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On the variation with modulation wave-length of the Curie temperature of a Heisenberg magnetic superlattice
Solid State Comuwnications, Printed”in Great Britain.
Vo1.55,No.6,
pp.499-500,
1985.
0038-1098/85 $3.00 + .OO Pergamon Press Ltd.
ON THEVARIATIONWITHMODULATION WAVE-L!XNGTH OF THE CURIETEMPERATURE OF A HEISl%IBERG MAGNETIC SJP!!YRLATTICEU Hong-ru Ma Institute
of Solid
State Physics,
Nanjing University,
Nanjing,, P. R. China
Chien-hua Tsei Institute
Department of Physics, Chiao-tung University, of Solid State Physics, Nanjing University, Reoeived
Shanghai and Nanjing, P. R. China
15 Feb. 1985 by W* Y* KUW
The variation with modulation wave-length of the Curie temperature of a Aeisenberg magnetic superlattice is studied in the moleoular field approximation for the cases of square-wave and sinusoidal modulation& The results are tentatively comparable with what are observed experimentally.
Magnet&u superlattices
J, the exchange integrals betveen nearest neighboring atomic pairs A-A, %B, and A-B,
aroused research
intaresidl-!’ 2 in recent years. We endeavour here for an elementary theoretical study of the variation with modulation wave length of
respectiveI>,
the Curie temperature4. We shall consider the magnetic superlattice as an Heisenberg spin system,of two species of atoms A and B vith a periodic modulation in composition along the z-direction, each period consisting of 1 atomic planes. Let Ci be the occupancy of an
With (4))
1, vhen (mir) is occupied 0, vhen (mix) is occupied
iiA i =
+
+ Ci_,JA<&_,> q
= 3 [GiJ+
+ Cit&I, tc
+
average of Stir
is
= ?~Q~(‘mi&
(5)
xmi&
Here ve assume Smir to take the discrete values,
5
B&x) is
mic’
~m$l,.
. . ,-smir,
and
= (2s)-‘i(z~~i~~~l)~2S~
the Brillouin
function,
vi th (6)
‘mir = d Smi&i~
by A(B) (2) by B(A)
Expli~iitly ,
t (I-Ci)J]
ci+,JA<$tl> + Wit,)
the-thermodynamic
< ‘mi>T
Cf>
z&iJ&>
(4)
by
given
(1) DA(B) -{ uhfle
the average
j+z = -&&iii.i~
atom A at a site in the ith atomic plane within a period. We give belov an elementary molecular field treatment of the problem. Let m be the supercell label and-r, the lattice vector within an atomic plane, then the mean field experienced by the atom at the (m,i,r) site is vhere
&nti< $ > and
spin on any site of the itb plane vhen that site is occupied, separately, by an atom A or B. ‘Ibe effective hamiltonian is
= SAW+,
x;,,
-
SBB(SB, x;,
(7)
vhere x; = /sA?,
J<$+,>
+ (I-Ci_,)J’S;_,>
(‘-Ci)JB<$>J
Near the Curie pdifit, (3)
smell.
+ (‘-Ci+,)JBG’ft,>
TC
Writing
pf(B)s
i_,J& + (l-c~_,)J&_,>
zI being the number of nearest neighbors of an atomic eite on a ~ipgle plane, JA, JB and * Project supported by the Science Chinese Academy of Sciences.
x; = p&F
replaced
by
composition
fund of the
499
riA(B) become
- pA(B) i {S, A(B)),
sA+1,‘;:/3
given PyQ
ve have (9)
sA(B)(sA(B)t,)cA(B) i
iTA(B)Lbeing I
(8)
/3T&
(10)
by Eqs. (3) vith @A(B) i ’ (P: = 1). ml- any dpeedific
modulation,
(9) and (10) constitute
CURIE TEMPERATUREOF A HEISENBERGMAGNETICSUPERLATTICE
500
a closedset of equationsto fix the Curie temperature. We have solvednumericallyEqed(9)and (10) in tvo typicalca8e8: (1)Squarcwavemodulation, i=1,2,...,1/2 ci = i:, i=(1/2)+1,(1/2)+2,...,1 with1 even, Ld (2)Sinusoidal modulation + &Acos[2R(i-lj/l] 'i = 'A
I .I
1.0 -
-L
J,= I,J,-0.01,J = 0.1
0.9 -
CA= 0.4 , A=0.7
We have computedTo, in both cases,for differentvaluesof lt The resultsare illustrated by the ourveeshoun in Figs.1and 2. It can be .soenthat T, rices Piret steeplyvith 1, and then saturatesafter 1 exceeds,say, 10 to 20. Hhir is qualitatively the obeervedbehavior
’ 0.71 0 4
1.5
Fig. 2.
