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The study of concentrational ferromagnetism in Hubbard model with infinite interaction
ELSEVIER
Synthetic
The study of concentrational
Metals
71 (1995)
ferromagnetism A.A.Ovchitmikov
hstttute
of
Cllem~cal
Pll!
1777-1779
in Hubbard model with infinite interaction and V. Ya.Krivtm
sits of
Kusst~ri~
117’977, Mosco\\.
Academ
of
Sciences.
Rttxm’
Abstract The coliz~ntrational tnrchatkm of ferrotnngnettc ordertnp has been comtdercd from the pomt of stem of the Hubbard tnodel with inlinitc: intctxtion. It has hem sl~owu that the crsatiott of ferrotnagttetic phase goes Ihrottgh a phase separation and both saturation mnpnetizntion and Curie temperature are proportional to a dettsity of doped electrons. Some experimental examples are discussed.
A complelclp different situation evolves it‘an extra electron is ad&d to ~ucli ii qstem (let us sii\, a~, tltc rssull of doptng). iirt: of the order oft iii such ‘Ilie resulting ~llag~letic iilteractiotis a CLLSC,i.e. much larger that1 t?TJ, and tltts Ibcilttntes and c;mses tlx parallel aligntiictit of all spins iii the systctn: i.e the mi~~imutit spilt in tltc ground state. Let us prove this. The Mamriltottian of the united svstetn H I~GIJ~bc writtett in !lte followtttg foi7i1
1. INTRODUCTION It has been found nxetttly, in connection Lvith the search for tlew non-trxlitiot~a1 (qxcially organic) materials that in some cases fctroti~agt~elistt~ appears as it result of tlis doping of substances (tiiolccttlar ctvlals, tiiagticticall\ indifferent polymers. k.) 1l-31. ‘The last exanq~le of this kind is tltc ferrotuagttettsm of fttllerene doped with orgatttc acceptors 141. ‘l’hc divsrsi t!, of the ttuttal substances and dopants tttdicates that uli these plt~‘~totnena have to be caused by some cottu~~ot~atld as yet ttttktio\~ 11 tlwcIlaii~~til of ferroniagtxtistn. Into the same catcgoq nix Ml hypothetical l~liotoferroti~agnettstii, the idea of which has bcctl l”or\vardcd earlier. l‘he ptii~~~~scof this work is to indicate such ;I tiiecltatiistii and to discus\ boilic‘ of its experimciital cotise~~tteiiccs. Our jxitiit of‘ refereticc is the assutiiptioti of the cotic‘ctttrattoii~il tiatttre of these plietiotiictia. The first tttdtcattoti ita this ditccllott ts givei) in the consideration b\ authors of prtxnt papci of Clir tiiagtisttc phse appearaticr tt; in system ot’ ditners, drscr~bcd b> the Hubbard model wttlt at1 inlitttte repulsive ittlcractiott [S]
I-I = l-l, + r-lz + 7 .
where IIt and lI? are the IIamiltottiat~s ot‘~111: isolated ceuters Al and A? and T. bctttg responsible for the hopping of electrons (or Iiolcs), has the tkaal form
and.
PROBLEM
* This \\orh \\;I\; supported No y:-03-i
b\, the IS’I’C ttndet- Grant No 0 I5 and tn
X(>~$J
0379-6779/95/$09.50 0 1995 Elsevier SSDI 0379-6779(94)03047-A
Science
S.A. All rights
reserved
dctitutton. tnq be constdereti wtthm a lxrturhnttott Wltc~t ati e\;tra elcctt-on ts at cetiter 2 the s\‘stcms LKiVL!
