Transport quantique dans les verres de spin

HAL Id: tel-00686330

https://tel.archives-ouvertes.fr/tel-00686330

Submitted on 9 Apr 2012

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Transport quantique dans les verres de spin

Thibaut Capron

To cite this version:

Thibaut Capron. Transport quantique dans les verres de spin. Systèmes mésoscopiques et effet Hall quantique [cond-mat.mes-hall]. Université de Grenoble, 2011. Français. �tel-00686330�

THÈSE

Pour obtenir le grade de

DOCTEUR DE L’UNIVERSITÉ DE GRENOBLE

Spécialité : Physique de la Matière Condensée et du Rayonnement

Arrêté ministériel : 7 août 2006

Présentée par

Thibaut CAPRON

Thèse dirigée par Laurent SAMINADAYAR

préparée au sein de l’Institut Néel dans l’École Doctorale de Physique

T

RANSPORT

Q

UANTIQUE DANS LES

V

ERRES DE

S

PIN

Thèse soutenue publiquement le 30 Mars 2011 devant le jury composé de :

Mr Eric VINCENT

DR CEA Saclay, Président du Jury

Mme Hélène BOUCHIAT

DR LPS Orsay, Rapporteur

Mr Giorgio PARISI

PR Université de Rome, Rapporteur

Mr David CARPENTIER

CR ENS Lyon, Membre

Mr Daniel ESTÈVE

DR CEA Saclay, Membre

Mr Laurent LÉVY

PR Université de Grenoble, Membre

Mr Laurent SAMINADAYAR

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:* #1&( (&#3$#&-*, "$'1 ()&* 2"%$C"( <"25 =$(# , $*4 &#( %-'$% 0$+*"#&G$#&-* &( G"2-9 :* $ ()&* +%$(( 1-;"<"2, #1" ()&*( $2" =2-G"* &* 2$*4-0 -2&"*#$#&-*( ."%-; Tg, $*4 #1"2" "C&(#( $ #$%&#

0$+*"#&G$#&-* mi ≡ hSii 6= 0 Hhi 4"*-#"( #1" #1"20$% $<"2$+&*+I9 ?-;"<"2 #1&( 0$+*"#&G$#&-*

&( 4&J"2"*# =-2 "$'1 (&#" i, $*4 #1" +%-.$% $<"2$+"4 0$+*"#&G$#&-* M &( G"2-9

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g9 B-*("M3"*#%5, #1" =2"" "*"2+5 %$*4('$)" $))"$2&*+ ."%-; Tg &( <&";"4

$( $ 0$*5/<$%%"5 )&'#32"9 N3'1 $ %$*4('$)" &( 4")&'#"4 -* >+32" O9P9

@# $*5 #"0)"2$#32" ."%-; Tg, #1" 0$*5/<$%%"5 (#23'#32" $))"$2(, ;&#1 <$%%"5( ;1&'1 $2"

(")$2$#"4 .5 .$22&"2( -= 4&J"2"*# 1"&+1#(9 D1" (5(#"0 &( #13( #2$))"4 &*#- $ ()"'&>' <$%%"5 α, β -2 γ9 D1&( .$22&"2 %$*4('$)" 4&($))"$2( ;1"* -*" +-"( #- T > TgQ #1" (5(#"0 &( 2"(#-2"4 &*

$ 3*&M3" <$%%"5 δ9

!"! #$%& '()**+*

!"#$% &'() !"#$%&'! ('!&)*# +, &"# "'#*%*!"'!%- .&*)!&)*# +, &"# $%/012%--#0 ("%.# .(%!# %. % ,)/!&'+/ +, &#$(#*%&)*#3 45&*%!&#6 ,*+$ 789:3

!"!# $%&'()* '+ %(),( -'('.,/,(

!"#"$%&&'( )$*"$"* +,%-"- .%# /" *"-.$0/"* /' %# )$*"$ +%$%1"2"$ 3,0., 0- 4"$) %/)5" 2," 2$%#-020)# 2"1+"$%26$"( %#* #)#4"$) /"&)37 8# -+0# 9&%--"-( 2," )$*"$ 0- $"&%2"* 2) 2," :$""40#9 ): 2," -+0#-7 ;/)5" Tg( 2," -+0#- $"&%< 5"$' :%-27 ="&)3 Tg( 2,"' %$" :$)4"#7 ; #%26$%& )$*"$

