Thermodynamique des magnons

à T≠0K: population de N magnons excités

à T=0K: état fondamental <nk>=0 (« vide ») 0 H 0 = E0 + 0 !knkk! 0 = E0

N(T ) = V8! 3

d3ke"#k !1"statistique de Bose-Einstein: g(!) densité d’état

de magnons

k ! !D

à basse température, seuls les magnons de basse énergie contribuent à N

g(!)d! =V8" 3 4"k

2dk g(!) = V2" 2 k

2 dkd!

g(!) = V4" 2

!1/2

D3/2 !!1/2dk

d!=

12 D!

N3D (T ) =V4! 2

1D3/2

"1/2d"e#" !10

+"

#

=g(!)d!e"!k !10

+"

#

En 3D:

x = !"

N3D (T ) =V4! 2

1D3/2

1" 3/2

x1/2dxex !10

+"

# N3D !T3/2

converge

! !

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

!"!"# $##"%

&!"### $"#'

!#$#

$!%"("

")

*!"#$##"#+

!2!3!4!3

5"*-#.1$*)-+1+&).6#17##)/#$$&%*0)#1+,.8!4!,9.*)':*$*%*0)#1+,.8!2!,9;

<*0)#1+,.:"*-#.1$*)-+1+&)-.*$#&/.=)'.&$'#$>.+;#;.<.,&)1+)?&?-@(0&#-.1&.A#$&.7"#).!,.+-*::$&*,"#';..

3$+1+,*@.#B:&)#)1-.'#-,$+6#.1"#.:$&:#$1+#-..&/.1"#.%*0)#1.)#*$.!,;.

"###$"!&

!"#$%&'#()*+,#&'#-./01#2&'#3&/4,#)'5#,26&%+,

"## #'$

Aimantation: loi de Bloch

N3D !T3/2

M (T ) = Stotalz = NS ! nk

k"

conséquence sur l’aimantation M(T)

M (0)!M (T ) = nkk" = N3D (T )#T

3/2

nombre total de magnons: N3D

en 2D: g(!) = cste

N2D (T )!Tdxex "10

+#

$ %#

loi de Bloch

diverge!

en 1D: g(!)!!"1/2

N1D (T )!T1/2 x"1/2dx

ex "10

+#

$ %#

diverge!

pas d’ordre magnétique en 2D et 1D à T≠0: théorème de Mermin-Wagner (1966)

effet de la dimensionalité

!

"#$ !B

B

mNS N d D

gk T

%

% &

&

S J J

Aq q

! ! !

!' %

!' %

(

)*" %

!

D D

D D

D D

D+ ,D -.(

D -.%

D -.!

& &

!

"#$ !B

m T D dm S

k T

& &

%/0...!B x

m dxk T

m e

!' %!' %

& &

!/0...!B x

m x dxk T

m e

!' %( ' %

& &( ' %

(/0...!

.............. .)*".

B x

B

m x dxk T

m e

k T 12345.267

"#$%&'()*+'#$,-.#/$#%0839.:";<"=>"9?.<@<*"A<.T) -.&839.D = %.6=B.D -.!

g(ω)

ω

N3D (T )!T3/2 x1/2dx

ex "10

+#

$

champ moyen: ! e"2T /Tc

Aimantation: loi de Bloch

N3D !T3/2 conséquence sur l’aimantation M(T)

standard method,12 M is measured as a function of H at fixedT . Then, the data are extrapolated back to H�0 to constructM (T ,0). Unfortunately, although the SQUID is highly sen-sitive, the uncertainty in M (T ,0) produced by this extrapo-lation in H dwarfs the �0.2% signal variation with tempera-ture in the range 5–25 K.At each field, 4–6 runs were taken without removing the

sample from the magnetometer. Typically, each run wouldbe displaced from the previous one by a random amountwithin �0.08% of M (0,H). Within the scatter of the data,however, all runs were parallel. Removing the sample fromthe magnetometer could produce a change in the measuredmagnetization of up to a few percent, owing to slightchanges in sample orientation. In each case, the three lowesttemperature points were discarded to negate any effects ofsystem startup transients, and the individual runs were aver-aged to form the final data set. Finally, a small backgroundsignal, measured at both fields with no sample in the mag-netometer, was subtracted off.For each field a second order polynomial was used to