(itttcttotl
ina\
bc
(N+l, Lz , n)
urtttcti
its
ct)t (N, L, , m) 1I)>,
(3)
where N is mitial number of electrons at the center A, Lt and I,? are tltc spms 01 state of the I-st and 2-nd cettkrs, II and m are tlleir projections ott iiti axis z (- Lt i tit I Lt - If < ii I Lz ), Let us abort’ that the ground stale A possessc~ a spm 1. + 1:2. the C;ISC ol‘ the \Wual ‘I‘lie oilier cases. atid parttcttlarl~ degradalion ol‘ the state:, \\,tth 1, + I 2 attd L - Ii2 cat be
To dcscr~bc: the above set of phenomena, let us consider the simplest silualtoti of two equivalent paratiiagnctic centers (A). where each ol‘ them possesses ii spin Ia, and they are at sonic distance lioitl 0111:another and their ititeractton is \\ea!i iii comprtson \\ ttli the ttitracetitct- correlations The spin state of rule: be descrtk.1 tqp tltc llic unit& \\ stem t11q. iIS a IHeiscttberg I Iatiiiltotitati 1%ith at1 antifen-otti;retiettc cxcitange ttttcgral 1’,IJ u here 1 i tilt:hopping iiitegral of electrons j is small m cottilx~risc~ti 1)ith the ititmcettter corrrlatioti parntiictcr U (rchttve encr-g,v of a charged pair A’ - A’). Consqucntly, the resulting spit1 of the qxtetn will be zero.
C;rant
by
tlicorj
01
2. TWO CENTER
(11
collsidClCJ
t,>
th!
s;IlllL’ \\a,
tltc Wipticr-L&t tltcorcnt, oti< litids tliat the matrix elctnettt of 11~ opet-ator -I- bet\\cm the skites j I > ad j 2 > , the tirst of u:lticli corrcspotids to iitt elc~lr~nl iit i.lie cctttcr At aid the second to another at the center Al , IS txp11 to Usitlg
= v C(L2’Ii ; L, C(I.~tt;L,‘tn’,a/2).K,.K**,
part
h!
111, o/2)
the ISIc trndcr Gtxttt No.M IO000 and h the IIFFII
(4)
ttndcr
1778
AA.
Ovchinnikov,
VTYa. Krivnov / Synthetic Metals 71 (1995) 1777-1779
C(L2 Ii . I,, 111.0/2j a11d C(L? 11 _ I.! 111 .d2) XL’ ClcbsclKiw da11 coclficlcllts. J,! L, , 11anll 111 are lllc: sp111s and their lxqections l’or the centers 2 and ! aller ai1elcctroil trader; K: :1u1 K, arc tlx rcdwcd matrix rlenm~ts \Vlwx
I lI,, = Hz? = (I. II,? = 7, I-12,= JY ”
(5) H= -3,
u ;IVC timctlon is then the subvector Y = (‘4’1,~~: ~‘?,UO 1; ~‘I,,,,, corresponds to an electron at Al wtl1 spn at A? \\ith projection5 II aill: ill, aid Yl,ul, to an electron pro:jectiolu 11 ;uid lli Tlic solution of the Scluodinger cqiiation is tlicn gi\ en III the iorlli
CC,, c,~ + cIo+Icloj.
,
Cl(l)
the qxtcms
‘f,, = ‘1‘22=(! .1‘i2= Tz,* = - (K, Kz’ (J+1/2):(2L+1/2) of which the cigcnvaltles 1; = ii K,
(7)
xc
I\:, t (.I + l/2) (21, + 1) ,
(8)
I IlKI! have lllS values 21, I- 1,2. 2L - l/2 . . . ....1/2. Ncwrth~lcsz. the n~~smwn possible .I = 2L i l/2 \\,1tll 1: = -t 1K, Kz i ~m~ponds to the ground state of the s!stem l‘lus is the niain r wilt. which allou us to state tliaii iiii extra clcctroii h~ilitatei aliglulient of all spins of the qsteiii iii the siliw diruztl~lil
3. MANY CENTERS PROBLEM WITH LNFLNITE LNTERACTION
- HUBBARD
MODEL
