$j8$8I&596&9q':.(e:t):)0;t*;+;,a<,/>!0(?|0?0?A1Y@1@+2#Ax2A2A3TBx3B4Ce4C4C.5PD-5Da5E5E6E6KFE6Fr6G6|G6G6HHB7Hs7I_7xI7I7CJ7J8K48tK@8K;8?L^8L8 M8oM8M8:N8N9@1  xD BAC0DEFf'00A0 @   {D BIC0DE F(00f00$000B000`00|0}}0x0<x0]0D0Z,0&0*0w!006///S / / /p x/ ^// !/ / .L . . .j . Y.);.--F\-)-,d{,(,"|+Z+*@o*)u)^( ('|&%:$#"W#!_ OtM3Qmn$H-   K> 6h'!0D  b  4 !t!!="$""[# # $ y$$6%%%T&j&'r''0(( (pN))a *fl*g**+p++bG,,-d--#...A/W/ /% ^0 0n1m |1 1- ;22 2FX33'4v44m45Y55R667o77-8X8M8HK99 :i::';B ; ;!E<"$>%>%??<&?&? ']@'@(A(zA(A(9Bt)B)B9*UCM*C*D*sD*D*+2Ef+E+E+OF4,F@,Gi,mG,G,,H,H,H3-II-I-@  |D B8CqDEHFP_ |(O;Yu1L4GVX8q%(@{   D BJC'+DE F('+ +~++?+*+a**"***C***e**%{*j*W*GK*C*-*i!*)) ) ) )K ) ) V)m 8) (- ( ( e(O =( ('q'$'1&&q&S &y%$u"$#5T#"!W^!l y92 [2|e >?}_%  7  B cI$NeF1g6t( ] J!! "j"f",#y#8#uN$$& %9 m% %I /&W &% & Q'W ') ( r(n (d3) )) U* *T +J v+ + 7,4 , ,c X- -5 .5 z.$ . ;/E / / \0 0 1r ~1 1 ?2 2 3 `3v 3 !44r4C556d66&7?77F8D89ch99*:^::J;l;G <`l > ? p? ?C!1@!@!@f"RA"A*#BR#tB#BD$5C$C$C %VD %D%E%xE!&E>&9F]&F&F&[G&GG'HZ'|H'H'=I'I'I'_J(J@(@  D B"CNDE<FD`D["z~Cd &*.F"N @z   D BOC['DEFfM'TK'J'=['K'(0'' ' '&p&&Z&&D&&-M&-& &% %w % %` X% %J $ $3 {$ $ # W#"|""f!w!P!P 9}#T lUe@q)Tq r  [ E%/pqxob#K 5! !_"" #~##h$n$P%%:&&#''(N((_m))!W**a@+#+G*,,---\r.s.F]// F0 0, 01) 1l 2 2Y 3 x3 34 b44_L5566p6 7 7 888vh939R::x;;;&<$<=C==Vn>3>X?Q?B@@+ANABB8 B~ uC C9!^Dv!D!HE"E^"1F"F"G&#Go#H#zH#H#dID$If$NJ$J$8K$K$!L%%L>% MH%Mi%Mw%jNy%N%TO%O%O%cd@b ^ D .BOCH#DEFH#;A#8###"d""B"R"0"!!k!e!IM! &  l r> O  ',  n y  `U  3  4\:h6ocA%j H $  q q b Nv X+n XwaUc2B~\d9 . Z!!+!@b""I@##$$]$h%%NE&Y&y"''(Hn(H(kL)))*o*+Zu++S,,0-~-#.%|.|._Z/$ / 70 0[ 1T 1 1 a2> 2=33Y444wh5 5D6,6"77'7n898K99)::;t;;R{>>Y?j?6@@AiAAC`BxB 0K$' 2& ' ` | c $A- ??`   | BD? (  kSingle-particle DOS  | < `  \T = 0K( @B | C I | 6 P  W22K& @B  | C fI  | 6t@ X 44K& @B  | C I  | 6Դ0 X 66K& @B  | C I | 64  X 88K& @B | C fI | 6  Z 222K& @B | C pIph<F  | p<B | # <B |B #  | 6  V 0& <B | #   <B |B #   <B | #   <B |B #    | 6T   X 0.1& <B | #   <B |B #   <B | # "#<B |B # "# | 6 X 0.2& <B  | # <B !|B # <B "| # )*<B #|B # )* $| 6 X 0.3& <B %| # <B &|B # <B '| # 12<B (|B # 12 )| 6t X 0.4& <B *| # { |<B +| # {1|N ,| 6Է3a V-4& <B -| #  <B .| # 1N /| 64ea V-2& <B 0| #  <B 1| #  1 N 2| 6 a  V 0& <B 3| #  <B 4| # 1N 5| 6aD V 2& <B 6| # C D<B 7| # C1DN 8| 6Tav V 4& B 9| 3 1 :| BHC,CDE F(B%BBB?BBBZBBBtBB0BuBhBKZBNBLBfEB+B!B BB<B BAV A A Ap A A- A vA bAG 6A A Ab @ @@|x@<@8@??S?E?>m>_>)#>=I=D<|<<_;}:9x9Y8576J5O3H2 ;1j/.&J+L)'@5%"$[zu 1|  LVg"  =!!! W"m "#q##z.${!$&$(+G%0%4&8b&<&T?'A|'B',C8(C(sB(@S)=):*F7m*3*n/)+`++'+$D,2!,-^-K-. x..*5///O00 1<j11&22H2@33y3E [4 4""5#u5$5%16'6Q(6!)L7)7*8+g8,8-"9.9/920=:0:1:2W;z3;37>7?o8c?8?9@9}@9@k:9A:AL;Ah;TB;B;Cb_FI>FI>G>yG>G?6H?HY?Hr?@y2 ;| BCDEPFX^<iy4O #j;Z&@),@yr   <| BNC:DE F(f::}::H:::y::D:::ty:f:@l:l: h:pQ:B:;9:::l9:6 9 9 9g 9 93 9 9 e9c /9 9. 8 8 8_x8Q8*877[876%6m6"6V55"k43P3R210/.N-,[+f*G(J'$C#z E7avEA&  q/=|nm8  !Mi!!4""' "e##0$$$l!_%w$%'+&s*&-&.\'/'A0&(b0(u0(/W).),"*^+*{(*%R+#+ ,,O,N-e-._..eJ//0z0R0F1f1~2w2G2A3K38 4r44,=5H576Un66a 97!7"8e$j8$8I&596&9q':.(e:t):)0;t*;+;,a<,/>!0(?|0?0?A1Y@1@+2#Ax2A2A3TBx3B4Ce4C4C.5PD-5Da5E5E6E6KFE6Fr6G6|G6G6HHB7Hs7I_7xI7I7CJ7J8K48tK@8K;8?L^8L8 M8oM8M8:N8N9@1  =| BAC0DEFf'00A0 @   >| BIC0DE F(00f00$000B000`00|0}}0x0<x0]0D0Z,0&0*0w!006///S / / /p x/ ^// !/ / .L . . .j . Y.);.--F\-)-,d{,(,"|+Z+*@o*)u)^( ('|&%:$#"W#!_ OtM3Qmn$H-   K> 6h'!0D  b  4 !t!!="$""[# # $ y$$6%%%T&j&'r''0(( (pN))a *fl*g**+p++bG,,-d--#...A/W/ /% ^0 0n1m |1 1- ;22 2FX33'4v44m45Y55R667o77-8X8M8HK99 :i::';B ; ;!E<"$>%>%??<&?&? ']@'@(A(zA(A(9Bt)B)B9*UCM*C*D*sD*D*+2Ef+E+E+OF4,F@,Gi,mG,G,,H,H,H3-II-I-@ " ?| B8CqDEHFP_ |(O;Yu1L4GVX8q%(@{   @| BJC'+DE F('+ +~++?+*+a**"***C***e**%{*j*W*GK*C*-*i!*)) ) ) )K ) ) V)m 8) (- ( ( e(O =( ('q'$'1&&q&S &y%$u"$#5T#"!W^!l y92 [2|e >?}_%  7  B cI$NeF1g6t( ] J!! "j"f",#y#8#uN$$& %9 m% %I /&W &% & Q'W ') ( r(n (d3) )) U* *T +J v+ + 7,4 , ,c X- -5 .5 z.$ . ;/E / / \0 0 1r ~1 1 ?2 2 3 `3v 3 !44r4C556d66&7?77F8D89ch99*:^::J;l;G <`l > ? p? ?C!1@!@!@f"RA"A*#BR#tB#BD$5C$C$C %VD %D%E%xE!&E>&9F]&F&F&[G&GG'HZ'|H'H'=I'I'I'_J(J@(@   A| B"CNDE<FD`D["z~Cd &*.F"N @z   B| BOC['DEFfM'TK'J'=['K'(0'' ' '&p&&Z&&D&&-M&-& &% %w % %` X% %J $ $3 {$ $ # W#"|""f!w!P!P 9}#T lUe@q)Tq r  [ E%/pqxob#K 5! !_"" #~##h$n$P%%:&&#''(N((_m))!W**a@+#+G*,,---\r.s.F]// F0 0, 01) 1l 2 2Y 3 x3 34 b44_L5566p6 7 7 888vh939R::x;;;&<$<=C==Vn>3>X?Q?B@@+ANABB8 B~ uC C9!^Dv!D!HE"E^"1F"F"G&#Go#H#zH#H#dID$If$NJ$J$8K$K$!L%%L>% MH%Mi%Mw%jNy%N%TO%O%O%cd@b d C| 4BOCH#DEFH#;A#8###"d""B"R"0"!!k!e!IM! &  l r> O  ',  n y  `U  3  4\:h6ocA%j H $  q q b Nv X+n XwaUc2B~\d9 . Z!!+!@b""I@##$$]$h%%NE&Y&y"''(Hn(H(kL)))*o*+Zu++S,,0-~-#.%|.|._Z/$ / 70 0[ 1T 1 1 a2> 2=33Y444wh5 5D6,6"77'7n898K99)::;t;;R{>>Y?j?6@@AiAAC`BxB 0K$' 2& ' `  c $A- ??`    B? (  kSingle-particle DOS   6T`  \T = 0K( :B  3 I  0P  W10K& :B  3 fI   0@ X 20K& :B   3 I   0t0 X 30K& :B   3 I   0Խ  X 40K& :B  3 fI  04  Z 100K& :B  3 pIph<F   p<B  # <B B #   6  V 0& <B  #   <B B #   <B  #   <B B #     6   X 0.1& <B  #   <B B #   <B  # "#<B B # "#  6T X 0.2& <B  # <B  B # <B ! # )*<B "B # )* # 6 X 0.3& <B $ # <B %B # <B & # 12<B 'B # 12 ( 6 X 0.4& <B ) # { |<B * # {1|N + 6t:3a V-4& <B , #  <B - # 1N . 6:ea V-2& <B / #  <B 0 #  1 N 1 62 a  V 0& <B 2 #  <B 3 # 1N 4 62aD V 2& <B 5 # C D<B 6 # C1DN 7 6T3av V 4& B 8 3 1 9 BHC,CDE F(B%BBB?BBBZBBBtBB0BuBhBKZBNBLBfEB+B!B BB<B BAV A A Ap A A- A vA bAG 6A A Ab @ @@|x@<@8@??S?E?>m>_>)#>=I=D<|<<_;}:9x9Y8576J5O3H2 ;1j/.&J+L)'@5%"$[zu 1|  LVg"  =!!! W"m "#q##z.${!$&$(+G%0%4&8b&<&T?'A|'B',C8(C(sB(@S)=):*F7m*3*n/)+`++'+$D,2!,-^-K-. x..*5///O00 1<j11&22H2@33y3E [4 4""5#u5$5%16'6Q(6!)L7)7*8+g8,8-"9.9/920=:0:1:2W;z3;37>7?o8c?8?9@9}@9@k:9A:AL;Ah;TB;B;Cb_FI>FI>G>yG>G?6H?HY?Hr?@y2 : BCDEPFX^<iy4O #j;Z&@),@yr   ; BNC:DE F(f::}::H:::y::D:::ty:f:@l:l: h:pQ:B:;9:::l9:6 9 9 9g 9 93 9 9 e9c /9 9. 8 8 8_x8Q8*877[876%6m6"6V55"k43P3R210/.N-,[+f*G(J'$C#z E7avEA&  q/=|nm8  !Mi!!4""' "e##0$$$l!_%w$%'+&s*&-&.\'/'A0&(b0(u0(/W).),"*^+*{(*%R+#+ ,,O,N-e-._..eJ//0z0R0F1f1~2w2G2A3K38 4r44,=5H576Un66a 97!7"8e$j8$8I&596&9q':.(e:t):)0;t*;+;,a<,/>!0(?|0?0?A1Y@1@+2#Ax2A2A3TBx3B4Ce4C4C.5PD-5Da5E5E6E6KFE6Fr6G6|G6G6HHB7Hs7I_7xI7I7CJ7J8K48tK@8K;8?L^8L8 M8oM8M8:N8N9@1  < BAC0DEFf'00A0 @   = BIC0DE F(00f00$000B000`00|0}}0x0<x0]0D0Z,0&0*0w!006///S / / /p x/ ^// !/ / .L . . .j . Y.);.--F\-)-,d{,(,"|+Z+*@o*)u)^( ('|&%:$#"W#!_ OtM3Qmn$H-   K> 6h'!0D  b  4 !t!!="$""[# # $ y$$6%%%T&j&'r''0(( (pN))a *fl*g**+p++bG,,-d--#...A/W/ /% ^0 0n1m |1 1- ;22 2FX33'4v44m45Y55R667o77-8X8M8HK99 :i::';B ; ;!E<"$>%>%??<&?&? ']@'@(A(zA(A(9Bt)B)B9*UCM*C*D*sD*D*+2Ef+E+E+OF4,F@,Gi,mG,G,,H,H,H3-II-I-@ " > B8CqDEHFP_ |(O;Yu1L4GVX8q%(@{   ? BJC'+DE F('+ +~++?+*+a**"***C***e**%{*j*W*GK*C*-*i!*)) ) ) )K ) ) V)m 8) (- ( ( e(O =( ('q'$'1&&q&S &y%$u"$#5T#"!W^!l y92 [2|e >?}_%  7  B cI$NeF1g6t( ] J!! "j"f",#y#8#uN$$& %9 m% %I /&W &% & Q'W ') ( r(n (d3) )) U* *T +J v+ + 7,4 , ,c X- -5 .5 z.$ . ;/E / / \0 0 1r ~1 1 ?2 2 3 `3v 3 !44r4C556d66&7?77F8D89ch99*:^::J;l;G <`l > ? p? ?C!1@!@!@f"RA"A*#BR#tB#BD$5C$C$C %VD %D%E%xE!&E>&9F]&F&F&[G&GG'HZ'|H'H'=I'I'I'_J(J@(@   @ B"CNDE<FD`D["z~Cd &*.F"N @z   A BOC['DEFfM'TK'J'=['K'(0'' ' '&p&&Z&&D&&-M&-& &% %w % %` X% %J $ $3 {$ $ # W#"|""f!w!P!P 9}#T lUe@q)Tq r  [ E%/pqxob#K 5! !_"" #~##h$n$P%%:&&#''(N((_m))!W**a@+#+G*,,---\r.s.F]// F0 0, 01) 1l 2 2Y 3 x3 34 b44_L5566p6 7 7 888vh939R::x;;;&<$<=C==Vn>3>X?Q?B@@+ANABB8 B~ uC C9!