extrapolate M (T ,H) back to T�0. Care is required duringthis step since variations in M (0,H) can affect the results.Using this method, M (0,H) was determined to within0.006%, and values of M (0,H) agreed with a fully alignedspecies consisting of 0.7 Mn3� ions �spin 2� and 0.3 Mn4�

ions �spin 32� per unit cell. Given the dimensions of our par-

ticular sample, the saturation field was 0.55 T.If we assume that the low-temperature magnetization fol-

lows the form of

M �0,H ��M �T ,H ���const��T�, �3�

where � is to be measured, then the result is similar to thatdepicted in Fig. 1. Figure 1 shows the magnetization forH�1 T on a logarithmic scale. A weighted fit to a simplepower law is seen to be a reasonable characterization of thedata, and produces a value of ��1.4�0.1. At first glance,this appears to be a confirmation of the Bloch law.However, we have yet to properly take account of the

presence of the field. Indeed, if a similar analysis is per-formed on the 3-T data, then the fit is equally good, but �

turns out to be 1.7�0.1. The key is to understand the effectof applied field on the spin-wave dispersion relation in Eq.�1�. In Eq. �1� the gap energy � is given by

���0�g�B�H�NM �, �4�

where �0 is the intrinsic gap, g�2 is the Lande g factor, �Bis the Bohr magneton, N is the demagnetization factor, andM is the measured magnetization of the sample. In our casewe can simply take M as M 0 , as the sample was essentiallyfully magnetized at all times. The derivation of the Blochlaw assumes that ��0. Indeed, neutron scattering has placedan experimental upper limit on �0 of 40 �eV for other mem-bers of the La1�xAxMnO3 family,7 and a direct measurementof the anisotropy field in our sample found it to be less than0.02 T, or �0�2.5 �eV.13 So the assumption that �0�0 isvalid, but the application of H forces a nonzero �, in whichcase the Bloch law must be modified. For temperatures be-low �0.2Tc , the momentum of thermally excited spin-wavesis low enough that Eq. �1� is a good approximation to thedispersion relation. Using the standard spin-wave picture, wefind that in this limit the magnetization becomes

M �0,H ��M �T ,H ��g�B� kBT4�D � 3/2f 3/2��/kBT �, �5�

where kB is Boltzmann’s constant, and f 3/2(y) is given by

f P�y �� �n�1

� e�ny

nP . �6�

Equation �5� reduces to the Bloch law when H and � arezero.Our sample is fairly flat, being a chip of dimensions

�3�3�0.5 mm3, and from the saturation field we estimatedthe demagnetizing factor to be N�80%�4��. With a magne-tization of 95 emu/g, assuming a lattice spacing of 3.92 Ågives a total demagnetizing field NM of about 0.6 T. Thus,�/kBT ranges from 0.02–0.7 in our experiment, and is notnecessarily small.Over the restricted range of temperature in our experi-

ment, Eq. �5� is well approximated by Eq. �3� with someeffective exponent �eff . Figure 2 shows the expected form of�eff as a function of applied field, compared with the twovalues measured from the data. As can be seen, at high fields�eff takes on the value 2. This could be one reason whyhigh-field magnetization measurements on SrRuO3 show aT2 dependence.14The existence of spin-waves in the La1�xAxMnO3 family

is not in doubt. They have been unambiguously detected ininelastic neutron scattering, both at low momentum7 andthroughout the Brillouin zone,15 and in spin-wave resonance8experiments. However, the existence of well-defined spinwaves in metallic ferromagnets does not insure that theBloch T3/2 law is followed,10 particularly in cases where fer-romagnetism is weak.16 Weak ferromagnets are character-ized by a low ratio of saturated moment ps to the total effec-tive moment peff of the system. The latter is determined fromthe Curie constant,12 and weak ferromagnets typically haveps/peff�0.2. It should be noted that both La1�xAxMnO3�ps/peff�1� and SrRuO3 �ps/peff�0.87 �Ref. 14�� are strongferromagnets.

FIG. 1. Magnetization M vs temperature T for H�1 T. The lineis a weighted best fit to Eq. �3� of the text. The 3-T data are similar,but a best fit indicates T1.7.