‘l‘lic anal\ 5is of the situation that xc several centcrs and sewral cxti:i cloclroils is iiiorc coiiiplicated and depends 011tlic COllCletc gci)llletl-\ paI-mete1-s and other pzmliariti~s 01‘ in sK5tL’111. I?rst 01‘all ne give ail esprcssion for the Hniiiiltonian describmg ;~bow ph~sl~11 situation. This Hamiltoniuu has ;I
wlwc clG+= :I,~+ i l- 11,..~)is the Ftmni olxmtor acting in the Slmx \v1tl1110dollblc occllplcd sites ‘l‘k ~iiodel ! 10 j is not solved in general cast at preszlit Wc wnwlcr hcre smqAlticd quasi 1-D model \\hcre the problem of !hc ground state iniilliplicit~ can hc solved This model represents ;I ladder consistiiip of t\! o-silt: segnients and 1s cliaractcnred b\ mm- and iiitcrsrgiiiciit Iioppiiig integrals t and [: lksidcs. u = I t, 1I << I) \\lllCll ~IllOV s use Ol’ the lxturbatioii tlleo~-~ in Q. hi scald order 111c( the model ( 10) rcili~ccs to ali dlktlve Hari~iltoii~ai~ 1I,,, the form of which depends on 111~electron density p = N, N (‘N, 1s the number of elcctrom and N 1s the number of segments It turns out [5] that for pll 2 tlw ground state is il singlet. ‘l‘lic cast of I 2 p li2 is inore iiitcrestiiig. hi this case there can be single one electron or couple of electrons at each segniciit. The uaw filllctiol~ of the s\ sttzl~~ contamlll~ M couples and (N - M) mgic ones depends on (N - M) spm variahlcs 6, = -t 1 2. and M variables ?.I = 1. -- I . 0. OS.iorresponclily to wlucs S al1d S, ol‘couples ( 1 1; I ,-I, 0.0: 0.0) ‘l‘lic I lamiltollian I I,g is clctennii~cd by its action upon spm 1ariablcs 01 iicigliboring s~yiicnts
(11)
Il,try’( ..ii, ) if*, ...r=-t?~~It/.(S.S,.,\I,( ci,,ci,,, )
1-I). (12)
pro~ectioii operator on state with spin L -t I/2 ground state of isokltcd ixiitcr ii \\itli one cstra electroll \vhlch are qua1 to
‘I‘lie alerg! ( 1I 1 IS iihinliil at iiiauiiial total spin (S=32) of 1~0 n~ightxxiiig qmrnts. ululc l:qi I 12i walls in the slliflct state. l’lius, Eql15 ( 1I ) and ( 12) describe coolpetiiig iiitcractions of fcITO- and alltlklTomagllctic t!‘pe5 For M= I (011e couple) the colnpet1t1on of ( 11) and ( 12) of the region with ferroniagiietic leads to tlic: foniiatioi~ ordering. i.e. the region of size N’ round the couple, the latter orlmtatcs the spins of slnglc ones parallel to Its ov.u S~III. Tlus pl~zllo~mmm is similar to the Connatloll of a spill polaron in N mgwtlc superconductors To estimate N‘ Icr us minmue the total t’nrrg of the swtcni
I’,‘.” ? = (1. 7~1 + 2 !j L) (21, + 1j
I:=-jt;/.IZcos(71N’)+(N-N’).C.
101111
li=
-t
\\here
\_: !I:‘.
i !.(il,n’;l,“+a~o+ill~).P,~*l’*
P,‘,” ’
depcndeiill!
is
011 the
I’,!.-“’ = (1, .- 7 5 L) 1,2L + 1j
)
(9)
c‘=1112
(14)
A.A. Ovchinnikov, KYa. Krivnov I Synthetic Metals 71 (1995) 17774779
\vlirre the first term is connected with the motion of ths couple in a polaron and the second one is the singlet ground state energy of the remaining part of the system It follows from IJqn.( 13) that N’=(2j$
(‘)I’.