+%$%1"2"$ 0- 2,6- 2," *'#%10.%& -+0# .)#>96$%20)#7 ?,0- 3%- 0#2$)*6."* /' @*3%$*- %#* ;#*"$-)# ABCD

qEA = lim

t→∞N →∞lim hSi(t0)Si(t0+ t)ii EB7F7FG

qEA 0- 2," @*3%$*-H;#*"$-)# )$*"$ +%$%1"2"$7 ?," &0102 )# N *"-09#%2"- 2," 2,"$1)*'H

#%10. &0102( t 2," 201" %#* t0 % $":"$"#." 201"7 ?,0- !""#$%&'!( !) &*# +,'( !(-./"%&'!(+ 30&&

.&"%$&' /" 4"$) :)$ +%$%1%9#"20. -+0#-( %#* 30&& /" #)#4"$) 0: 2," .)#>96$%20)# 0- :$)4"#7 I,"# 2," -+0# 9&%-- 0- .))&"* *)3# /"&)3 Tg( 02 0- 2$%++"* 0# % 5%&&"' -2%2"7 8# 2,0- -2%2"(

2," -+0# .)#>96$%20)# 0- :$)4"#( %#* 0- .)#-0*"$"* 0#*"+"#*"#2 ): 201"7 ?," )$*"$ +%$%1"2"$ qEA 2,6- 1"%-6$"- 2," -0#9&"H5%&&"' 1"%# -J6%$" &).%& 1%9#"204%20)#7

8: )#" ,%- % 1%#'H5%&&"' +,%-" -+%."( 02 0- 0#2"$"-20#9 2) %-K #)2 )#&' %/)62 qEA 0# % -0#9&"

-2%2" /62 %&-) %/)62 2," .)$$"&%20)# /"23""# -2%2"-7 L)3 %$" 2,"' &0#K"* /"23""# 2,"1M ?," !0#"$%, qαβ 0- *">#"* /"23""# *0-20#.2 5%&&"'- α %#* βN qαβ = 1 N X i hSiαS β ii EB7F7OG P6" 2) 2," .)1+&"<02' ): 2," +,%-" -+%."( qαβ 1%' 2%K" 1%#' 5%&6"-7 ?,"$":)$" )#" .%#

.)#-0*"$ 02- *0-2$0/620)# P (q) ABQD7 R)$ % -'-2"1 302, )#&' 23) 5%&&"'- E&0K" % :"$$)1%9#"2G( q .%# )#&' 2%K" 23) 5%&6"-N 2," 5%&6" ): 2," .)$$"&%20)# ): 2," 5%&&"' α 302, 2," )2,"$ 5%&&"' β( )$ 302, 02-"&: α7 ?,6-( 2," -,%+" ): 2," *0-2$0/620)# P (q) +"$102- 2," 0#5"-209%20)# ): 2," -2$6.26$" ): 2," +,%-" -+%."7

!"#$%& '( )#*+ ,-"..%./ " 012%- 31& ,-"..4 .4.$%0.

!" #$%&'( )$* (+,- .'/(( (0(1&2(

!"#$%&'% &' (&)*#("#"( )+),"-)