55 5641BRIEF REPORTS

ln M0 !M (T )[ ] " 32lnT

Plan du cours

  Magnétisme sans interaction   Magnétisme atomique   Moments magnétiques localisés   Environnement

  Magnétisme localisé en interaction   Interactions d’échange   Modèle de champ moyen du ferromagnétisme   Anisotropie: hystérésis

  Au delà du champ moyen   Hamiltonien d’Heisenberg: du classique au quantique   Magnons: approximation harmonique   Renormalisation de l’état de Néél par les magnons AF

  Magnétisme itinérant   Paramagnétisme d’un gaz d’électrons libres   Instabilité magnétique de Stoner   Effets Hall quantiques

Holstein-Primakov sur réseau AF

J

état de Néél

2 sous réseaux A et B: 2 opérateurs bosoniques a et b

•  on ne connaît pas l’état fondamental

•  on part de l’état de Néél et on regarde comment les magnons le modifient

SAj+ = 2S 1!

aj+aj2S

"

#$$

%

&''

12

aj

SAj! = 2Saj

+ 1!aj+aj2S

"

#$$

%

&''

12

SBj! = 2S 1!

bj+bj2S

"

#$$

%

&''

12

bj

SBj+ = 2Sbj

+ 1!bj+bj2S

"

#$$

%

&''

12

SjAz = S ! aj

+aj

SjBz = !S + bj

+bj

attention: incrémenter Sz avec S+ correspond à une création de boson pour B mais une destruction pour A

Magnons AF 2 opérateurs de magnons: c et d

aj =2N

e!ik!. j!

ckk"

aj+ =

2N

eik!. j!

ck+

k!

bj =2N

e!ik!. j!

dkk"

bj+ =

2N

eik!. j!

dk+

k!

SjAz = S ! aj

+aj

Sj+!,Bz = !S + bj+!

+ bj+!Hzz =

J2

Sj,Az Sj+!,B

z

j,!!

Hzz =J2

Sj,Az Sj+!,B

z

j,!! " #

Nz J S2

2+J S2

aj+aj + bj+!

+ bj+!j,!!

Hzz = E0 + J Sz ck+ck + dk

+dkk! diagonal

expression de H en fonction des opérateurs de magnons

N/2 termes

E0: énergie de l’état de Néél

(néglige les termes en a+ab+b)

Magnons AF

SAj+ ! 2Saj = 2

SN

e"ik!. j!

ckk#

SB, j+!! " 2Sbj+! = 2

SN

e!ik!.( j!+!!)dk

k#

approx. magnons libres

Sj,A+ Sj+!,B

!

j,!" =

4SN

e!i j!.(k!1+k!2 )e!ik

!2 .!!

ck 1dk2j,!,k1,ké

"

H ± =J4

Sj,A+ Sj+!,B

! + Sj,A! Sj+!,B

+

j,!"

! k =1z

eik!."!

"

! Sj,A+ Sj+!,B

!

j,!" = 4Sz " kckd!k

k"

H ± = J Sz ! kckd!kk" +! kc!k

+ dk+

pas diagonal ! k = !!ksi centre d’inversion

= 4S eik!.!!

ckd!kk,!"

Magnons AF

H = E0 + J Sz ck+ck + dk

+dkk! +! k (ckd"k + c"k

+ dk+ )

couplage entre sous-réseau sous réseaux indépendants

diagonalisation: transformations de Bogoliubov (cf théorie BCS)

ck = uk!k + vk"!k+

dk = uk!k + vk"!k+ avec !,!+!" #$= "," +!" #$=1

nouveaux opérateurs bosoniques

ck,ck+!" #$=1

conditions sur les uk vk

uk!k + vk"!k+ ,uk!k

+ + vk"!k"# $%= uk2 !k,!k

+"# $%+ vk2 "!k

+ ,"!k"# $%=1

uk2 ! vk

2 =1

(2) Éliminer les termes croisés (αβ) pour diagonaliser H:

(1)

2ukvk +! k uk2 + vk

2( ) = 0

uk = cosh(!k ) vk = sinh(!k )et

sinh(2!k )+" k cosh(2!k ) = 0 ! k = ! tanh(2"k )