N’>>l
(19
The gain of the energ\ owing to the polaron fonuation is equal
1779
WllCll
M=(N’a).(3rc. C 21”-’ the polaron length is equal to the ladder length N and the total spin S = S,,,,, Thus: at l/2 < p < pc , wllere pc is equal to pc =I!2 + (2Jq“ .(3x. C/2)‘:”
(1X)
l0
E = - 2 / t_ I + 3 I 1_i “(7t.C 2)‘j3
(16)
Calculations shoa that when there are M couples, the formation of one polaron with the ferromagnetic ordering of sin& ows and couples 1s enerprtically favorable. The size of this polaroll cm be estunalcd by the equation similar to (14) E=-2it
1 \N' x)~sm(xMM’) - 1tLl .(N - N’)C
(17)
The length ok’the ferromagnetic region N’ is equal to
the phase separation occurs: the bulk magnetic moment develops in the antiferromagnetic background. At p > pc the system becomes ferromagnetic. This phenomenon takes place for arbitrary values of a as it is shawl in our work [6] but corresponding analysis is more complicated. Saturation magnetization as a f~mctioii of p (rate of doping) is shown iii Fig I As to Curie tcmpcrature of this fcrromagnet it could be cstlmatcd roughI\ and it 1s proportional to boih t, and p ‘I‘,=t .p
(19)
CONCLUSION and It liiiearl! increases with the grout11 of couples number. The total spiii S is equal to S = (N’ + Mj 2
The considerations above may be the foundation for the search for iiw magnetic materials For csaiiiple, a simple prediction. Let 11stake some dintonuc pxamagnetic crystals: 02, NO, CO, etc. They arc in the solid state at temperatures = 30 K and tlicy are ~uitilcrroiii~~giidtsat temperatures = 1 K So, if one dopes these crystals \vith strong donors or acceptors, then at sufticirntly high dopiq levels ( 1% f lO%), the crh stals shall beconw ferromapixts i\ ltli Curit temperature = 100 K.
REFERENCES 1, A.Narutyunyan~ L.S Grygoryan, E.G Shuro~~an,MatSci.
2 3. 4. 5 0
Fig. 1. The ilcpeiidriice of s(p) = S( p)!S,,,,(p) for the ground state of the ladder model. S,,,, (p) is the maximal value of the total spin for ;I fired value of p.
14 (1988) 131. A.A.Ovcliiniiikov, Physics 1,ettcrs A 16-l ( 1992) 21 X. l:.L.Nagaev, JITl’: 57 ( 1969) I-‘.M.Allcmand et al. Science: 253 (199 I J 301. V.Ya.Krluiw, A.A.Ovchiimiliov and V 0 Chcrmiovskil, Sy,nthetlc Mclals. 33 ( I980) 65 V Ya Krivnov.and A A.OVCII~ILI~I~OL, 1’11) s,liu B 45 ( 1992) I2W.
Synthetic
The study of concentrational
Metals
71 (1995)
ferromagnetism A.A.Ovchitmikov
hstttute
of
Cllem~cal
Pll!
1777-1779
in Hubbard model with infinite interaction and V. Ya.Krivtm
sits of
Kusst~ri~
117’977, Mosco\\.
Academ
of
Sciences.
Rttxm’
Abstract The coliz~ntrational tnrchatkm of ferrotnngnettc ordertnp has been comtdercd from the pomt of stem of the Hubbard tnodel with inlinitc: intctxtion. It has hem sl~owu that the crsatiott of ferrotnagttetic phase goes Ihrottgh a phase separation and both saturation mnpnetizntion and Curie temperature are proportional to a dettsity of doped electrons. Some experimental examples are discussed.
A complelclp different situation evolves it‘an extra electron is ad&d to ~ucli ii qstem (let us sii\, a~, tltc rssull of doptng). iirt: of the order oft iii such ‘Ilie resulting ~llag~letic iilteractiotis a CLLSC,i.e. much larger that1 t?TJ, and tltts Ibcilttntes and c;mses tlx parallel aligntiictit of all spins iii the systctn: i.e the mi~~imutit spilt in tltc ground state. Let us prove this. The Mamriltottian of the united svstetn H I~GIJ~bc writtett in !lte followtttg foi7i1
1. INTRODUCTION It has been found nxetttly, in connection Lvith the search for tlew non-trxlitiot~a1 (qxcially organic) materials that in some cases fctroti~agt~elistt~ appears as it result of tlis doping of substances (tiiolccttlar ctvlals, tiiagticticall\ indifferent polymers. k.) 1l-31. ‘The last exanq~le of this kind is tltc ferrotuagttettsm of fttllerene doped with orgatttc acceptors 141. ‘l’hc divsrsi t!, of the ttuttal substances and dopants tttdicates that uli these plt~‘~totnena have to be caused by some cottu~~ot~atld as yet ttttktio\~ 11 tlwcIlaii~~til of ferroniagtxtistn. Into the same catcgoq nix Ml hypothetical l~liotoferroti~agnettstii, the idea of which has bcctl l”or\vardcd earlier. l‘he ptii~~~~scof this work is to indicate such ;I tiiecltatiistii and to discus\ boilic‘ of its experimciital cotise~~tteiiccs. Our jxitiit of‘ refereticc is the assutiiptioti of the cotic‘ctttrattoii~il tiatttre of these plietiotiictia. The first tttdtcattoti ita this ditccllott ts givei) in the consideration b\ authors of prtxnt papci of Clir tiiagtisttc phse appearaticr tt; in system ot’ ditners, drscr~bcd b> the Hubbard model wttlt at1 inlitttte repulsive ittlcractiott [S]
I-I = l-l, + r-lz + 7 .