!"#$%&#'(( )#*+%$,-'( (.'-)"/ 0'"*,+'( )#*% *1' *1'%+'*)-"/ *+'"*&'#* )# (*"*)(*)-"/ .12()-(3 4( 5' 1"6' +"#$%& ."+"&'*'+( {J} -%&)#7 )#*% *1' .+%8/'&9 %#' "../)'( (*"*)(*)-"/ .12()-( *% *1')+ $)(*+)8,*)%#3 :1'+&%$2#"&)- ;,"#*)*)'( -"# 8' $'+)6'$ 0+%& *1' 0+''<'#'+72 *1"* -"# 8' '=.+'(('$ 0+%& *1' ."+*)*)%# 0,#-*)%# Z9 51)-1 $'.'#$( %# *1' $)(%+$'+ ."+"&'*'+(3 :% +'-%6'+ *1' &"-+%(-%.)- 8'1"6)%+9 *1' 0+''<'#'+72 1"( *% 8' "6'+"7'$ >#%*'$ [ ]av? %# *1' $)(*+)8,*)%# %0 {J}3 :1' $)@-,/*2 "+)('( )# (,-1 "# "6'+"7)#7 %0 F = [F {J}]av = −kBT [ln Z{J}]av9 "( ln Z{J} )( #%* "# '=*'#()6' ;,"#*)*23 :1' 5)$'/2 ,('$ !"#$%& '!()*+ "//%5( *% %6'+-%&' *1)( $)@-,/*23 :1' "6'+"7' -"# 8' .'+0%+&'$ 82 $'-%&.%()#7 *1' (2(*'& )#*% n )$'#*)-"/ !"#$%&, 0%+ " 7)6'# +'"/)A"*)%# %0 *1' $)(%+$'+ {J}9 "#$ *1'# *"B)#7 *1' /)&)* n → 03 C# *1)( &'*1%$9 *1' .+%8/'& )( (2&&'*+)- ,#$'+ *1' .'+&,*"*)%# %0 +'./)-"( < "// *1' +'./)-"( "+' ';,)6"/'#*3

./" 0/"##&'%,*'12&#34$,#&53 -*("6

D1'#%&'#%/%7)-"//29 +"#$%&#'(( "#$ 0+,(*+"*)%# 1"6' 8''# )$'#*)E'$ *% 8' *1' &")# )#7+'< $)'#*( %0 (.)# 7/"(( 8'1"6)%+3 :1' &%$'/ )#*+%$,-'$ 82 F$5"+$( "#$ 4#$'+(%# >F4? GHIJ )# H IK -%&'( 0+%& *1)( )$'"3 C#(*'"$ %0 *"B)#7 *1' +"#$%&#'(( )# *1' (.)#( "*,$($*-9 %#' *"B'( *1' +"#$%&#'(( )# *1' $-(! &%($*-, 8'*5''# (.)#( /2)#7 %# " +'7,/"+ /"**)-'3

4 &'"#<E'/$ 6'+()%# %0 *1)( &%$'/ 5"( *1'# )#*+%$,-'$ 82 L1'++)#7*%# "#$ M)+B."*+)-B >LM? GHNJ3 C* )( " 7'#'+"/)A"*)%# %0 *1' F4 &%$'/ 0%+ )#E#)*'<+"#7' )#*'+"-*)%#( "#$ C()#7 (.)#(3 :1' O"&)/*%#)"# )( HSK = − 1 2 X i,j JijSizSjz + h X i S_iz >H3P3H? 51'+' Sz i )( *1' >C()#7? (.)# "* ()*' i "#$ h " &"7#'*)- E'/$ "../)'$ "/%#7 *1' z "=)(3 Jij

+'.+'('#*( *1' +"#$%& -%,./)#7 8'*5''# *1' (.)#( i "#$ j3 :1' (,& )( .'+0%+&'$ %# "// *1' #')718%+(3 4// *1' +"#$%&#'(( /)'( )# *1' $)(*+)8,*)%# %0 Jij3

C# &'"#<E'/$ &%$'/(9 *1' $)&'#()%# "8%6' 51)-1 &'"#<E'/$ *1'%+2 .+'$)-*( *1' '=)(*'#-' %0 " .1"(' *+"#()*)%# "* E#)*' *'&.'+"*,+' )( -"//'$ *1' /%5'+ -+)*)-"/ $)&'#()%# dl3 C* 1"( 8''#

1'"6)/2 $'8"*'$ )# *1' ."(*9 "#$ )( (*)// #%* .+'-)('/2 B#%5#3 C* &"2 8' /2)#7 8'*5''# Q "#$ P 0%+ C()#7 (2(*'&(9 "#$ "8%,* dl ≃ 3 0%+ O')('#8'+7 (2(*'&( 5)*1 "#)(%*+%.2 GH 9 QRJ3

:1' LM &%$'/ -"# 8' (%/6'$ 82 ,()#7 *1' +'./)-" &'*1%$9 "#$ ('/0<-%#()(*'#* ';,"*)%#( -"# 8' $'+)6'$ 0%+ q = [m2

i]av3 :1)( (%/,*)%# -".*,+'( 5'// *1' -,(. )# (,(-'.*)8)/)*2 "* Tg9 *1'