^Dv!D!HE"E^"1F"F"G&#Go#H#zH#H#dID$If$NJ$J$8K$K$!L%%L>% MH%Mi%Mw%jNy%N%TO%O%O%cd@b d B 4BOCH#DEFH#;A#8###"d""B"R"0"!!k!e!IM! &  l r> O  ',  n y  `U  3  4\:h6ocA%j H $  q q b Nv X+n XwaUc2B~\d9 . Z!!+!@b""I@##$$]$h%%NE&Y&y"''(Hn(H(kL)))*o*+Zu++S,,0-~-#.%|.|._Z/$ / 70 0[ 1T 1 1 a2> 2=33Y444wh5 5D6,6"77'7n898K99)::;t;;R{>>Y?j?6@@AiAAC`BxB 0K)$ 2  `  c $A. 8?? P 8`  c $A, 9??`Y  9H  0t4?   ,N=400, T=1/8 (28K for n/nC=1), a=0.3, MC=103x-      C A0hT:\PPT\referat Rackeve (Coulomb glass)\dane\cg4.gifp`H  0޽h ? ff3333f C;(    048 ? lPair distribution (T > 0K)$ 2  `  c $A. :?? P :`  c $A, ;??`Y  ;  C A2hT:\PPT\referat Rackeve (Coulomb glass)\dane\cg4.gifp`G  0t7?   +N=400, T=1 (222K for n/nC=1), a=0.3, MC=103x,    H  0޽h ? ff3333fU  }( 8 E  08 ?  Outline 0. Example of disordered system with long-range interactions 1. Simulation procedure for computation of single particle DOS 2. The dynamics of the Coulomb gap 3. The Coulomb glass and the glass transition 4. Phononless AC conductivity in Coulomb glass 29=^6     ! H  0޽h ? ff3333f aY0'(    0T9 ? 0The dynamics of the Coulomb gap: the time scales21 2! 1 d  < ?@$F   @ l  < ?rB  B D?@rB  B D?0rB  B D? rB   B D? P PrB   B D? p   <7?0`* QConduction band     <:?0v XImpurity band (donors)  pB  H D?@@@l O   O ,$D0~B  N Do?000   <:?  K Acceptors   ~B  N Do?P0P  <;?pO  N Valence band     <;?p,$D0 >Averaged thermal activation time (T = 7K, DE=31.27 meV): 104 s~?"  **   Kz P >v $ P >v,$D 0n  s *AO ??@a v n  s *AP ?? @MP   B<?@>: Vthe microscopic time    B=?P ;J ALifetime of donor (inverted transfer rate up to conduction band):BB B Jz 0*  ' 0* ,$D0Z ! Bt=?p 0* ,$D0 TQuestion 1: What must be the temperature to keep the electron in the imputity band?UU E    B   T D?00  B=?p XDE( 2 & B4>?`@ > H  0޽h ? ff3333f  %   (   t%     0IJ ? 0The dynamics of the Coulomb gap: the time scales21 2! 1 d   < ?@$F    @ l   < ?rB   B D?@rB   B D?0rB   B D? rB   B D? P PrB   B D? p   <?0`* QConduction band     <D?0v XImpurity band (donors)  vB   N D?   s *AQ ??0  & 8 $D0pB   H D?@@@z P @ %  P @,$D 0n   s *AM ??2@    B?P ' J ~&Miller-Abrahams transfer rate for VRH:''       <d?pn XDE(    <ĵ?p,$0 >Averaged thermal activation time (T = 7K, DE=31.27 meV): 104 s~?"  **   l ` *  # ` * ,$0s   B$?p * ,$D0 WQuestion 2: How long does it takes to transfer electron from donor i to empty donor j?@XD X B   T D?` ` h "  c $AS ??    2 $  B?@  > H   0޽h ? ff3333f% @(A( w ( ( 0 ? 0The dynamics of the Coulomb gap: the time scales21 2! 1 d ( < ?@$F  ( @ l ( < ?rB ( B D?@rB ( B D?0rB ( B D? rB  ( B D? P PrB  ( B D? p  ( <$?0`* QConduction band    ( <?0v XImpurity band (donors)  vB  ( N D? ( s *AQ ??0  & 8 $0pB ( H D?@@@ ( <D?pn XDE(  ( <?p,$0 >Averaged thermal activation time (T = 7K, DE=31.27 meV): 104 s~?"  **    ( B? w ,$0 KConclusion: For Si:P (n = 69% of nC) during 103s electron travel only 0.03AbL    6    ) |B ( T D?`  ` f ( s *AS ??    ( <? gi ,$D0 h< size of atom"3  ( <$? ,$D0 __One need to decrease the n/nC and/or wait very long The Coulomb glass is an isolator (n/nC < 1)`# 6  ;   d2 ( < ?  2 ( B?@  > H ( 0޽h ? ff3333f`D CCPGJ|C( Phiw   0 ? 2The dynamics of the Coulomb gap: the gap evolution.3 2! 3  / BD@`T  gSingle-particle DOS&  C@8 W IWBB  3 [\BB B 3 [\  < v T 0$ BB  3 z { BB B 3 z {    <@   V 0.1$ BB   3 BB  B 3    <d`  V 0.2$ BB   3 BB B 3   <d- V 0.3$ BB  3 BB B 3   <M V 0.4$ BB  3 L[BB  3 BB  3 i>j[BB  3 ij  <$!W T-4$ BB  3 L [BB  3  BB  3  > [BB  3     <_  W T-2$ BB  3 G LH [BB  3 G H BB  3  > [BB   3    ! <4  W T 0$ BB " 3 L[BB # 3 BB $ 3 %>&[BB % 3 %& & <XW T 2$ BB ' 3 L[BB ( 3 BB ) 3 d>e[BB * 3 de + <tW T 4$ BB , 3 L[BB - 3 H . C [ 1 <$p  ~R0 = 0.1H  HB 2 C @ 3 B"QCDEFEE:QiN&, b   8<rDAL&{]sm>L7 5m!"#%F&}'() +*W,Y-^./01}g234 6A7Jx8f9:<R= >??z@+BbCGDEG(K0)D)`**+',,(W--.l//xM00U|122C3$3s4? 55:6&6ji788S099`::-;'3 ? ? M@ @ }A Bp B DC|CsD; EE;FtFiG'HPH0II`JJK]'LnLVMM(NNOOMPPU"Qa@   7 <p W1.2& HB 8 C `?a 9 rB"QC2%DEF2%S%%*%$$n$$F$$s$>$I$`$#8###{b#F#S""+ " Y" "m ! q!E I!  z f ` { 680{ZR?n*L lDd_27 zS QR  )z EmoDgT^(6Wz}Q  )!! "l""HC## $$$^%L%k 4&s &M ' x' ' O(i ( ') ) ) j* * A+ +2 , , ,M ]- -P 4. . / w//rO0;0d &11 1 j2z 2~ A3m 3 4 4 4E \5 5 36 6 7w77N88z%9c9+9i::A;;<<<[=s=2>[> ?v??N@D@v%AA]AhBB@CCEDhDD[E>E3FF GM uG G MH!H !$I3!Iw!I!hJ!J3"@KL"Km"L"L"L"ZM"M#2N0#N!# OL#tOk#O#LP#P#"Q#@5 : <Dq W1.4& HB ; C f3P?Q < B]JCh3DE F(f3Z3\Y3h3]3{S3Q3:Q3C3A3ZI3L3A3y=3738!332W222v2252o2V2U J2 ?2 02s 2 13 1 1 1R l1 G1 (1q 0 010~0F0O0/]/n...4.-;-M -, +l"+*,*J)(J'& v%i $"(! ?Hbgy&rE  e $AC#b! B @!!!`"" # ~# # >$!$$n]%%l&Q{&&M!;'o"'h#'#Y([$($$)#x)#)#8*!* *W++p, v,,6--m- T../Ps//20&0Y0"R11*2q220333O4>4-5o5>5.666M7\7 8l8e8+9B99J:d :! ;"j;?"; #)<#&g>'>'&?(?(?)F@)@S*AG*dA*A*$Bv+Bz+B,CCE,C,D,bD-D>-!E-E-E.AFD.F.G._G.G/HA/~HW/H/=I/I/I20]JD0@C8 = BCDEPFX_C>]7|PM<X|Z),@   > <q W2.0& HB ? C f3@?A @ B{CC>DE F(f3>>k>>>q>>>v>>%>|>>*>l>j>/g>U>J>5O>;>8>:9>1>.>@+>>>F > = =J = = =P = = f=V _= 2= =[ = < (E;(;(v;D):)9)7J*5*[3*0O+-+ ++'T,$,"-Z--._.Q./e//0k0091o112u22#3{33.)444/55f5\ 36>!6r"6P"97J$7$7&?8&8'8(D95)95*9+I:+:,:,O;t-;t.;.TK2`>2>3 ?3d?34?e4@4j@X5@5A5oAV6A6B6uB6BS7$C7{C7@HJ A B C^DEF**VG\?X agVvm1qfA Tw%}3+DGN1 u  5   ;   +A 2 3 :F X ] ^UX@H!  B <r W5.0& HB C C 0@0 D BNC@DE F(@N@@@~@@I@@@y@@D@@@t@@?}@z@ x@oi@c@:S@A@ $@k @ @5 ? ? ?f ? ?0 s? P? ,?a ? ?,>>>\j>=>'+>==Wl=="<O<;RW; ;S:99M;876}54I320xw/-D+p)(t$/#>f  o9%H j65 e ! 0!o!!3a"Z" +# #.#[$ $T!&%&%*%]0U&4&8!'[;'$>'?P(@(@)?)_>)F{4>4>5K?6?6@7{@7@7FAt8A8B;9vB9B9BC9C<: D:qD:D-;K)>KF>^L]>Lz>)M>M>M>YN>N ?@wI E BcCuDE F( e"09d[`ucu@  H F C [f G s *A- ??P p  J <dr?0p* HT = 0K H  0޽h ? ff3333f$ `0$@(  $ $ 0s ? 2The dynamics of the Coulomb gap: the gap evolution.3 2! 3  e$ <$spp0  }Single-particle DOS& F `pG {$ p`GBB I$ 3 HIBB J$B 3 HI K$ <Dt k V 0& BB L$ 3 e f BB M$B 3 e f  N$ <t` P  X 0.1& BB O$ 3 BB P$B 3  Q$ <u`+P X 0.2& BB R$ 3 BB S$B 3  T$ <du`MP X 0.3& BB U$ 3 BB V$B 3  W$ <u`pP X 0.4& BB X$ 3 ,HBB Y$ 3  Z$ <$vLG V 5& BB [$ 3 ,HBB \$ 3  ]$ <v G X 2.5& BB ^$ 3  , HBB _$ 3    `$ <v  G U1& BB a$ 3 j,kHBB b$ 3 jk c$ <DwG X 0.5& H d$ C H` j$ 0BPNCZDEdFl8 $ c  Q %N i!x" z#t$Kd% L& ,'{(+t+/Y4M?9g@FEZPNU34@v F` m$ 0BPNC DEdFl  ciQj 4 %)N i!x"vz#t$d%)L&O,' ( +g+/646?9g@E6PN34@! ` p$ 0BPNC3DEdFl3 31c/-Q+*v'%l%N "i!yx"z#t$jd%L& ,'@ (5 ++/4?9g@EPN34@O vT @0@ x$# d d g$ 6wp0 YEnergy:$   h$ 6x W S0$ BB i$ 3 @ k$ 0dx@S V-0.5$ BB l$ 3 @@@ n$ 6x[] T-1$ BB o$ 3 @ q$ 6$ya[ T-2$ BB r$ 3 @` s$ 0BPNC DEdFl   c v Q[ %N gi!x"z#t$d%L&,'(++/5 4 ?9 g@ El PN 34@ 7 H t$ C Hl w$ 0AU S??  SH $ 0޽h ? ff3333f `Xp668( w 8 8 0y ? 2The dynamics of the Coulomb gap: the gap evolution.3 2! 3  8 <ypp0  }Single-particle DOS& F `pG 8 p`G<B 8 # HI<B 8B # HI 8 6z k V 0& <B 8 # e f <B  8B # e f   8 6{` P  X 0.1& <B  8 # <B  8B #   8 6d{`+P X 0.2& <B 8 # <B 8B #  8 6`MP X 0.3& <B 8 # <B 8B #  8 6`pP X 0.4& <B 8 # ,H<B 8 #  8 6ܡLG V 5& <B 8 # ,H<B 8 #  8 6dܡ G X 2.5& <B 8 #  , H<B 8 #    8 6ܡ  G U1& <B 8 # j,kH<B 8 # jk 8 6$ݡG X 0.5& B  8 3 H` !8 0BPNCZDEdFl8 $ c  Q %N i!x" z#t$Kd% L& ,'{(+t+/Y4M?9g@FEZPNU34@v F` "8 0BPNC DEdFl  ciQj 4 %)N i!x"vz#t$d%)L&O,' ( +g+/646?9g@E6PN34@! ` #8 0BPNC3DEdFl3 31c/-Q+*v'%l%N "i!yx"z#t$jd%L& ,'@ (5 ++/4?9g@EPN34@O vT @0@ $8# d d %8 6ݡp0 YEnergy:$   &8 6ݡ W S0$ BB '8 3 @ (8 0Dޡ@S V-0.5$ BB )8 3 @@@ *8 6ޡ[] T-1$ BB +8 3 @ ,8 6ߡa[ T-2$ BB -8 3 @` .8 0BPNC DEdFl   c v Q[ %N gi!x"z#t$d%L&,'(++/5 4 ?9 g@ El PN 34@ 7 B /8 3 Hf 08 s *AU X??  