where IIt and lI? are the IIamiltottiat~s ot‘~111: isolated ceuters Al and A? and T. bctttg responsible for the hopping of electrons (or Iiolcs), has the tkaal form
and.
PROBLEM
* This \\orh \\;I\; supported No y:-03-i
b\, the IS’I’C ttndet- Grant No 0 I5 and tn
X(>~$J
0379-6779/95/$09.50 0 1995 Elsevier SSDI 0379-6779(94)03047-A
Science
S.A. All rights
reserved
dctitutton. tnq be constdereti wtthm a lxrturhnttott Wltc~t ati e\;tra elcctt-on ts at cetiter 2 the s\‘stcms LKiVL!
(itttcttotl
ina\
bc
(N+l, Lz , n)
urtttcti
its
ct)t (N, L, , m) 1I)>,
(3)
where N is mitial number of electrons at the center A, Lt and I,? are tltc spms 01 state of the I-st and 2-nd cettkrs, II and m are tlleir projections ott iiti axis z (- Lt i tit I Lt - If < ii I Lz ), Let us abort’ that the ground stale A possessc~ a spm 1. + 1:2. the C;ISC ol‘ the \Wual ‘I‘lie oilier cases. atid parttcttlarl~ degradalion ol‘ the state:, \\,tth 1, + I 2 attd L - Ii2 cat be
To dcscr~bc: the above set of phenomena, let us consider the simplest silualtoti of two equivalent paratiiagnctic centers (A). where each ol‘ them possesses ii spin Ia, and they are at sonic distance lioitl 0111:another and their ititeractton is \\ea!i iii comprtson \\ ttli the ttitracetitct- correlations The spin state of rule: be descrtk.1 tqp tltc llic unit& \\ stem t11q. iIS a IHeiscttberg I Iatiiiltotitati 1%ith at1 antifen-otti;retiettc cxcitange ttttcgral 1’,IJ u here 1 i tilt:hopping iiitegral of electrons j is small m cottilx~risc~ti 1)ith the ititmcettter corrrlatioti parntiictcr U (rchttve encr-g,v of a charged pair A’ - A’). Consqucntly, the resulting spit1 of the qxtetn will be zero.
C;rant
by
tlicorj
01
2. TWO CENTER
(11
collsidClCJ
t,>
th!
s;IlllL’ \\a,
tltc Wipticr-L&t tltcorcnt, oti< litids tliat the matrix elctnettt of 11~ opet-ator -I- bet\\cm the skites j I > ad j 2 > , the tirst of u:lticli corrcspotids to iitt elc~lr~nl iit i.lie cctttcr At aid the second to another at the center Al , IS txp11 to Usitlg
part
h!
111, o/2)
the ISIc trndcr Gtxttt No.M IO000 and h the IIFFII
(4)
ttndcr
1778
AA.