()7#"*,+' %0 *1' &"7#'*)- 0+''A)#7 %0 *1' (.)#(3

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χ3

! !"! #$%&'( )$* (+,- .'/(( (0(1&2( "#$ %&'()*)&+ ,-./)&)-. ,'. ($ ,'*,0*'&$/ '% ' 10.,&)-. -1 &$23$4'&04$ T './ ). &#$ 34$%$.,$ -1 ' 2'5.$&), 6$*/ H7 8& +)$*/% ' %&'()*)&+ *).$ ). &#$ 9:" 3*'.$ ,'**$/ &#$ ;:" *).$7 ;(-<$ &#$ ;:" *).$= &#$ >? %-*0&)-. )% %&'(*$= './ &#$ 34$<)-0% 2-/$* /$%,4)($% ,-44$,&*+ &#$ 3'4'2'5.$&), 3#'%$7 @$*-A &#)% *).$ #-A$<$4= &#$ %-*0&)-. )% .-& <'*)/ '.+2-4$7 B.$ ,'. &4'.%3-%$ &#)% )%%0$ -1 0./)%&).50)%#'(*$ 4$3*),'% &- &#$ ,-23*$C)&+ -1 &#$ 3#'%$ %3',$7 8. &#)% 3$,0*)'4 *'./%,'3$= &#$ <'**$+% 2'+ .-& ($ $D0)<'*$.&= -.$ #'% &- (4$'E &#$ %+22$&4+ -1 &#$ 4$3*),'% -1 &#$ %+%&$27 "#$ ;:" *).$ )% &#0% &#$ *).$ ($*-A A#),# &#$ 4$3*),'% '4$ .-& %+22$&4),7

!"! #$%&'() *+,,$-.+ /.$)0'123 -4$ 5).'*' *6&7-'61

"#$ -4/$4 3'4'2$&$4 ,-.&').).5 n 4$3*),'% )% 4$34$%$.&$/ '% ' 2'&4)C Qab= A#$4$ a './ b

'4$ &#$ 4$3*),'% )./),$%7 8. &#$ >? %-*0&)-.= '** &#$ 4$3*),'% '4$ $D0)<'*$.&= $D0'* &- q0= $C,$3&

&#$ /)'5-.'* &$42%= A#),# '4$ F$4-7 "#)% %+22$&4+ #'% &- ($ (4-E$.= (0& #-AG

"#$ %-*0&)-. #'% ($$. 34-3-%$/ (+ H'4)%) IJJK ). LML7 N$ /$&')* #$4$ ' %&$3 -1 %0,# ' O$3*),' >+22$&4+ @4$'E).5 PO>@Q 34-,$/04$7 R-.%)/$4 &#$ Pn × nQ 4$3*),' %+22$&4), 2'&4)C 6**$/ A)&# q0 P1-4 )**0%&4'&)-. A$ &'E$ n = 8 -. 6504$ 7SQ7 B.$ &#$. /)<)/$% )& ).&- Pn/m1Q

(*-,E% -1 %)F$ Pm1 × m1Q7 "#$ -T:/)'5-.'* (*-,E% '4$ *$1& 0.,#'.5$/ A)&# q07 "#$ /)'5-.'*

(*-,E% '4$ 2-/)6$/= &'E).5 &#$ %'2$ %&40,&04$ '% &#$ Pn × nQ 2-&#$4 2'&4)C= A)&# F$4- /)'5-.'* <'*0$ './ q1 -T:/)'5-.'* <'*0$7

!"#$% &'() !!"#$%&$'() (* $+, Q -&$%'. %,/!'0& #1--,$%1 2%,&3')45 &$ #$,/# K = 1 &)6 27

B.$ ,'. 4$3$'& &#)% O>@ 34-,$/04$ '5'). PK )% &#$ .02($4 -1 O>@:%&$3% 3$41-42$/Q= './ &#$ ($%& %-*0&)-. A)** ($ 1-0./ &- ($ &#$ 10**:%&$3 O>@ IJU= JVK PK → ∞Q7