X` 18 c $AT Y??e X  Y 28 0ߡ?    R0(t0) = 1.25L    pB 38 H D?@ @ 48 <?   JFitting:    58 <? 4 ,$D0 J1.26  68 6D? @ ,$D0 Cb H 8 0޽h ? ff3333f .:< ( w < < 0d ? 2The dynamics of the Coulomb gap: the gap evolution.3 2! 3  < <pp0  }Single-particle DOS& :B < 3 HI:B <@ 3 HI < <$ k V 0& :B < 3 e f :B  <@ 3 e f   < < `P  X 0.1& :B  < 3 :B  <@ 3   < <+`P X 0.2& :B < 3 :B <@ 3  < <DM`P X 0.3& :B < 3 :B <@ 3  < <p`P X 0.4& :B < 3 ,H:B < 3  < <LG V 5& :B < 3 ,H:B < 3  < <d G X 2.5& :B < 3 , H:B < 3   < < G U1& :B < 3 ,jkH:B < 3 jk < <$G X 0.5& @  < C HX !< 0BPNCZDEdFl8 $ c  Q %N i!x" z#t$Kd% L& ,'{(+t+/Y4M?9g@FEZPNU34@v F48 ?d  7<d?  %< <?df$ YEnergy:$   &< <Y+ S0$ HB '< C  @ /< C Hf 0< s *AU [??  [` 1< c $AT \??e X  \ 2< 0?    R0(t0) = 1.25L    pB 3< H D?@ @ 4< <B ?   JFitting:    5< <C ? 4 ,$0 J1.26 8 % ] :<@0 h2 8< <C ? S  kYu (SCE+numeric, 1999): b = 1   9< <4D ?% ] x*Malik and Kumar (analytical., 2004): b = 2++ & H < 0޽h ? ff3333fx! !!C  (    0E  ? 8The dynamics of the Coulomb gap: the experiment proposal29 2! 9 d  < ?p@ $F   P@l  < ?rB  B D?@rB  B D?0rB  B D? rB   B D? P PrB   B D? p   <TE ?` QConduction band     <F ?0vz XImpurity band (donors)  vB   N D?P pl 0 &  0  & ,$D0~B  N D? 0 0 n  s *AK ??0 &    <G ? TT = 300K 3  l    ,$D 0  <TH   1  ~R0 = 0.1H  BB  3   BB B 3 BB   3  e eBB !B 3 eeBB " 3  BB #B 3   BB $ 3  , , BB %B 3 , , BB & 3  BB 'B 3   BB ( 3   BB ) 3  BB * 3 { { BB + 3 { { BB , 3 eeBB - 3 e e BB . 3 OPBB / 3 O P BB 0 3 :;BB 1 3 : ; BB 2 3 %%BB 3 3 % % BB 4 3 BB 5 3   BB 6 3 BB 7 3   BB 8 3 BB 9 3   BB : 3 BB ; 3   BB < 3 BB = 3   H > C   ? B"QCDEFEE:QiN&, b   8<rDAL&{]sm>L7 5m!"#%F&}'() +*W,Y-^./01}g234 6A7Jx8f9:<R= >??z@+BbCGDEG 2  BI ?  > 2  B4J ? @ >   <J ?0 * ]Random occupations of sites  l @00 ,$D0l  < ?@00  <TK ?p0 JT = 7K   <L ?0v*  n/nC = 8%L   H  0޽h ? ff3333f ]U= ( w   0tL  ? 8The dynamics of the Coulomb gap: the experiment proposal29 2! 9 d  < ?p@ $F   P@l  < ?rB  B D?@rB  B D?0rB  B D? rB   B D? P PrB   B D? p   <D ?` QConduction band     <M ?0vz XImpurity band (donors)  vB   N D?P vB  N D?0 0 f  s *AK ??0  &  F     BB A 3   BB BB 3 BB C 3  e eBB DB 3 eeBB E 3  BB FB 3   BB G 3  , , BB HB 3 , , BB I 3  BB JB 3   BB K 3   BB L 3  BB M 3 { { BB N 3 { { BB O 3 eeBB P 3 e e BB Q 3 OPBB R 3 O P BB S 3 :;BB T 3 : ; BB U 3 %%BB V 3 % % BB W 3 BB X 3   BB Y 3 BB Z 3   BB [ 3 BB \ 3   BB ] 3 BB ^ 3   BB _ 3 BB ` 3   H a C   b B"QCDE,F4MZ}; DtF ;ti~#1  Q`  e /'  "W ~ =BM }   D & tx  G ;ej19a<w(W.M  i}!""D##t$s %7%.:&&Ei'E(>(K0)D)`**+',,(W--.l//xM00U|122C3$3s4? 55:6&6ji788S099`::-;'3 ? ? M@ @ }A Bp B DC|CsD; EE;FtFiG'HPH0II`JJK]'LnLVMM(NNOOMPPU"Qa@ H c C    <TN   1  ~R0 = 1.0H    <?   oRelaxing (1st hour) ...*     pB  H D?@2  B?` > 2  Bd?@p > 2  Bħ? @ >   B$?  JT = 7K   <䨡?0w *,$0 \Random occupation of sites    < ?0@` ,$D0  <?0v*  n/nC = 8%L   H  0޽h ? ff3333f rj; ( w   0d ? 8The dynamics of the Coulomb gap: the experiment proposal29 2! 9 d  < ?p@ $F   P@l  < ?rB  B D?@rB  B D?0rB  B D? rB   B D? P PrB   B D? p   <$?` QConduction band     <䫡?0vz XImpurity band (donors)  vB   N D?P vB  N D?0  0 f  s *AK ??0  &  F     BB f 3   BB gB 3 BB h 3  e eBB iB 3 eeBB j 3  BB kB 3   BB l 3  , , BB mB 3 , , BB n 3  BB oB 3   BB p 3   BB q 3  BB r 3 { { BB s 3 { { BB t 3 eeBB u 3 e e BB v 3 OPBB w 3 O P BB x 3 :;BB y 3 : ; BB z 3 %%BB { 3 % % BB | 3 BB } 3   BB ~ 3 BB  3   BB  3 BB  3   BB  3 BB  3   BB  3 BB  3   H  C    rB"QC2%DEF2%S%%*%$$n$$F$$s$>$I$`$#8###{b#F#S""+ " Y" "m ! q!E I!  z f ` { 680{ZR?n*L lDd_27 zS QR  )z EmoDgT^(6Wz}Q  )!! "l""HC## $$$^%L%k 4&s &M ' x' ' O(i ( ') ) ) j* * A+ +2 , , ,M ]- -P 4. . / w//rO0;0d &11 1 j2z 2~ A3m 3 4 4 4E \5 5 36 6 7w77N88z%9c9+9i::A;;<<<[=s=2>[> ?v??N@D@v%AA]AhBB@CCEDhDD[E>E3FF GM uG G MH!H !$I3!Iw!I!hJ!J3"@KL"Km"L"L"L"ZM"M#2N0#N!# OL#tOk#O#LP#P#"Q#@ t H  C    <  1  ~R0 = 1.2H    <d?   pRelaxing (2nd hours) ...*     pB  H D?@2  Bĭ?` > 2  B$?@p > 2  B? @ >   B䮡?  JT = 7K   <?0v*  n/nC = 8%L   H  0޽h ? ff3333f9!   <0 M ( w   0d ? 8The dynamics of the Coulomb gap: the experiment proposal29 2! 9 d  < ?p@ $F   P@l  < ?rB  B D?@rB  B D?0rB  B D? rB   B D? P PrB   B D? p   <?` QConduction band     <?0vz XImpurity band (donors)  vB   N D?P vB  N D?0  0 f  s *AK ??0  &  .8    BB  3   BB B 3 BB  3  e eBB B 3 eeBB  3  BB B 3   BB  3  , , BB B 3 , , BB  3  BB B 3   BB  3   BB  3  BB  3 { { BB  3 { { BB  3 eeBB  3 e e BB  3 OPBB  3 O P BB  3 :;BB  3 : ; BB  3 %%BB  3 % % BB  3 BB  3   BB  3 BB  3   BB  3 BB  3   BB  3 BB  3   BB  3 BB  3   H  C    B]JCh3DE F(f3Z3\Y3h3]3{S3Q3:Q3C3A3ZI3L3A3y=3738!332W222v2252o2V2U J2 ?2 02s 2 13 1 1 1R l1 G1 (1q 0 010~0F0O0/]/n...4.-;-M -, +l"+*,*J)(J'& v%i $"(! ?Hbgy&rE  e $AC#b! B @!!!`"" # ~# # >$!$$n]%%l&Q{&&M!;'o"'h#'#Y([$($$)#x)#)#8*!* *W++p, v,,6--m- T../Ps//20&0Y0"R11*2q220333O4>4-5o5>5.666M7\7 8l8e8+9B99J:d :! ;"j;?"; #)<#&g>'>'&?(?(?)F@)@S*AG*dA*A*$Bv+Bz+B,CCE,C,D,bD-D>-!E-E-E.AFD.F.G._G.G/HA/~HW/H/=I/I/I20]JD0@ & 2  BCDEPFX_C>]7|PM<X|Z),@H  C    <D  1  ~R0 = 1.4H    <d5?   oRelaxing (3rd hour) ...*     pB  H D?@2  B5?` > 2  B$5?@p > 2  B5? @ >   B5?  JT = 7K   <5?0v*  n/nC = 8%L   H  0޽h ? ff3333fQ" !!?m!( w   05 ? 8The dynamics of the Coulomb gap: the experiment proposal29 2! 9 d  < ?p@ $F   P@l  < ?rB  B D?@rB  B D?0rB  B D? rB   B D? P PrB   B D? p   <d5?` QConduction band     <5?0vz XImpurity band (donors)  vB   N D?P vB  N D?0 p0 f  s *AK ??0  &  BF 0 Z2   <B  # 0 1A 2<B B # I1Y2<B  # 0 A <B B # IY<B  # 0 A <B B # I Y <B  # 0 \ A \ <B B # I\ Y\ <B  # 0 A <B B # I Y <B  # 0 )0 1<B  # 0 0 <B  #  " 1<B  #   <B  # )1<B  #   <B  # "1<B  #   <B  # )1<B  #   <B  # "1<B  #   <B  # )1<B  #   <B  # "1<B  #   <B  # )1<B  #   <B  # o"p1<B  # o p <B  # Y)Z1<B  # Y Z B  3 0 Y1B  3 0 Y1  BNC@DE F(@N@@@~@@I@@@y@@D@@@t@@?}@z@ x@oi@c@:S@A@ $@k @ @5 ? ? ?f ? ?0 s? P? ,?a ? ?,>>>\j>=>'+>==Wl=="<O<;RW; ;S:99M;876}54I320xw/-D+p)(t$/#>f  o9%H j65 e ! 0!o!!3a"Z" +# #.#[$ $T!&%&%*%]0U&4&8!'[;'$>'?P(@(@)?)_>)F{4>4>5K?6?6@7{@7@7FAt8A8B;9vB9B9BC9C<: D:qD:D-;K)>KF>^L]>Lz>)M>M>M>YN>N ?@0  '  65 1  ~R0 = 5.0H  l ` k  ` k ,$D0N @[   @[ h  c $AI ??p[    <5? @0 : W1/2&3   <d5?` k  SE s formula for DC conductivity (gap in g(E ) around the Fermi level):LH) 9    pB  H D?@2  B5?@p > 2  B$5? @ >   <5?0X *,$D0 \Pseudo-grand state reached    BD5?  JT = 7K   <5?0v*  n/nC = 8%L   H  0޽h ? ff3333f JB@4 ( w 4 4 0$5 ? 8The dynamics of the Coulomb gap: the experiment proposal29 2! 9 d 4 < ?p@ $F  4 P@l 4 < ?rB 4 B D?@rB 4 B D?0rB 4 B D? rB  4 B D? P PrB  4 B D? p  4 <5?` QConduction band    4 <D5?0vz XImpurity band (donors)  vB  4 N D?P vB 4 N D?0 p0 f 4 s *AK ??0  &  pB :4 H D?@2 ;4 B5?@p > 2 <4 B 5? @ >  =4 <5?0X *,$0 \Pseudo-grand state reached   >4 <D5?  JT = 7K X ?4 <5? z |Change from 1/4-law (Mott) to 1/2-law (SE) not because of the cooling of the sample, but because it relaxed for 104 s (3h).,}s     d  @4 <d5?0v*  n/nC = 8%L   H 4 0޽h ? ff3333fC BBGG.B(    05 ? 2The dynamics of the Coulomb gap: the gap evolution.3 2! 