Ovchinnikov,
VTYa. Krivnov / Synthetic Metals 71 (1995) 1777-1779
C(L2 Ii . I,, 111.0/2j a11d C(L? 11 _ I.! 111 .d2) XL’ ClcbsclKiw da11 coclficlcllts. J,! L, , 11anll 111 are lllc: sp111s and their lxqections l’or the centers 2 and ! aller ai1elcctroil trader; K: :1u1 K, arc tlx rcdwcd matrix rlenm~ts \Vlwx
I lI,, = Hz? = (I. II,? = 7, I-12,= JY ”
(5) H= -3,
u ;IVC timctlon is then the subvector Y = (‘4’1,~~: ~‘?,UO 1; ~‘I,,,,, corresponds to an electron at Al wtl1 spn at A? \\ith projection5 II aill: ill, aid Yl,ul, to an electron pro:jectiolu 11 ;uid lli Tlic solution of the Scluodinger cqiiation is tlicn gi\ en III the iorlli
CC,, c,~ + cIo+Icloj.
,
Cl(l)
the qxtcms
‘f,, = ‘1‘22=(! .1‘i2= Tz,* = - (K, Kz’ (J+1/2):(2L+1/2) of which the cigcnvaltles 1; = ii K,
(7)
xc
I\:, t (.I + l/2) (21, + 1) ,
(8)
I IlKI! have lllS values 21, I- 1,2. 2L - l/2 . . . ....1/2. Ncwrth~lcsz. the n~~smwn possible .I = 2L i l/2 \\,1tll 1: = -t 1K, Kz i ~m~ponds to the ground state of the s!stem l‘lus is the niain r wilt. which allou us to state tliaii iiii extra clcctroii h~ilitatei aliglulient of all spins of the qsteiii iii the siliw diruztl~lil
3. MANY CENTERS PROBLEM WITH LNFLNITE LNTERACTION
- HUBBARD
MODEL
‘l‘lic anal\ 5is of the situation that xc several centcrs and sewral cxti:i cloclroils is iiiorc coiiiplicated and depends 011tlic COllCletc gci)llletl-\ paI-mete1-s and other pzmliariti~s 01‘ in sK5tL’111. I?rst 01‘all ne give ail esprcssion for the Hniiiiltonian describmg ;~bow ph~sl~11 situation. This Hamiltoniuu has ;I
wlwc clG+= :I,~+ i l- 11,..~)is the Ftmni olxmtor acting in the Slmx \v1tl1110dollblc occllplcd sites ‘l‘k ~iiodel ! 10 j is not solved in general cast at preszlit Wc wnwlcr hcre smqAlticd quasi 1-D model \\hcre the problem of !hc ground state iniilliplicit~ can hc solved This model represents ;I ladder consistiiip of t\! o-silt: segnients and 1s cliaractcnred b\ mm- and iiitcrsrgiiiciit Iioppiiig integrals t and [: lksidcs. u = I t, 1I << I) \\lllCll ~IllOV s use Ol’ the lxturbatioii tlleo~-~ in Q. hi scald order 111c( the model ( 10) rcili~ccs to ali dlktlve Hari~iltoii~ai~ 1I,,, the form of which depends on 111~electron density p = N, N (‘N, 1s the number of elcctrom and N 1s the number of segments It turns out [5] that for pll 2 tlw ground state is il singlet. ‘l‘lic cast of I 2 p li2 is inore iiitcrestiiig. hi this case there can be single one electron or couple of electrons at each segniciit. The uaw filllctiol~ of the s\ sttzl~~ contamlll~ M couples and (N - M) mgic ones depends on (N - M) spm variahlcs 6, = -t 1 2. and M variables ?.I = 1. -- I . 0. OS.iorresponclily to wlucs S al1d S, ol‘couples ( 1 1; I ,-I, 0.0: 0.0) ‘l‘lic I lamiltollian I I,g is clctennii~cd by its action upon spm 1ariablcs 01 iicigliboring s~yiicnts
(11)
Il,try’( ..ii, ) if*, ...r=-t?~~It/.(S.S,.,\I,( ci,,ci,,, )
1-I). (12)
pro~ectioii operator on state with spin L -t I/2 ground state of isokltcd ixiitcr ii \\itli one cstra electroll \vhlch are qua1 to
‘I‘lie alerg! ( 1I 1 IS iihinliil at iiiauiiial total spin (S=32) of 1~0 n~ightxxiiig qmrnts. ululc l:qi I 12i walls in the slliflct state. l’lius, Eql15 ( 1I ) and ( 12) describe coolpetiiig iiitcractions of fcITO- and alltlklTomagllctic t!‘pe5 For M= I (011e couple) the colnpet1t1on of ( 11) and ( 12) of the region with ferroniagiietic leads to tlic: foniiatioi~ ordering. i.e. the region of size N’ round the couple, the latter orlmtatcs the spins of slnglc ones parallel to Its ov.u S~III. Tlus pl~zllo~mmm is similar to the Connatloll of a spill polaron in N mgwtlc superconductors To estimate N‘ Icr us minmue the total t’nrrg of the swtcni
I’,‘.” ? = (1. 7~1 + 2 !j L) (21, + 1j
I:=-jt;/.IZcos(71N’)+(N-N’).C.