;1&$4 &#)% 34-,$/04$ )% /-.$= -.$ /$6.$% P (q) '% &#$ 14',&)-. -1 &#$ 2'&4)C $*$2$.&% &#'& '4$ &'E).5 &#$ <'*0$ q P&#'& )% &#$ /)%&4)(0&)-. *'A -1 -<$4*'3 <'*0$% qQ7

N#$. K )% <$4+ *'45$= -.$ ,'. $C34$%% !"#$"%!%&'( &#$ <'*0$% qi A)&# ' 10.,&)-.

qi → q(x) with 0 ≤ x ≤ 1 P 7U7JQ

'./ &#$ /)%&4)(0&)-. P (q) )% 5)<$. (+

P (q) = dx(q)

dq P 7U7UQ A#$4$ x(q) )% /$6.$/ %0,# &#'& q(x(q)) = q7

"#$ 10**:%&$3 O>@ H'4)%) %-*0&)-. 4$2-<$% &#$ *-A &$23$4'&04$ .$5'&)<$ $.&4-3+ './ -.$ 4$,-<$4% S(T → 0) = 07 W-4$-<$4= &#$ 2'5.$&), %0%,$3&)()*)&+ ,'. ($ /$4)<$/ 14-2 q(x) I SK7 "#$ 10**:%&$3 O>@ %-*0&)-. ,'3&04$% A$** &#$ 2'5.$&), %0%,$3&)()*)&+ ($#'<)-4 IJXK7

!"#$%& '( )#*+ ,-"..%./ " 012%- 31& ,-"..4 .4.$%0. !"#$%&' $()*+,+*)&)$-( -. )!* /01

!" #$ %&'$ ($)*+,)$- ./$'+,0"123 *%$ ,'$/1&. 4$*#$$) '&11$2" 5+'$" +)6,/(&*+,) ,) *%$ "*/07*0/$ ,6 *%$ .%&"$ ".&7$8 9$* 0" 7,)"+-$/ & ".+) 51&"" #%+7% +" 7,,1$- -,#) *, & 1,#: *$(.$/&*0/$ "*&*$8 ;%+" $<0+1+4/+0( "*&*$ α ='&11$2> +" 7%&/&7*$/+?$- 42 & ".+) 7,)@50/&*+,) Sα

i 8 ;%$ "&(.1$ +" %$&*$- 0. *, T > Tg *, 7%&)5$ *%$ +)+*+&1 ".+) 7,)@50/&*+,)8 ;%$ 7,,1+)5

-,#) ./,7$"" 7&) *%$) 4$ /$.$&*$-8 ;%$ (&)2:'&11$2 .+7*0/$ "055$"*" *%&* ,)$ 6&11" +)*, & -+A$/$)* "*&*$ Sβ

i8 !) +)*$/$"*+)5 <0&)*+*2 +" *%$ ,'$/1&. qαβ3 -$@)+)5 42 %,# (07% *%$ ".+)

7,)@50/&*+,)" &/$ -+A$/$)* 6/,( $&7% ,*%$/B qαβ = 1 N n X i=1 hSiαS β ii = 8C8D>

;%$ (&E+(0( '&10$ 6,/ qαβ +" qEA &)- *%$ (+)+(0( '&10$ 7&) 4$ 0 +6 *%$ *#, 7,)@50/&*+,)"

&/$ 7,(.1$*$12 0)7,//$1&*$-8 ;%$ -+"*/+40*+,) ,6 *%$"$ ,'$/1&." +" 5+'$) 42 P (q) =X

αβ

wαwβδ(qαβ− q) = 8C8F>

#%$/$ wα &)- wβ &/$ *%$ "*&*+"*+7&1 #$+5%*" ,6 *%$ '&11$2"8

G) ".+) 51&""$"3 +* %&" 4$$) "%,#) H IJ *%&* *%+" 60)7*+,) P (q) +" +)-$$- *%$ ,)$ -$@)$-6/,( *%$ /$.1+7&:"2(($*/2 4/$&K+)5 ",10*+,)3 *%&* +"3 *%$ 6/&7*+,) ,6 $1$($)*" ,6 *%$ (&*/+E Qab