3   <$5@`T  gSingle-particle DOS&  ?F W  W<B  # [\<B B # [\  6D5 v T 0$ <B  # z { <B  B # z {    65@   V 0.1$ <B   # <B  B #    65`  V 0.2$ <B  # <B B #   6d5- V 0.3$ <B  # <B B #   6Ą5M V 0.4$ <B  # L[<B  # <B  # i>j[<B  # ij  6$5!W T-4$ <B  # L [<B  #  <B  #  > [<B  #     65_  W T-2$ <B  # G LH [<B  # G H <B   #  > [<B ! #    " 65  W T 0$ <B # # L[<B $ # <B % # %>&[<B & # %& ' 6D5XW T 2$ <B ( # L[<B ) # <B * # d>e[<B + # de , 65W T 4$ <B - # L[<B . # B / 3 [ 0 6d5  ~R0 = 0.1H  BB 1 3 @ 2 B"QCDEFEE:QiN&, b   8<rDAL&{]sm>L7 5m!"#%F&}'() +*W,Y-^./01}g234 6A7Jx8f9:<R= >??z@+BbCGDEG(K0)D)`**+',,(W--.l//xM00U|122C3$3s4? 55:6&6ji788S099`::-;'3 ? ? M@ @ }A Bp B DC|CsD; EE;FtFiG'HPH0II`JJK]'LnLVMM(NNOOMPPU"Qa@   6 6$5 W1.2& BB 7 3 `?a 8 rB"QC2%DEF2%S%%*%$$n$$F$$s$>$I$`$#8###{b#F#S""+ " Y" "m ! q!E I!  z f ` { 680{ZR?n*L lDd_27 zS QR  )z EmoDgT^(6Wz}Q  )!! "l""HC## $$$^%L%k 4&s &M ' x' ' O(i ( ') ) ) j* * A+ +2 , , ,M ]- -P 4. . / w//rO0;0d &11 1 j2z 2~ A3m 3 4 4 4E \5 5 36 6 7w77N88z%9c9+9i::A;;<<<[=s=2>[> ?v??N@D@v%AA]AhBB@CCEDhDD[E>E3FF GM uG G MH!H !$I3!Iw!I!hJ!J3"@KL"Km"L"L"L"ZM"M#2N0#N!# OL#tOk#O#LP#P#"Q#@5 9 65 W1.4& BB : 3 f3P?Q ; B]JCh3DE F(f3Z3\Y3h3]3{S3Q3:Q3C3A3ZI3L3A3y=3738!332W222v2252o2V2U J2 ?2 02s 2 13 1 1 1R l1 G1 (1q 0 010~0F0O0/]/n...4.-;-M -, +l"+*,*J)(J'& v%i $"(! ?Hbgy&rE  e $AC#b! B @!!!`"" # ~# # >$!$$n]%%l&Q{&&M!;'o"'h#'#Y([$($$)#x)#)#8*!* *W++p, v,,6--m- T../Ps//20&0Y0"R11*2q220333O4>4-5o5>5.666M7\7 8l8e8+9B99J:d :! ;"j;?"; #)<#&g>'>'&?(?(?)F@)@S*AG*dA*A*$Bv+Bz+B,CCE,C,D,bD-D>-!E-E-E.AFD.F.G._G.G/HA/~HW/H/=I/I/I20]JD0@C8 < BCDEPFX_C>]7|PM<X|Z),@   = 65 W2.0& BB > 3 f3@?A ? B{CC>DE F(f3>>k>>>q>>>v>>%>|>>*>l>j>/g>U>J>5O>;>8>:9>1>.>@+>>>F > = =J = = =P = = f=V _= 2= =[ = < (E;(;(v;D):)9)7J*5*[3*0O+-+ ++'T,$,"-Z--._.Q./e//0k0091o112u22#3{33.)444/55f5\ 36>!6r"6P"97J$7$7&?8&8'8(D95)95*9+I:+:,:,O;t-;t.;.TK2`>2>3 ?3d?34?e4@4j@X5@5A5oAV6A6B6uB6BS7$C7{C7@HJ @ B C^DEF**VG\?X agVvm1qfA Tw%}3+DGN1 u  5   ;   +A 2 3 :F X ] ^UX@H!  A 6D5 W5.0& BB B 3 0@0 C BNC@DE F(@N@@@~@@I@@@y@@D@@@t@@?}@z@ x@oi@c@:S@A@ $@k @ @5 ? ? ?f ? ?0 s? 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A. .h . -4 - - -e `- C-2 -,,c,e,0,,+ad+?+-****_)2)+((['\&&(r%$Z$ZT#p"&,!- $X<#fV4!%TT   !PW   N!!"B""(L##$~$$yJ% %&G|&& G' 't(z(&(;D)2)*u**B++n,ls,,k@-- .Qq..V>/F/ 0o00:11h2m2 2 83( 32 4 j4# 4 65E 5 6] h6 6 47;77Cf88<299U9*d::/;G;;a<]<-===^>>+???\@@)A_ A Am!ZB!BT"%C$#C$C-$XD%D>%#E%E%EN&VF&F'!G_'G'G)(SHO(Hs(I(I(I(QJc)J)K*K1*K*OL*L*M+Md+Mi+LN}+N+O+@P+ H  C    <*F?0@* N T = 0.2 d.u.  F  @  0@  6ā5  ~R0 = 0.1H  BB  3 @  6dxF W1.0& BB  3 p?q  6xF W1.2& BB  3 `?a  6$yF W1.4& BB  3 f3P?Q  6yF W2.0& BB  3 f3@?A  6yF W5.0& BB  3 0@0H  0޽h ? ff3333fm  (  ]  0{F ?  Outline 0. Example of disordered system with long-range interactions 1. Simulation procedure for computation of single particle DOS 2. The dynamics of the Coulomb gap 3. The Coulomb glass and the glass transition 4. Phononless AC conductivity in Coulomb glass 29=!*06     ! H  0޽h ? ff3333f 004!(    0{F ? }'Edwards-Anderson order parameter (EAOP)(( 2' ( 2  Bd{F? @  > X2  0?p `d2  <?` 0 2  B$|F?   > 2  B|F? p0 > 2  B|F?`   > 2   BD}F?   > X2   0?00 X2   0?0 ` X2   0?pX2   0? @pd2  <?  d2  <?pPd2  <?@@ d2  <?0P^  6? 0   0}F?p t2T = 0K - no transitions in the pseudo-ground state33 3   6d~F? RN = 10 K = 0.5 2 "8 `.   ` 2  B~F?@ >   0F?@ ROccupied donor 2  `2  0? @ l2  <? @   0DF?@  O Empty donor 2     0F?@ `.  UOccupied acceptor 2  [l p  p ,$D0  <āF?p/  kT = 6K - some transitions (VRH)  xB  H D?p`p xB  H D?   l p`@ -`p@,$D0 ! 0F?p`Z g%T = 100K - a lot of transitions (NNH)&& & x@ 0@ ,0@xB " H D? P@PxB # H D?@ @~B $ N D?~B % N D? @  ~B &B N D?` 0 ~B ' N D?00 ~B ( N D? P ~B ) N D?0 ` ~B * N D?p ~B + N D?` 0 ` / s *AF |??R8 |$D0tl  P0 4,$D 0xB 2 H D? P0xB 3 H D? P0H  0޽h ? ff3333fI gI_I@qPH(   b!  l 0:  0: ,$D0  B$F?0o * kT = 6K - some transitions (VRH)   @ @  J: ~B < N D?2 = HF?8@ >  > BF?X  Ftime  2 ? HDF?@ > 2 @ BF?@h > 2 A BF?(@ > 2 B BdF?x@  > 2 C BF? @X  > 2 D B$F? @  > 2 E BF?h @  > 2 F HF? @H > 2 G HDF?@ > 2 H HF?X@ > 2 I HF?@8 >   <dF?z ^ni.    0$F ? }'Edwards-Anderson order parameter (EAOP)(( 2' (   0F? t2T = 0K - no transitions in the pseudo-ground state33 3  8   : xB - H D?oo2  HF?0  >  . <DF?P Ftime  2 / BF?  > 2 0 BF? ` > 2 1 BdF?   > 2 2 BF?p   > 2 3 B$F? P  > 2 4 BF?   > 2 5 BF?`   > 2 6 BDF? @ > 2 7 BF?  > 2 8 BF?P  > 2 9 BdF? 0 >   <P? ^ni.   z       ,$D0 ! 6P? `  g%T = 100K - a lot of transitions (NNH)&& &  T  Z  # p  ~B L N D?  2 M HP?8 `  >  N BTP?X` Z  Ftime  2 O BP? `  > 2 P HP? h`  > 2 Q BtP?( `  > 2 R HP?x  `  > 2 S B4P? X `  > 2 T BP? `  > 2 U HP?h `  > 2 V BTP? H`  > 2 W HP? `  > 2 X HP?X `  > 2 Y BtP? 8`  >   BP?   ^ni.  l `0  `0 ,$D0z  c y `0,$D0r w B ?Pn pw  v# @c,$D0B Z T DԔ?Ow O~B ] N D?p__ _ B4P?( Ftime  B k T DԔ?w B l T DԔ?w B m T DԔ??w ?B n T DԔ?w B o T DԔ? w B p T DԔ?/ w / B q T DԔ? w  B r T DԔ? w B s T DԔ?w B t T DԔ?ow oB u T DԔ?w   p   0C ,$D0r { B ?p |N p   p B } Z DԔ?O O0~B ~ N D?p  BP?( Ftime  B  Z DԔ? 0B  Z D3Ԕ? B  Z D3Ԕ?P PB  Z D3Ԕ? B  Z D3Ԕ? B  Z D3Ԕ?@ @ B  Z D3Ԕ? B  Z DԔ? 0B  Z DԔ? 0B  Z DԔ?o o0B  Z DԔ? 0   0    0 ,$D0r  B ? 0|N       B  Z DԔ? h~B  N D?@  BP?  Ftime  B  Z DԔ?` ``B  Z D3Ԕ?B  Z D3Ԕ?@@@B  Z D3Ԕ?@B  Z D3Ԕ?P 7P B  Z D3Ԕ? 7 B  Z D3Ԕ?@B  Z DԔ? `B  Z DԔ? hB  Z DԔ? hB  Z DԔ? `@l  ,$D0h  c $AJ p?? ` p  <TP? m+Order parameter (per analogy to spin glass),, ,   <P? Uq = 1.0    <P? Uq = 0.8    <tP?P J  Uq = 0.1  H  0޽h ? ff3333f P;*(    0P ? fGlass transition( 2  _  0P? Davies, Lee, Rice (lattice model, no actual acceptors): glass transition from random (T > 0.3K) to ordered (T = 0K) system of {ni}8     2 " BQ?` `  > X2 # 0?`p ^2 $ 6?@ ` 2 % BdQ?@ p  > 2 & BĀQ? p @ > 2 ( B$Q?` > d2 ) <?`` X2 * 0? ` @X2 + 0? p` ^2 - 6?@ p ^2 . 6? p@^2 / 6?p ` ^ 1 6? 0 X2 2 0?`2 3 BQ?@   > 2 4 BQ? @ > 2 5 BDQ?`  > X2 6 0?` d2 7 <?@  ^2 8 6? @d2 9 <?` vB : N D? @  ; 0Q?  Oa  H  0޽h ? ff3333fF ` t(      0Q ? fGlass transition( 2  _   0Q? Davies, Lee, Rice (lattice model, no actual acceptors): glass transition from random (T > 0.3K) to ordered (T = 0K) system of {ni}8        <$Q?  MYu: glass transition in Coulomb glass is the phase transition of second orderNN L B   s *޽h ? ff3333f   pT(    0DQ ? fGlass transition( 2  _  0Q? Davies, Lee, Rice (lattice model, no actual acceptors): glass transition from random (T > 0.3K) to ordered (T = 0K) system of {ni}8       <dQ?  MYu: glass transition in Coulomb glass is the phase transition of second orderNN L   C ALT:\Wydawnictwa\Wyklad TIDS\Prezentacja - Coulomb glass\img\Davies_tytul.bmp `K<  c $޽h ? ff3333f$ X(    0dQ ? fGlass transition( 2  _  0ĆQ? Davies, Lee, Rice (lattice model, no actual acceptors): glass transition from random (T > 0.3K) to ordered (T = 0K) system of {ni}8       <$Q?  MYu: glass transition in Coulomb glass is the phase transition of second orderNN L   C ANT:\Wydawnictwa\Wyklad TIDS\Prezentacja - Coulomb glass\img\Davies_wykres1.bmp0X<  c $޽h ? ff3333fF  t(    0Q ? fGlass transition( 2  _  0DQ? Davies, Lee, Rice (lattice model, no actual acceptors): glass transition from random (T > 0.3K) to ordered (T = 0K) system of {ni}8       <Q?  MYu: glass transition in Coulomb glass is the phase transition of second orderNN L B  s *޽h ? ff3333f  r j   (    0dQ ? fGlass transition( 2    0ĉQ? ?Our model: random positions of sites, actual acceptors present@@ @ 8   @  w 2  H$Q?p   > f2  6?  r2   B? @ 8  @  ~B  N D? @   0Q?   Oa  Tl  z  z,$D0N 0p   0p 2  HQ?pp > f2  6?0`  r2  B?     <DQ? ( @But closely locates groups are not present in the lattice modelAA A +  <Q? @ z uThe crowded group has higher energy than surrounding and preserve its occupation unchanged even for high temperaturesvv v N  @   0 ~B  N D? @   0P?   Oa  $  <4P?6  vThe lattice model well describes remote donors interaction<< < H  0޽h ? ff3333f$; ::I <:( L{X  H H 0F ? fGlass transition( 2  ' H 0F? Our model: EAOP has the same value for N=100 and N=500, glass transition has long exponential tail (nonzero values even for T > 300K)   H B   fEA order parameter&   H < s lt [d.u.]6   H <įQ  i our model& ZL AF ;I# 7pBB H 3 AF I BCDEF @  "I <Q@ Davies, Lee, Rice&BL P8Ux :I# 8 %xBB  I 3 PXUYH 7I C 8x4B CI # 4B DI@ #  EI 6Q; S $ 4B FI #   4B GI@ #    HI 6DQ? -w  V 0.2$ 4B II #   4B JI@ #    KI 6QC -w  V 0.4$ 4B LI # 4B MI@ #  NI 6QG-w  V 0.6$ 4B OI # 4B PI@ #  QI 6dQK-w  V 0.8$ 4B RI # 4B SI@ #  TI 6IJQOw T 1$ 4B UI # p4B VI #  WI 6$Qm X 0.001$ 4B XI # |4B YI # 4B ZI # |4B [I # 4B \I # |7 8 4B ]I # 7 8 4B ^I # p| } 4B _I # | }  `I 6QL m W 0.01$ 4B aI # |T U 4B bI # T U 4B cI # |r s 4B dI # r s 4B eI # |  4B fI #   4B gI # pJ K 4B hI # J K  iI 6Q m V 0.1$ 4B jI # |" # 4B kI # " # 4B lI # |@A4B mI # @A4B nI # |4B oI # 4B pI # p4B qI #  rI 6DQzm T 1$ 4B sI # |4B tI # 4B uI # | 4B vI #  4B wI # |4B xI # 4B yI # p4B zI #  {I 6QVtm U 10$ 4B |I # |4B }I # 4B ~I # |4B I # 4B I # |mn4B I # mn4B I # p4B I #  I 6Qmm V 100$ : I 3 4 I BSCCEDEXF` Z G !T &+%.!-t257]= :B I 3 h i @ I C J d :B I 3 q: ; r@ I C   :B I 3  @ I C  !t :B I 3  JK @ I C c A :B I 3   @ I C HD:B I 3 onop@ I C a:B I 3 : I 3 H H 0޽h ? ff3333ff" !!T!( aZ~0%   0ĵQ ? fGlass transition( 2   P BQ   fEA order parameter&  F `   `  R <DQ  i our model& hT AF S# 7pBB T 3 AF U BCDEF @ V 6Q`  Davies, Lee, Rice (B = 2)&BT P8Ux W#  8%xBB X 3 PXUYH Y C 8x4B \ # 4B ]@ #  ^ 6dQv5 T 0$ 4B _ # ; < 4B `@ # ; <  a 6ĸQ 5  V 0.1$ 4B b #   4B c@ #    d 6$Qc 5"  V 0.2$ 4B e # ()4B f@ # () g 6Q5 V 0.3$ 4B h # 4B i@ #  j 6QO5 V 0.4$ 4B k # 4B l # 4B m # YZ4B n # YZ4B o # 4B p # 4B q # `a4B r # `a4B s # 4B t #  u 6DQl2  T 1$ 4B v #  4B w #  4B x # M N 4B y # M N 4B z # z{4B { # z{4B | # cd4B } # cd4B ~ # "#4B  # "#4B  # 4B  # 4B  # OP4B  # OP4B  # 4B  # 4B  # 9:4B  # 9:  6Q U 10$ 4B  # 4B  # :  3   BVC>DE F( f+/ 7y9:F=V>@  BC)DEF)) @ :B  3   BC)DEF)) @# :B  3   BC)DEF)) @  :B  3    BC)DEF)) @_ y :B  3     BC)DEF)) @ !w>:B  3 LM  BC)DEF)) @L8:B  3 scdt  |BECY)DEF }''EY) @9@  C  c :B  3 ; < @  C \" x :B  3 M N @  C d:B  3 9::  3   6K s lt [d.u.]6    6dK h  T 3$ '  0K? Our model: EAOP has the same value for N=100 and N=500, glass transition has long exponential tail (nonzero values even for T > 300K)  H  0޽h ? ff3333f# P#H#UW"( w   0K ? fGlass transition( 2    <DK   fEA order parameter&  F `  W `   < K  i our model& hT AF # 7pBB  3 AF   BCDEF @   6 K`  Davies, Lee, Rice (B = 2)&BT P8Ux  #  8%xBB   3 PXUYH   C 8x4B  # 4B @ #   6$!Kv5 T 0$ 4B  # ; < 4B @ # ; <   6!K 5  V 0.1$ 4B  #   4B @ #     6!Kc 5"  V 0.2$ 4B  # ()4B @ # ()  6D"K5 V 0.3$ 4B  # 4B @ #   6"KO5 V 0.4$ 4B  # 4B  # 4B  # YZ4B   # YZ4B ! # 4B " # 4B # # `a4B $ # `a4B % # 4B & #  ' 6#Kl  W 100K$ 4B ( #  4B ) #  4B * # M N 4B + # M N 4B , # z{4B - # z{4B . # cd4B / # cd4B 0 # "#4B 1 # "#4B 2 # 4B 3 # 4B 4 # OP4B 5 # OP4B 6 # 4B 7 # 4B 8 # 9:4B 9 # 9: : 6d#Kr X 1000K$ 4B ; # 4B < # : = 3  > BVC>DE F( f+/ 7y9:F=V>@ ? BC)DEF)) @ :B @ 3  A BC)DEF)) @# :B B 3  C BC)DEF)) @  :B D 3   E BC)DEF)) @_ y :B F 3    G BC)DEF)) @ !w>:B H 3 LM I BC)DEF)) @L8:B J 3 scdt K BECY)DEF }''EY) @9@ L C  c :B M 3 ; < @ N C \" x :B O 3 M N @ P C d:B Q 3 9:: R 3  S 6#K `s iT [K]6  T 6$$K h W 300K$ & U <$K?b : Si:P n/nC = 8%\ ,' V 0D%K? Our model: EAOP has the same value for N=100 and N=500, glass transition has long exponential tail (nonzero values even for T > 300K)  H  0޽h ? ff3333f# Q#I#UW"(    0&K ? fGlass transition( 2    <&K   fEA order parameter&  F `  W `   <'K  i our model& hT AF # 7pBB  3 AF   BCDEF @   6D(K`  Davies, Lee, Rice (B = 2)&BT P8Ux  #  8%xBB   3 PXUYH   C 8x4B  # 4B @ #   6(Kv5 T 0$ 4B  # ; < 4B @ # ; <   6K 5  V 0.1$ 4B  #   4B @ #     6Kc 5"  V 0.2$ 4B  # ()4B @ # ()  6dK5 V 0.3$ 4B  # 4B @ #   6ĘKO5 V 0.4$ 4B  # 4B  # 4B  # YZ4B   # YZ4B ! # 4B " # 4B # # `a4B $ # `a4B % # 4B & #  ' 6$Kl  W 197K$ 4B ( #  4B ) #  4B * # M N 4B + # M N 4B , # z{4B - # z{4B . # cd4B / # cd4B 0 # "#4B 1 # "#4B 2 # 4B 3 # 4B 4 # OP4B 5 # OP4B 6 # 4B 7 # 4B 8 # 9:4B 9 # 9: : 6Kr X 1965K$ 4B ; # 4B < # : = 3  > BVC>DE F( f+/ 7y9:F=V>@ ? BC)DEF)) @ :B @ 3  A BC)DEF)) @# :B B 3  C BC)DEF)) @  :B D 3   E BC)DEF)) @_ y :B F 3    G BC)DEF)) @ !w>:B H 3 LM I BC)DEF)) @L8:B J 3 scdt K BECY)DEF }''EY) @9@ L C  c :B M 3 ; < @ N C \" x :B O 3 M N @ P C d:B Q 3 9:: R 3  S 6K `s iT [K]6  T 6DK h W 590K$ ' U <K?b  Si:P n/nC = 69%\ ,' V 0dK? Our model: EAOP has the same value for N=100 and N=500, glass transition has long exponential tail (nonzero values even for T > 300K)  H  0޽h ? ff3333f C;0 (  =  0ěK? - introduced to demonstrate that the Coulomb fields induce sine ordering at low temperatures - in our (random site) model it depends on N!"b,    0$K ? )Modified Edwards-Anderson order parameter(* 2) * `  c $A ??@\   <K?@PH:  \normal  spin     <K?@ X  L spin in system with no interactions'' '   0K?   }the difference report the contribution to EA order parameter related to presence of the Coulomb interaction within the system~~ ~ H  0޽h ? ff3333fF FyFF( w   0dK?  r0Goes to zero faster than the EA order parameter.11 1   0ĞK ? )Modified Edwards-Anderson order parameter(* 2) *   <K5  omodified EA order parameter&    0DKxo 8 gour model (N = 100)&    0KX g mDavies, Lee, Rice (B = 2)&  L P8Ux # BB   3 PXUYH   C 8x<F !   <B   # <B  B #   6dK`  T 0$ <B  #   <B B #     6ġK! }  V 0.2$ <B  #   <B B #     6$K!@   V 0.4$ <B  #   <B B #     6K! ]  V 0.6$ <B  # pq<B B # pq  6K!! V 0.8$ <B  # <B B #   6DK> T 1$ <B  # PQ<B B # PQ   6N! V 1.2$ <B ! # <B " #  # 6N W 0.01$ <B $ # <B % # <B & # <B ' # <B ( #  ! <B ) #  ! <B * #   <B + #   <B , #   <B - #   <B . #  ! <B / #  ! <B 0 # ] ^ <B 1 # ] ^ <B 2 #   <B 3 #   <B 4 #   <B 5 #    6 6DN   V 0.1$ <B 7 #   <B 8 #   <B 9 #   <B : #   <B ; # ? @ <B < # ? @ <B = #   <B > #   <B ? #   <B @ #   <B A # @A<B B # @A<B C # }~<B D # }~<B E # <B F # <B G # <B H #  I 6N T 1$ <B J # !"<B K # !"<B L # <B M # <B N # _`<B O # _`<B P # <B Q # <B R # <B S # <B T # _`<B U # _`<B V # <B W # <B X # <B Y # <B Z # <B [ #  \ 6Nt< U 10$ <B ] # AB<B ^ # AB<B _ # <B ` # B a 3   b BVC9DEDFLn  < ES- (m$m,q*1)1\96@&8*E8J8R9V9#$@ c BC)DEF)) @_BB d 3  e BC)DEF)) @BB f 3  g BC)DEF)) @$BB h 3  i BC)DEF)) @ ] RBB j 3 2 *3 + k BC)DEF)) @ x BB l 3    m BC)DEF)) @ * yBB n 3  Q R o BC)DEF)) @ ?  BB p 3    q BC)DEF)) @' | BB r 3 R S  s BC)DEF)) @ m 6 BB t 3     u BC)DEF)) @c  BB v 3    w BC)DEF)) @^gBB x 3 3@4A y BC)DEF)) @FcBB z 3 qr { BC)DEF)) @sVBB | 3 +, } BC)DEF)) @~@BB ~ 3   BC)DEF)) @(~BB  3 ST  BRC5BDE4F< n hN-, (C$1Y4\9V<@RA*EAJAR5B@5SBB  3 67BB  3 ^BB  3 DEBB  3 lBB  3   BB  3   CBB  3   BB  3   BB  3  q? rBB  3  J BB  3 ' c } d BB  3 R < S BB  3  ! BB  3   BB  3  ^ BB  3 3 4 BB  3 FBB  3 qbrBB  3 VBB  3 +u,BB  3 @BB  3 zBB  3 (~BB  3 ST  B{1C0DE(F0  `[ !n"$(j*,{10@H  C yBB  3 H  C HBB  3 pqH  C  ` BB  3  H  C  P K BB  3  w ! x H  C    BB  3  H  C  P BB  3 % & H  C  x BB  3  H  C   bBB  3  ; <H  C BB  3 B  3   6N  lt [d.u.]6  `F /~  BB  3 RS  BC)DEF)) @q/~  0Nho ( gour model (N = 500)&  :B  3 pB  H D?XXH  0޽h ? ff3333fz  )ݐ (    0N ? }5New order parameter related to the electron diffusion6 26 6   0DN?  