Magnons AF

H = E0 + J Sz ck+ck + dk

+dkk! +! k (ckd"k + c"k

+ dk+ )

couplage entre sous-réseau sous réseaux indépendants

H = E0 !N J Sz2

+ J Sz !k (1+"k+"k +

k" #k

+#k )

!k = 1!" k2

H = E0 + J Sz (1+!k+!k +

k! "k

+"k )(uk2 + vk

2 + 2# kukvk )"1

Forme diagonale de H (à vérifier):

!k = cosh 2!k( )+! k sinh 2!k( )

! k = ! tanh(2"k ) = !sinh(2"k )1+ sinh2(2"k )

sinh(2!k ) = !" k1!" k

2

Expression de l’énergie des magnons en fonction de γk

cosh(2!k ) =11!" k

2

ωk : énergie des magnons

E0: énergie de l’état de Néél

relation de dispersion des magnons AF

!k = JSz 1!" k2magnons AF: 2 modes dégénérés

! k =1z

eik!."!

"

! =12cos(ka)+ cos("ka)[ ] = cos(ka)en 1D: 2 premiers voisins

-a +a

!k = JSz 1!" k2 = JSz sin(ka)

124

!"""""2

#!"""""2

ka

$k

Figure 7.9. Spinwave (magnon) dispersion for an antiferromagnet. For small k thedispersion is linear.

with Ak = 1 and Bk = cos k, for a one dimensional spin chain. Thus thespinwave energy is

!k = 2SJ!

A2k ! B2

k

= 2SJ sin(ka)" 2SJka (when k # 0) (7.61)

and the Hamiltonian is, up to a constant

HLSW ="

k

!k(nk +12) (7.62)

as in the case of phonons, see Fig. 7.9. The spectrum is linear for smallwavenumber k and thus we know from the results of the phonons, thatquantum fluctuations destroy magnetic ordering of the antiferromagnetin 1D (and in 2D at any non-zero temperature). All of this is in accor-dance with the Mermin-Wagner theorem.

If such quantum melting takes place our linear spin wave approxima-tion is not valid anymore. The number of bosons in the groundstatewill become large and the linearized form of the spinwave equation willnot be justified. Actually the problem is even much worse: in the HPtransformation we assumed that there is long-range antiferromagneticorder to start with. The magnetic order prescribes how we define thequantization axis of the spins and thus the HP bosons. Clearly, if af-ter the spin wave calculation it turns out that Neel order is destroyed,the bosonization that we started with is not justified. But if there isordering and the Neel state is stable, spin rotation symmetry is broken.The gapless Goldstone mode that corresponds to the breaking of thiscontinuous symmetry is the magnon.

We observe that when the groundstate is an eigenstate of a quantumHamiltonian, as we have for the ferromagnet, or for the non-interacting

relation périodique kBZ=π/a

zone de Brillouin magnétique ≠ cristal dispersion linéaire à basse énergie (ferro: quadratique)

!k ! JSz ka

2JS

relation de dispersion des magnons AF

M=(0,0)

dispersion carré 2D (z=4) !k = 4JS 1! 14cos(kxa)+ cos(kya)"# $%

2

!("2, "2) = 4JS !("

2,0) = 2 3JS

Γ=(π/2,π/2) X=(π,π)

La2-xSrxCuO4: supraconducteur à haute température critique

Plan Cuivre AF (S=1/2)

La2CuO4

fondamental AF dans l’approximation harmonique

H = E0 !NJSz2

+ JSz !k (1+"k+"k +

k" #k

+#k )

état fondamental !k+!k = "k

+"k = 0

EAF = H = E0 !NJSz2

+ JSz !kk" = E0 ! JSz

N2! 1!! 2k

k" )

#

$%

&

'(

! k =1z

eik!."!

"

! <1 1!! 2k "1>0

EAF = !NJS2z2

! JSz (1! 1!! 2k )k"

correction en 1/S

!k = 1!" k2

L’état de Néél est renormalisé par les magnons: fluctuations de point zéro

EAF = !NJS2z2

1+ 2NS

(1! 1!! 2k )k"

#

$%

&

'(

< E0