*&K+)5 *%$ '&10$ q8 ;%$ !"#$%& &)- *%$ '&##!( +)-+7$" &/$3 +) *%+" /$".$7*3 +-$)*+7&18 ;%+" +" *%$ .%2"+7&1 $E.1&)&*+,) ,6 *%$ LMNB +) *%$ ".+) 51&"" .%&"$3 *%$ '&11$2 "*&*$" &/$ ),* $<0+'&1$)*8

;%$/$6,/$3 *%$ -+"*/+40*+,) P (q) 5+'$" +)6,/(&*+,) &4,0* *%$ "*/07*0/$ ,6 *%$ .%&"$ ".&7$ ,6 *%$ "2"*$(8 ;2.+7&1 $E&(.1$" ,6 *%$ P (q) -+"*/+40*+,) &/$ 5+'$) +) @50/$ 8O8

!" *%$ *$(.$/&*0/$ +" -$7/$&"$- 4$1,# Tg3 *%$ .%&"$ ".&7$ +" 6/&5($)*$- +)*, "04 '&11$2"

"$.&/&*$- 42 $)$/52 4&//+$/"8 ;%$ LMN ./,7$-0/$ 5+'$" &) +)"+5%* +) %,# *%+" %&..$)"3 1$&-+)5 *, & %+$/&/7%+7&1 "*/07*0/$ ,6 *%$ $)$/52 1&)-"7&.$8 !4,'$ Tg3 &11 *%$ ,'$/1&." &/$ $<0&1 *, q0

=0)+<0$ '&11$2>8 ;%$ "04:-+'+"+,) ./,7$"" 4/$&K" *%+" '&11$2 +)*, *#, "(&11$/ 4/&)7%$" q13 #%+7%

+) *0/) &/$ 4/,K$) +) 4/&)7%$" q23 !)%8 ;%+" %+$/&/7%+7&1 "*/07*0/$ +" 7&11$- *%$ P%+$/&/7%+7&1

*/$$P &)- +" "%,#) ,) @50/$ 8Q8

G) "07% & %+$/&/7%+7&1 ".&7$3 $&7% '&11$2 =,/ /$.1+7&> 7&) 4$ 1&4$1$-3 &)- *%$ '&10$ ,6 *%$ ,'$/1&. qαβ 7&) 4$ ,4*&+)$- 42 */&7+)5 4&7K *%$+/ 7,((,) &)7$"*,/8 ! -+"*&)7$ 4$*#$$) *#,

"*&*$" 7&) 4$ -$@)$- &)- &) +(.,/*&)* 6$&*0/$ ,6 "07% & %+$/&/7%+7&1 ".&7$ +" *#) &+!) $%$)(B *%$ -+"*&)7$" d 4$*#$$) *%/$$ "*&*$" α3 β &)- γ &/$ "07% *%&*

dαβ ≤ max (dαγ, dβγ) = 8C8I>

($&)+)5 +) *$/(" ,6 ,'$/1&."

qαβ ≥ min (qαγ, qβγ) = 8C8O>

;%+" 01*/&($*/+7 ".&7$ +" & 7,)"$<0$)7$ ,6 *%$ LMN ./,7$-0/$ &)- +" *%$/$6,/$ &) +(.,/*&)* ./,.$/*2 ,6 *%+" ($&):@$1- ",10*+,) 6,/ ".+) 51&""$"8

! !"! #$%&'( )$* (+,- .'/(( (0(1&2(

!"#$% &'() !" #$%&'() P (q) *&+#,&-.#&"/+0 "%1 #23 ,3%)&'( +$443#,$ &+ -,"53/6 P (q) &+ !&*3)$ *&+#,&-.#3* .% #" ( 4(7&4.4 8().3 qEA0 9"##"41 &/ #23 ,3%)&'(:+$443#,&' '(+36 q &+ +&/;)3:8().3*0

!"#$% &'*) 23 2&3,(,'2&'() #,330 23 "83,)(% -3#!33/ ( %(&, "< +#(#3+ α (/* β =)3(83+ "< #23 #,33> *3%3/*+ "/ 2"! 4(/$ )383)+ "/3 4.+# ;" .% #" ?/* ( '"44"/ (/'3+#",0 23 *&@3,3/'3 -3#!33/ qmax

(/* qmin *3%3/*+ "/ #23 #34%3,(#.,30

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