KThe quest for the phase transition parameter in the Coulomb glass (Lee, Yu)LL H     <N?,$D0 Analysis of  Binder g suggests that the glass transition is the phase transitionRR R .  <N?,$D0 LNeed for the parameter which rapidly goes to zero. What mechanism behind it?MM M 2l 0  ݐ0 ,$D02  BdN? @  > `2  0?p  `l2  <?0 ` 2  BN?   > 2  B$N? p0 > 2  BN?`   > 2  BN?   > `2  0?00 `2  0?0 ` `2  0?p`2  0? @pl2  <?  l2 Ð <?Ppl2 Đ <?@ @ l2 Ő <?0Pf Ɛ 6? 0  ǐ 6N? RN = 10 K = 0.5 2 8N `.  Ȑ ` 2 ɐ BN?@ >  ʐ 0N?@ ROccupied donor 2  `2 ː 0? @ l2 ̐ <? @  ͐ 0N?@  O Empty donor 2    ΐ 0DN?@ `.  UOccupied acceptor 2  ~B Ґ N D? P?P~B Ӑ N D?? @~B Ԑ N D?~B Ր N D? @  ~B ֐B N D?_ 0 ~B א N D?/0 ~B ؐ N D? P ~B ِ N D?/ ` ~B ڐ N D?o ~B ې N D?_ 0 `n ܐ <N? ,$D0 Our idea is to trace the electron instead of the site s occupation!EE E H  0޽h ? ff3333fh    .( Ow@   0N ? }5New order parameter related to the electron diffusion6 26 6 R  0dN?J The quantities which may be a base for a new order parameter: - the distance of the electron from the final site to the initial one - the total hops length - the number of hops   * <N?r ,$D0 0The value of the new order parameter may be the percentage of all electrons for which: - the distance between final and initial site is smaller than ... - the total hops length is smaller than ... - the number of hops is smaller than ...   + <$N? z,$D0 We just measure the percentage of the electrons which stay for all simulation at the initial position. Thus the new order parameters = 1 only if the EA order parameter = 1.  D - 6dN?  ,$D0 fThe disadvantage: its value is related to the measurement time (stronger than the EA order parameter)g 2g g ^ . 6ĎN?,$D0 The advantage: its value more rapidly goes to zero => the phase transition apply to the electron diffusion in the Coulomb glass 2  H  0޽h ? ff3333fC dC\C0B( w   0$N ? }5New order parameter related to the electron diffusion6 26 6 4B   #  2 4B  @ #  o    6N ]K V 0& 4B  #  2 4B @ #  o   6N -  X 0.2& 4B  #  2 4B @ #  o   6DNb   X 0.4& 4B  # y2z4B @ # yoz  6ND  X 0.6& 4B  # [2\4B @ # [o\  6N&  X 0.8& 4B  # =2>4B @ # =o>  6dN] V 1& 4B  #   4B  # =V  6đNu Z 0.001& 4B   #   4B ! # =J4B " #   4B # # =J4B $ #   4B % # =J4B & #   4B ' # =V ( 6$Nu  Y 0.01& 4B ) #  4B * # = J4B + #   4B , # =  J4B - #  4B . # = J4B / #  4B 0 # = V 1 6Nt d  X 0.1& 4B 2 #  4B 3 # = J4B 4 #  4B 5 # = J4B 6 #  z{ 4B 7 # =z{J4B 8 #   4B 9 # =V : 6N V 1& 4B ; #   4B < # =J4B = #   4B > # =J4B ? #  ^_ 4B @ # =^_J4B A #   4B B # =V C 6DNJ W 10& 4B D #   4B E # =J4B F #   4B G # =J4B H #  BC 4B I # =BCJ4B J #   4B K # =V L 6N) X 100& : M 3 = < P BMC1?DE\FdE? V<cZO<S sF d"]%'"*h).0477:G<=WE?M1?/0@H  R BEC6DEF6E6 @.b:B S 3 HI T BFC7DEF7F7 @8Hl:B U 3 RcdS V BFC6DEF6F6 @]# :B W 3 w x X BFC6DEF6F6 @  :B Y 3   Z BEC7DEF7E7 @M :B [ 3 h i  \ BEC6DEF6E6 @ :B ] 3   ^ BFC7DEF7F7 @ ( N:B _ 3 4  5 ` BFC6DEF6F6 @f, b :B a 3 G H  b BFC6DEF6F6 @ :B c 3   d BEC6DEF6E6 @:  m:B e 3 S T f BFC6DEF6F6 @ A :B g 3 & '  h BEC6DEF6E6 @< s :B i 3 W X  j BFC7DEF7F7 @ :B k 3   l BFC6DEF6F6 @1 g Q:B m 3 7L M 8 n BEC6DEF6E6 @ , :B o 3    p BEC7DEF7E7 @ FH :B q 3 . +,/  r BEC6DEF6E6 @E y :B s 3 _ `  t BFC6DEF6F6 @z  :B u 3    v BFC7DEF7F7 @ K( :B w 3  01  x BEC6DEF6E6 @n * :B y 3    z BEC6DEF6E6 @ v :B { 3    | BFC6DEF6F6 @   :B } 3   <  BMC>DE\FdE: HVc< s'10P6"<% >'D>*>.Z>4>7>G<>WEr>Mh>/0@G :B  3 -a:B  3 -a:B  3 7Hk:B  3 7Hk:B  3 d# :B  3 d# :B  3   :B  3   :B  3 M /:B  3 M /:B  3 M :B  3 M :B  3 / ( b:B  3 . ( b:B  3 ", c V:B  3 ", c V:B  3  2:B  3  1:B  3    :B  3    :B  3 6 A i :B  3 5 A i :B  3 ; < s o :B  3 ; < s n :B  3 I } :B  3 I } :B  3  1 g :B  3  1 g :B  3  :B  3  :B  3  F :B  3  F :B  3   :B  3   :B  3   :B  3   :B  3  K :B  3  K :B  3  * :B  3  * :B  3  v :B  3  v :B  3    :B  3    8 `  `  N 6dN uP fEA order parameter&  @ r $BB O 3  ~ BFC6DEF6F6 @irBB  3   6$N` Pu kthe new order parameter&  &@ p BB  3 BB  3 pBB  3 p:  3 =    <N%  oEA and new order parameters&     6N  lt [d.u.]6  l 0p  p0 ,$D0  <dN?p  c!about 1% of electrons got stacked"" " xB B H D?0  H  0޽h ? ff3333fp7 77@ p6( w   0ėN ? -Glass transitions versus Coulomb gap smearing(. 2- .   <N>`  Single-particle DOS for E = 0H  :B 0 3  %C :B 1@ 3    2 <Nq Q V 0& :B 3 3 o %Cp :B 4@ 3 o p  5 <DN5   X 0.1& :B 6 3 4%C5:B 7@ 3 45 8 <N X 0.2& :B 9 3 %C:B :@ 3  ; <Nk X 0.3& :B < 3  %& :B = 3 %& > <:S  Y 0.01& :B ? 3  ~ :B @ 3 ~ :B A 3  GH :B B 3 GH :B C 3   :B D 3  :B E 3  EF :B F 3 EF :B G 3   :B H 3  :B I 3   :B J 3  :B K 3  / 0 :B L 3 / 0 :B M 3  i j :B N 3 i j :B O 3  :B P 3   Q <:S $   X 0.1& :B R 3  :B S 3  :B T 3  :B U 3  :B V 3  O P :B W 3 O P :B X 3  :B Y 3  :B Z 3    :B [ 3   :B \ 3  e f :B ] 3 e f :B ^ 3  :B _ 3  :B ` 3  :B a 3  :B b 3   :B c 3  d <D;S D V 1& :B e 3  no :B f 3 no :B g 3  89 :B h 3 89 :B i 3   :B j 3  :B k 3  67 :B l 3 67 :B m 3   :B n 3  :B o 3   :B p 3  :B q 3    :B r 3   :B s 3  Z[ :B t 3 Z[ :B u 3   :B v 3  w <;S $ W 10& :B x@ 3 45:B z@ 3 o p :B |@ 3   @ ~ C %   BCC&DE,F4 &&&v %e#Z D#,^C@<9   BDC]DEF]D] @ Y @B  C  9:   BEC]DEF]E] @ q @B  C     BEC^DEF^E^ @l   @B  C     BDC]DEF]D] @ 0 @B  C     BEC]DEF]E] @ @B  C    BEC]DEF]E] @F * @B  C c d   BEC]DEF]E] @ @B  C    BEC]DEF]E] @B @B  C c d   BEC]DEF]E] @/ Ji@B  C L*+M  BEC]DEF]E] @Y@B  C <=  BZC=DEF ==l2 'd Z @SE @B  C  %f @B  C d EF @B  C  @B  C  @B  C  + @B  C  @B  C C D@B  C & a@B  C S T@B  C 6 p8 . .  6S5   W0.5&  } <>Sq  W1.0& T  1 #  p .   <d?S  fEA order parameter&  HB  C x1  <$@S  kthe new order parameter&  HB  C p1qT %E  # %E   BCC%DE@FH%c%$ 0$ d#!x3$<!( -3:q|>qCC!$@%G 8  BCC%DEPFX%ct%$ J# "f x2h!Z3$5!(N-(3 7|>C),@%E @  C %   6@Sp 8  lt [d.u.]6  H  0޽h ? ff3333f? a?Y?P>( w   0DAS ? *Glass transitions vs Coulomb gap evolution(+ 2*      <BS?`  Single-particle DOS for E = 0H 4B  #  %C 4B @ #     6dBSq Q V 0& 4B  # o %Cp 4B @ # o p    6BS5   X 0.1& 4B   # 4%C54B  @ # 45   6$CS X 0.2& 4B   # %C4B @ #   6CSk X 0.3& 4B  #  %& 4B  # %&  6CS  Y 0.01& 4B  #  ~ 4B  # ~ 4B  #  GH 4B  # GH 4B  #   4B  #  4B  #  EF 4B  # EF 4B  #   4B  #  4B  #   4B  #  4B  #  / 0 4B   # / 0 4B ! #  i j 4B " # i j 4B # #  4B $ #   % 6DDS $   X 0.1& 4B & #  4B ' #  4B ( #  4B ) #  4B * #  O P 4B + # O P 4B , #  4B - #  4B . #    4B / #   4B 0 #  e f 4B 1 # e f 4B 2 #  4B 3 #  4B 4 #  4B 5 #  4B 6 #   4B 7 #  8 6DS D V 1& 4B 9 #  no 4B : # no 4B ; #  89 4B < # 89 4B = #   4B > #  4B ? #  67 4B @ # 67 4B A #   4B B #  4B C #   4B D #  4B E #    4B F #   4B G #  Z[ 4B H # Z[ 4B I #   4B J #  K 6ES $ W 10& 4B L@ # 454B M@ # o p 4B N@ #   : O 3 %  P BCC&DE,F4 &&&v %e#Z D#,^C@<9  Q BDC]DEF]D] @ Y :B R 3  9:  S BEC]DEF]E] @ q :B T 3    U BEC^DEF^E^ @l   :B V 3    W BDC]DEF]D] @ 0 :B X 3    Y BEC]DEF]E] @ :B Z 3   [ BEC]DEF]E] @F * :B \ 3 c d  ] BEC]DEF]E] @ :B ^ 3   _ BEC]DEF]E] @B :B ` 3 c d  a BEC]DEF]E] @/ Ji:B b 3 L*+M c BEC]DEF]E] @Y:B d 3 <= e BZC=DEF ==l2 'd Z @SE :B f 3  %f :B g 3 d EF :B h 3  :B i 3  :B j 3  + :B k 3  :B l 3 C D:B m 3 & a:B n 3 S T:B o 3 6 pF . p . q 0dES0 l g(0)8  BB r 3 I.P$ s 6$FS $g(0) (Grannan and Yu; lattice model)8%#6     BB t 3 @.AD z %0 u %0,$0 v <Spp0e  fEA order parameter&   w 6S W0.0&  x 6DS5   W0.5&  y 6Sq  W1.0& T  1 z#  p .  { 6dS  fEA order parameter&  BB | 3 x1 } 6$S  kthe new order parameter&  BB ~ 3 p1qT %E  # %E   BCC%DE@FH%c%$ 0$ d#!x3$<!( -3:q|>qCC!$@%G 8  BCC%DEPFX%ct%$ J# "f x2h!Z3$5!(N-(3 7|>C),@%E :  3 % l  ,$D 0B  H D? ,$D0  <S?,$D0 g!electrons leave the initial sites"" " xB  H D?P Pl     ,$D 0xB  H D?    <S?   `almost none electron rests  xB  H D?  Il P P,$D 0xB  H D?   <dS?P Ycomplete randomness  xB  H D? l p`E  `pE ,$D0  <$S?p`E Z Xgap starts to form  xB  H D?PPPl P  P ,$D 0~B  N D?   BS?P  Tgap is formed  xB  H D? H  0޽h ? ff3333f' }'u'`Ud0 &( 3   0S ? }'The gap transition with time limitation(( 2' (   <dS`@  gEA order parameters&  :B  3 ():B @ 3 (r):B   3   :B  @ 3  r    <S   X 0.2& :B   3   :B @ 3  r   <$S   X 0.4& :B  3  :B @ 3 r   <S X 0.6& :B  3 :B @ 3 r  <Sx X 0.8& :B  3 :B @ 3 r  <DS]m V 1& :B  3  &:B  3    <Si Z 0.001& :B  3 rs&:B  3 rs:B  3 &:B  3 :B   3  !&:B ! 3  !:B " 3  ef&:B # 3 ef  $ <Si Y 0.01& :B % 3 <=&:B & 3 <=:B ' 3 X Y &:B ( 3 X Y :B ) 3  &:B * 3  :B + 3  / 0 &:B , 3 / 0  - <dSi  X 0.1& :B . 3   &:B / 3   :B 0 3 " # &:B 1 3 " # :B 2 3  &:B 3 3  :B 4 3   &:B 5 3   6 <Si 0  V 1& :B 7 3  &:B 8 3  :B 9 3 &:B : 3 :B ; 3 }~&:B < 3 }~:B = 3  &:B > 3   ? <Vif. W 10& :B @ 3 &:B A 3 :B B 3 &:B C 3 :B D 3 FG&:B E 3 FG:B F 3  &:B G 3   H <D Vi, X 100& @ I C & J < V0 p [R = 0.3&  L BJCDE,F4 s!Dm)U,2':BJ@ M <!V0 P [R = 0.5&  O BJCDE,F4 *s!Pm) ,25':B~J@z P <!V` 0]  \R = 0.75&   R BJC-DE8F@!Aj$s! c$m)#-?$,2)':x,B-J- @w  S <$"V0 0   [R = 1.0&  U BJC CDEHFP&g@jX Ps!c$m)8+,2;':-A/?BBBJ C%(@  V <"V p  \No limit&  B X BJCGDE\FdF ;a:=C j rs!+c$R!{&'m).#-6,2>5B':7EBFJG/0@& [ <#V`  lt [d.u.]6   ] 0d$V?  d"The gap starts to form for R > 1.2## # & ^ <$V?  ,$D0 ,, while the order is established for R < 1.5,-* - l .  1 d. 1,$D0B ` N D?. . ,$D0B c H D?  1,$D0H  0޽h ? ff3333fT* ))pU|)( w   0%V ? }'The gap transition with time limitation(( 2' (   <D&V*?  hnew order parameters&   P 6'V`  lt [d.u.]6   Q 0'V?  d"The gap starts to form for R > 1.2## # & R <$(V?  ,$0 ,, while the order is established for R < 1.5,-* - $F ."  ."BB X 3 ; < BB YB 3 k; <  Z <(V   X 0.2& BB [ 3 ) * BB \B 3 k) *  ] <(V   X 0.4& BB ^ 3 BB _B 3 k ` <D)V X 0.6& BB a 3 BB bB 3 k c <)V X 0.8& BB d 3 BB eB 3 k f <*Vlm V 1& BB g 3 1BB h 3   i <d*V u" Z 0.001& BB j 3 w%x1BB k 3 wxBB l 3 %1BB m 3 BB n 3 #%$1BB o 3 #$BB p 3 hi1BB q 3 hi  r <*Vu," Y 0.01& BB s 3 ?%@1BB t 3 ?@BB u 3 Z %[ 1BB v 3 Z [ BB w 3  % 1BB x 3   BB y 3 0 1 1BB z 3 0 1  { <$+V u " X 0.1& BB | 3  % 1BB } 3   BB ~ 3 ! %" 1BB  3 ! " BB  3  % 1BB  3   BB  3   1BB  3     <+V u3 " V 1& BB  3  % 1BB  3   BB  3 %1BB  3 BB  3 y%z1BB  3 yzBB  3 1BB  3    <Vhu0" W 10& BB  3 %1BB  3 BB  3 %1BB  3 BB  3 ?%@1BB  3 ?@BB  3 1BB  3    <TVu." X 100& H  C 1  BJClDE,F4 ki!(P1z9BEJl@L  BJC^ DE,F4  6k!(01_ z9^ B JE @  BJC,DE8F@Tk! #(,$1`(z9|*B+J, @p "  BJC8EDEHFP,pqkp N !{$#2(?1Cz9Ds>DB8EJ8E%(@J  BJCQFDE\FdB qpl k{r ,d 6=!D#E&E(E, F12#6l@p"Q" @n<  B"QC6DE<FD IMd<w" &(.#,5.6>266R6l@6"Q 6 @nH  C W~@ 8  8 HB  C pqHB  C 00  B4V?8  IT = 0.1   BV?8  IT = 0.2 4B " # M N 4B #@ # M N  $ 6V =  V 0& 4B % # op4B &@ # op ' 6TV0  X 0.5& 4B ( # 4B )@ #  * 6V= V 1& 4B + # C M 4B , # 4B . # H M 4B / # 4B 0 # H M 4B 1 # 4B 2 # H  M 4B 3 #   4B 4 # H opM 4B 5 # op4B 6 # H M 4B 7 # 4B 8 # H M 4B 9 # 4B : # H CDM 4B ; # CD4B < # H xyM 4B = # xy4B > # C M 4B ? # 4B A # H M 4B B #  4B C # H M 4B D #  4B E # H   M 4B F #   4B G # H ~  M 4B H # ~  4B I # H M 4B J #  4B K # H   M 4B L #   4B M # H Q R M 4B N # Q R 4B O # H M 4B P #  4B Q # C M 4B R #  4B T # H M 4B U #  4B V # H M 4B W # 4B X # H '(M 4B Y # '(4B Z # H M 4B [ # 4B \ # H M 4B ] # 4B ^ # H $%M 4B _ # $%4B ` # H `aM 4B a # `a4B b # H M 4B c # 4B d # C M 4B e # 4B g # H M 4B h # 4B i # H M 4B j # 4B k # H 67M 4B l # 674B m # H M 4B n # 4B o # H M 4B p # 4B q # H 23M 4B r # 234B s # H opM 4B t # op4B u # H M 4B v # 4B w # C M 4B x # : z 3 M  { B"QC(DE<FD Iddw"\ &( #,.!>2J'6(l@'"QP& @ | B"QCz=DE<FD I dAw"- &J ( "#,7.;>2z=6=l@<"QP= @. : } 3 M    BtV?@ O EA order par.  ~ B4V?`t  Pnew order par.  lF 8   8 BB  3 pqBB  3 00  <V?8  IT = 0.1   <?X?8  IT = 0.2   6t@X P \R [d.u.]&  lF 8   p * BB  3 pqBB  3 00  <@X?8  IT = 0.1   <4AX?8  IT = 0.2   <AX? 8  M From random     <BX?% M From ground     <tCX?@U M From ground   H  0޽h ? ff3333f xP( ]`@ P P 0CX ?  xOutline 0. Example of disordered system with long-range interactions 1. Simulation procedure for computation of single particle DOS 2. The dynamics of the Coulomb gap 3. The Coulomb glass and the glass transition 4. Phononless AC conductivity in Coulomb glass a) Experimental results of AC conductivity measurements b) brief introduction to Shklovskii and Efros s model of zero-phonon AC hopping conductivity of disordered system c) results of computer simulations for T = 0K 29=!+,4z+x     y        p H P 0޽h ? ff3333f tP(  t t 0DX ? hExperimental results& 2  l t 0A<  ??    ` t 0DX?N M. Lee and M.L. Stutzmann, Phys. Rev. Lett. 87, 056402 (2001) E. Helgren, N.P. Armitage and G. Gru:ner, Phys. Rev. Lett. 89, 246601 (2002)@,K  l t 0A=  ??P0    < t c $޽h ? ff3333f |P(  | | 0EX ? hExperimental results& 2  ` | 0FX?N M. Lee and M.L. Stutzmann, Phys. Rev. Lett. 87, 056402 (2001) E. Helgren, N.P. Armitage and G. Gru:ner, Phys. Rev. Lett. 89, 246601 (2002)@,K  l | 0A>  ?? &   l | 0A?  ?? c   < | c $޽h ? ff3333f 0 (  0. 0 0tFX ? TShklovskii and Efros s model Pair of sites*+ 2*,     ~ 0 B A@  ? ?P@   0 64GX?` Z oHamiltonian of a pair of sites: 2    l  P J  0P J,$D0 0 B A  ? ?)Pz   0 6GX?  J HSite energy is determined by Coulomb interaction with surrounding pairsI 2I I F 0 Z  0 0 Z   0 BTHX?0 Z <Overlap of site s wavefunctionn  0 s *A  ??4 =   l i`   0` i ,$D0n  0 s *A  ??i    0 BHX? `   BNotice that because of overlap I(r)  intuitive states can be not good eigenstatesXT  G     0 c $A  ??@wn 8  $D0l @ 8 0 @8,$D0h 0 c $A  ??P&  h 0 c $A  ??  V 0 6IX?@  Anyway four states are possible a priori: " there is no electron, so no interaction and energy is equal to 0 " there is one electron at the pair (two states) " there are two electrons at the pair , 2   n 0 s *A  ??P    n 0 s *A  ??P08  r 0 B3Ԕ? 0H 0 0޽h ? ff3333fU    &8 m (  8 8 c $A  ??  8  $D0. 8 0tIX ? TShklovskii and Efros s model Pair of sites*+ 2*,      8 6IX?`Z HOnly pairs with one electron are interesting in context of conductivity:I 2I I ` 8 c $A  ??   8 B A  ? ?0 8  $D0l  %8,$D0n 8 s *A  ??`8    8 BJX? cThe isolated sites base    0 8# 0 ,$D0h 8 c $A  ?? 0p   8 <TKX? O Normalisation   8 c $A  ??  8  $D0!l pXF &8pXF,$D0h  8 c $A  ??pF  h "8 c $A  ?? X   #8 <X?   Qwhere  H 8 0޽h ? ff3333f@ &B@pD(  @ @ 6X?`  gEnergy which pair much absorb or emit to move the electron between split-states (from to ):h 2h h . @ 0tX ? TShklovskii and Efros s model Pair of sites*+ 2*,     ` @ c $A  ??2kK  l b ?@h,$D0 @ 64X? [Source of energy: photons  n @ s *A  ??b  ` @ c $A  ??@D  ` @ c $A  ??  l @  )@@ ,$D0h @ c $A  ??p Z   @ <X?p j  _And finally the conductivity:  l @ <Ԕ? P~B @ NDԔ?P p P  @ <TX?p @   AShklovskii and Efros formula for conductivity in Coulomb glassesBB,    . h  @ c $A  ??   h !@ c $A  ??0   xB #@ HDԔ?``P~B $@ NDԔ?`Pp P %@ <X?p _ &Numerical calculation (esp. for T > 0)'', l   ;@ ,$D0= @ <tX? KEnergy which must be absorbed by pairs in unit volume due to el. transition:L'3 =     @ '   :@'   +@ 0X ?9   EQ =   ,@ 0X ?0 8  'QM transition prob. (Fermi Golden Rule)((6       -@ 0TX ?p'  <prob. of finding  proper pair   .