101111
li=
-t
\\here
\_: !I:‘.
i !.(il,n’;l,“+a~o+ill~).P,~*l’*
P,‘,” ’
depcndeiill!
is
011 the
I’,!.-“’ = (1, .- 7 5 L) 1,2L + 1j
)
(9)
c‘=1112
(14)
A.A. Ovchinnikov, KYa. Krivnov I Synthetic Metals 71 (1995) 17774779
\vlirre the first term is connected with the motion of ths couple in a polaron and the second one is the singlet ground state energy of the remaining part of the system It follows from IJqn.( 13) that N’=(2j$
(‘)I’.
N’>>l
(19
The gain of the energ\ owing to the polaron fonuation is equal
1779
WllCll
M=(N’a).(3rc. C 21”-’ the polaron length is equal to the ladder length N and the total spin S = S,,,,, Thus: at l/2 < p < pc , wllere pc is equal to pc =I!2 + (2Jq“ .(3x. C/2)‘:”
(1X)
l0
E = - 2 / t_ I + 3 I 1_i “(7t.C 2)‘j3
(16)
Calculations shoa that when there are M couples, the formation of one polaron with the ferromagnetic ordering of sin& ows and couples 1s enerprtically favorable. The size of this polaroll cm be estunalcd by the equation similar to (14) E=-2it
1 \N' x)~sm(xMM’) - 1tLl .(N - N’)C
(17)
The length ok’the ferromagnetic region N’ is equal to
the phase separation occurs: the bulk magnetic moment develops in the antiferromagnetic background. At p > pc the system becomes ferromagnetic. This phenomenon takes place for arbitrary values of a as it is shawl in our work [6] but corresponding analysis is more complicated. Saturation magnetization as a f~mctioii of p (rate of doping) is shown iii Fig I As to Curie tcmpcrature of this fcrromagnet it could be cstlmatcd roughI\ and it 1s proportional to boih t, and p ‘I‘,=t .p
(19)
CONCLUSION and It liiiearl! increases with the grout11 of couples number. The total spiii S is equal to S = (N’ + Mj 2
The considerations above may be the foundation for the search for iiw magnetic materials For csaiiiple, a simple prediction. Let 11stake some dintonuc pxamagnetic crystals: 02, NO, CO, etc. They arc in the solid state at temperatures = 30 K and tlicy are ~uitilcrroiii~~giidtsat temperatures = 1 K So, if one dopes these crystals \vith strong donors or acceptors, then at sufticirntly high dopiq levels ( 1% f lO%), the crh stals shall beconw ferromapixts i\ ltli Curit temperature = 100 K.
REFERENCES 1, A.Narutyunyan~ L.S Grygoryan, E.G Shuro~~an,MatSci.
2 3. 4. 5 0
Fig. 1. The ilcpeiidriice of s(p) = S( p)!S,,,,(p) for the ground state of the ladder model. S,,,, (p) is the maximal value of the total spin for ;I fired value of p.
14 (1988) 131. A.A.Ovcliiniiikov, Physics 1,ettcrs A 16-l ( 1992) 21 X. l:.L.Nagaev, JITl’: 57 ( 1969) I-‘.M.Allcmand et al. Science: 253 (199 I J 301. V.Ya.Krluiw, A.A.Ovchiimiliov and V 0 Chcrmiovskil, Sy,nthetlc Mclals. 33 ( I980) 65 V Ya Krivnov.and A A.OVCII~ILI~I~OL, 1’11) s,liu B 45 ( 1992) I2W.