@ 0X ?0   M   /@ 0X ? 0  M   0@ 6tX ?` '   |.prob. of finding photon with energy equals to// + n 1@ s *A3  ??     2@ 0X ?p' >  M  h 5@ c $A3  ??0W l   l    >@  ,$D0 8@ B ??  ,$D0z @ 6A4  ???   tl    B@ ,$D0xB @@ H D?  xB A@ H D?  H @ 0޽h ? ff3333fz  (    0X ? lPair distribution (T = 0K)$ 2  `  c $A, n??`Y  n`  c $A. o?? P o  C A/hT:\PPT\referat Rackeve (Coulomb glass)\dane\cg3.gifp`  <TX?   &N=400, T=0K, NMonte-Carlo=1000, a=0.27*' 6 H  0޽h ? ff3333f @8 (    0X ? v$Pairs mean spatial distance (T = 0K)$% 2$ % 8 P| @P|,f   6A8?| h  c $A3  ??p U      BX?Pqp  @pair mean spatial distance     <X? >  vMott s formula       <TX?2c, M simulations     <k?J,$D0 In contradiction to the Mott s assumption the distribution of pairs distances is very wide]]   ?   <4 k?. CN=1000, K=0.5, 2500 realisations periodic boundary conditions, AOERDD D `  c $A1  ??p 00   H  0޽h ? ff3333fg$ ##@D#( fNf   0 k ? s!Pair energy distribution (T = 0K)$" 2! " `  c $A3 M ?? 0   M }8 (H  BH( <B   # : X <B  B #      6T!kg ;  T 0$  <B   # : X <B  B #     6!kh $  X 50000$  <B  # :X<B B #   6"k(  Y 100000$  <B  # :X<B B #   6t"k(v Y 150000$  <B  # :X<B B #   6"k(_ Y 200000$  <B  # :|X}<B B # |}  64#k(H Y 250000$  <B  # : ; <B  # :|;  6#k4   T 0$  <B  #   <B  # |   6#k}4   T 1$  <B ! # R S <B " # R|S # 6T$k 4   T 2$  <B $ #   <B % # | & 6$k4 '  T 3$  <B ' # j k <B ( # j |k  ) 6%k! 4  T 4$  <B * #  <B + #  |  , 6t%k 4 ?  T 5$  <B - #  <B . #  |  / 6%k: 4  T 6$  <B 0 #   <B 1 # | 2 64&k4 X  T 7$  <B 3 #   <B 4 # | 5 6&kR4   T 8$  <B 6 # ' ( <B 7 # '|( 8 6&k4 p  T 9$  <B 9 #   <B : # | ; 6T'kJ4   U 10$  B < 3 :|  ? BLCr7DE F(r7r7cL77)6>65S432{t10Bb.y,r*k@(%1v# 2ZC RI  t  r s j w8   fb < '  M Q"7zy@t<i0X ) f  HpHZ7`&Oyz< > !"hH##-$<%%W{&' ' S( (F!u)!* "c*n"*"U+5#+#F,#,]$-$q-$%-%.%r.M&.&/'`/v'/'/<('0(v0)0f)0)1+*P1*1*1U+1+2,D2~,t2,2D-2-2 .&3m.Z3.y33/3/3/3\0304"1;41N41h4K242434t3434:444555c5-5595*6O56]56q5R7}57~585|8585B9595:5k:5:51;6;6;6[<*6<36 =?6=K6=V6J>e6>p6?s6s?~6?69@6@6@6bA6A6(B6B6B6QC6C6D6{D6D6@E6E6F6jF6F60G6G6G6YH6H6I6I6I6IJ6J6K6qK6K68L7L7@: BB @ 3   B A 3 :| l  * D *,$D0xB  HD?   <(k?0* ]We work here!!!     <(k?PP $ 'N=500, T=0, K=0.5, aver. over 100 real.((L C 6)k?;  5Number of pairs  H  0޽h ? ff3333f  l d   (      0T*k ? eConductivity (T=0K)$ 2  `  c $A3 I ??  I ^  6A6?%  B+k?OPp  uConductivity (arb. un.)B   <jk?p  Helgren et al. (T=2.8K) n = 69%* ,   <kk?`P4 M simulations   v   <kk? . N=500, T=0, K=0.5, aver. over 25k real. (hw)=0.001 (blue), (hw)=0.01 (green)OO6*     Y   6Dlk?   Wn = 69% of nC means a = 0.27 [l69%] (in units of n-1/3)XX 2   0    K    <lk? m^  $Hfixed parameters for Si:P: a = 20, and nC = 3.521024 m-3 (lC = 65.7)lI*      L          0mk?~ d +There is no crossover in numerical results!,,3, H  0޽h ? ff3333f ?70( J   0mk ? eConductivity (T=0K)$ 2  `  c $A3  ??     Bnk?OPp  uConductivity (arb. un.)B v  <nk? . N=500, T=0, K=0.5, aver. over 25k real. (hw)=0.001 (blue), (hw)=0.01 (green)OO6*     4B   # c4B @ #   6Dok "b Y 1e-008$  4B  # - cr. 4B @ # - . 4B  #  cr 4B @ #   4B  # H crI 4B @ # H I 4B  #  cr 4B @ #   4B  #  cr 4B @ #   4B  #  cr 4B @ #   4B  # c crd 4B @ # c d 4B  # < cr= 4B @ # < = 4B   #  c 4B !@ #    " 6ok "i  Y 1e-007$  4B # # 5 cr6 4B $@ # 5 6 4B % #  cr 4B &@ #   4B ' # P crQ 4B (@ # P Q 4B ) #  cr 4B *@ #   4B + # cr4B ,@ # 4B - # cr4B .@ # 4B / # kcrl4B 0@ # kl4B 1 # DcrE4B 2@ # DE4B 3 # "c#4B 4@ # "# 5 6pk"q Y 1e-006$  4B 6 # =cr>4B 7@ # =>4B 8 # cr4B 9@ # 4B : # XcrY4B ;@ # XY4B < # cr4B =@ # 4B > # cr4B ?@ # 4B @ # cr4B A@ # 4B B # scrt4B C@ # st4B D # LcrM4B E@ # LM4B F # )c*4B G@ # )* H 6dpk"y Y 1e-005$  4B I # EcrF4B J@ # EF4B K # cr4B L@ # 4B M # `cra4B N@ # `a4B O # cr4B P@ # 4B Q # cr4B R@ # 4B S # cr4B T@ # 4B U # {cr|4B V@ # {|4B W # TcrU4B X@ # TU4B Y # 1c24B Z@ # 12 [ 6pk" Y 0.0001$  4B \ # cd4B ] # 1cd@4B ^ # 4B _ # 1@4B ` # 4B a # 1@4B b #   4B c # 1  @4B d # gh4B e # 1gh@4B f # 4B g # 1@4B h # 4B i # 1@4B j #  @A4B k # 1@AN l 6$qka X 0.001$  4B m # 4B n # 1@4B o #  4B p # 1 @4B q #  4B r # 1 @4B s #   4B t # 1  @4B u #  4B v # 1 @4B w #  4B x # 1 @4B y # 3 4 4B z # 13 4 @4B { # { | 4B | # 1{ | @4B } #  4B ~ # 1 N  6qka* L  W 0.01$  4B  # ab4B  # 1ab@4B  # XY4B  # 1XY@4B  #  4B  # 1 @4B  # 4B  # 1@4B  # 4B  # 1@4B  # ]^4B  # 1]^@4B  # 4B  # 1@4B  # 4B  # 1@4B  #  674B  # 167N  6qka V 0.1$  4B  # 4B  # 1@:  3 1c  6rk`? ` _ simulations&   :B  3   B 2C &DE F( & &l#"?   r Tv/sJU'e,SL({>c;&Cgp,H=w$7h]; }  K J=!({!!!-"`e"i"]"& #?#s#{#:# $ :$ j$ $ $u $~ #%P P%: {% % % % $&r N&E v&1 & & &# ' 8' ]' ' ' 'Z ' ( 5( W( y( (6 ((()=)^)3})()))m)Q*51*N*k*i*;*'***+/+I+d+}~+H++)+++,0,MI,a, z,,,,,q,^-D-76-CL-Rc-Ey-/-!-----.&.;.P.e.y...q.v..i.U/D/+/>/Q/d/w////]/E/?/9/$ 010K.0=@05Q0c0t00000000`0K 1h1},1U>1/N1$^1"n1'~1V1O141!1.1]1?11 2@kcd   B0C1DE F(1-<JYhvpr]Q`wwti|!/<JXero (5AO[zhbteA42%'3>JuUaliwZ3<8# ! (3=HR]gr|l '1;ENXbl|vsdkSIA,"% &/8oAPJ5S*\en v 98" "*3;CKS\dlt|ytqzWg)  !!(#0@bdk  BCDE F(&-5<DKRZbjqxdgdMQG.8+8#*07>*EL&S"Y`gnt{h,C8 '# & ,29?ELRX^ekqw}viXE}y\V%U+j1Q7^<IB#HNSY_ejpv{xrrjF 4$"'-27:=#B)G/MAR>WE\Mb9g.l!qw|(, ,  @m  BkCDE F( #(-27<BGLQV[`zeij{oyt\y^}R[ON.<WaD9% o `SH,#&(',B1*61:"?C HLQVZ_ch{lrq\uUzU~b]glcbjKKIE640)ww y`b Q%R)K-_1U5<:@>9BBFFJPOPSHWR[B_BdBhQl]pttwx\|RDA=-2  !%(z,Z0)4'8"<.@5D.GKOSW[^%bgk@  BCDE F( "&)-158<@DGKORVZ^aeilptw{ r lo]YL;"F&Z)M-70N3z7T:D>CA8E+H0K8O2RV,Y\`cg"jmq tx{~~{oWPb_\lufG9440%#     w#`&X)\,T/X3E6G9H<5?6B"EIL ORU X[_1b6e=hTkcnNqOtSwLz7~IDH@F4-)& @N4  B+CDE F(  "%(|+v.j1[4K7G:^=]@SCPF_IVLWONRDU.X3[5]L`Vc=f>iEl@o6r8v)y@|00.'+!##&463(33&03@<0OEFM@=00 zp|zwvaIP GK7BB:,,!*$%',)&,)/#2 47 : =?B EHJM PS&UX[ ^ `cfhknqsvy{~xvm`livv|xsiqpppwvmsninVH@1# .9:5('0.IED:+"  +6 -'!$&)+"@4S  BCGDE F(GD88 7 2@*!$&)+.0358:=?BDGILNQSVX[]`bdgilnqsvx{}zshnwp{x~k8@?55)%(22%4#)#%   !#%(*,/1358:<?ACEHJLOQSUXZ\_acehjdmXoPrXtIvIxN{O}VIIXJI0)$%  8&!$&(&'.*   %!@  BCHDE F(//44 5 H @@8218@4.!&#(%,'7)7+.04274<6B8F:E=1?7A8C9E0G#I9L6N>P8R=T7V#XZ \_acegikmortvxz|~lsyzh`YSKRLG7,/+      {meSUD!J#Y%`'W*Q,b.b0g2s4k6`8R:W<S=I?LAFCCEFG7I:K>MFO0Q&S1U/W"Y[]_#ac)d*f*h;j,l/n*p!r/t2vxz|~+7942,,$(& @+R  BgCDE F(  !#$&(*,.024579;=?ABDFHJLNPQSUWY[\^`bdfhikmoqstvxz|~~}|uloohgeZZRNbbgmf\QOVVYOTQILQE??8<A80!%&+&     ynke\d e!\$Q&F(M)S+W-K/O0Z2J4Q5J7N9S;J<B><@?AAC9E0G6H5J;L4M6O6Q+S%T#V&X Y[]^ `bdeg@R.  B@CnDE F(^acgcW W acnld\UQPRRO M"W#\%_'\(S*L,J-M/J1B2Q4O6P7H9G;E<:>B@BABCEEBF8HAJLKBM?O3P-R)T0U*W-Y2Z(\!]!_ abd fgiklnoqstvxy{}~w{mnnqm_f]a[JM@?:BH@784/(2:25/-,77:?:<2:9>2492=HU\\TNOA? A 7 :51/:FMDLX_R N!Y#L$J&='>)7*/,-1/01*214%5!78&:";=>@@  B C9DE(F0 94.!    @  ~B`MCO2DEFeeO2O21K1Y0 G0/B/y.A>. -:-,a 6,* + 2+ * .*J )*)(%(k'2!'&&%R%$$#s#;" "!!\ # |}D y !uc",#q#$m%L&h''d(m)_4**[+,WU-.S./Ou0>1L22H3^4D&55?6~7;F8 97 9 :3 f; / O?' @@"ApB7CDDEXF GGH xIAJKKL`M@Lc   6dsk``  Helgren 69% Si:P&,  B1C\DEF\1\ @p H :B  3  gh   B1C\DEF\1\ @[  :B  3 y z   B1C]DEF]1] @   :B  3     B1C\DEF\1\ @  :B  3     B1C]DEF]1] @D n ~ :B  3 a b   B1C\DEF\1\ @ ; x :B  3  Z [   B1C\DEF\1\ @c :B  3    B1C]DEF]1] @ I ?:B  3 "* + #  B1C\DEF\1\ @  Z:B  3 = >  B0C\DEF\0\ @C:B  3 ab  B0C\DEF\0\ @p:B  3 :  3 1cjB  BD? 0   0$tk?@r  a crossover (?)3   6tk? E  + a = 0.36., 2' , H  0޽h ? ff3333f @0( w   0tk ? 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