1.4-
Vol. 55, No. 6
’
8
1
I2
’
I6
1.
20
1
24
1
26
1
32
’
36
1
4C
Tc versus1 of a einueoidally modulated auperlattice with CA=0.4,A=O.7,
1.3-
JA=l, JB=O.Ol,J=O.l and SA=SB-,. JA=I,JB=0.2,J ~0.5
1.2-
of kk
I.1 -
IO
Fig.1.
0
I
I
I
I
4
6
12
16
I 20
I 24
I 26
32
Tc vereue1 of a square-vave modulated superlattice with JA=l, JB=O.2,J=O.5 end SA.SB+
vere~a
modulationvave-le.qth4.
For metallicsuperlattices, the exchange integralswill have valuesdifferentfrom that in pure metalsor uniformalloys.We must also considerthe contributions from Itinerant electronsto the superlattice magnetiem.Therefore, the presentsimpletheory16 not directly comparablewith measurements on, e.g., Ni/Cu superlattices. Nevertheless, the resultsare eatiefactory.
REFERENCES 1.
2.
3. 4. 5. 6.
E.M. Gyorgyet al, Phys. Rev. Lett.U 57 (1980). T. Jarlborgand A.J. Freeman,Phys. Rev. Lett. &, 653 (1980). J.Q. Zhen et al, Appl. Phya. Lett.,j& 424 (1981 k. J.Q. Zheng et al, J. Appl. Phys.22, 3150 (1982). N,K. Fleuariset al, J. APPL. Phys.2, 2493 (1982). E.M. Gyorgyet al, Phys. Rev. 2B, 6739 (1982).
7. 8.
9. 10. 11. 12.
R.E. Camleyet al, Phye. Rev. ZB, 26i (1983). P. Grunbergend K. Mika,khye. Rev. GB, 2955 (1983). M. Grim&itch et al, Phye. Rev. Lett. 51, 498 (1983). A. Kueny et al, Phya Rev. aB, 2879 (1984). S.V. Lyo, Solid State Cow il, 709 (1984). H.R. Ma and C.H. Teai,ChinesePhya. Lett.1, 92 (1984).
Vo1.55,No.6,
pp.499-500,
1985.
0038-1098/85 $3.00 + .OO Pergamon Press Ltd.
ON THEVARIATIONWITHMODULATION WAVE-L!XNGTH OF THE CURIETEMPERATURE OF A HEISl%IBERG MAGNETIC SJP!!YRLATTICEU Hong-ru Ma Institute
of Solid
State Physics,
Nanjing University,
Nanjing,, P. R. China
Chien-hua Tsei Institute
Department of Physics, Chiao-tung University, of Solid State Physics, Nanjing University, Reoeived
Shanghai and Nanjing, P. R. China
15 Feb. 1985 by W* Y* KUW
The variation with modulation wave-length of the Curie temperature of a Aeisenberg magnetic superlattice is studied in the moleoular field approximation for the cases of square-wave and sinusoidal modulation& The results are tentatively comparable with what are observed experimentally.
Magnet&u superlattices
J, the exchange integrals betveen nearest neighboring atomic pairs A-A, %B, and A-B,
aroused research
intaresidl-!’ 2 in recent years. We endeavour here for an elementary theoretical study of the variation with modulation wave length of
respectiveI>,
the Curie temperature4. We shall consider the magnetic superlattice as an Heisenberg spin system,of two species of atoms A and B vith a periodic modulation in composition along the z-direction, each period consisting of 1 atomic planes. Let Ci be the occupancy of an
With (4))
1, vhen (mir) is occupied 0, vhen (mix) is occupied
iiA i =
+
+ Ci_,JA<&_,> q
= 3 [GiJ+
+ Cit&I, tc
+
average of Stir
is
= ?~Q~(‘mi&
(5)
xmi&
Here ve assume Smir to take the discrete values,
5
B&x) is
mic’
~m$l,.
. . ,-smir,
and
= (2s)-‘i(z~~i~~~l)~2S~
the Brillouin
function,
vi th (6)
‘mir = d Smi&i~
by A(B) (2) by B(A)
Expli~iitly ,
t (I-Ci)J
ci+,JA<$tl> + Wit,)
the-thermodynamic
< ‘mi>T
Cf>
z&iJ&>
(4)
by
given
(1) DA(B) -{ uhfle
the average
j+z = -&&iii.i~
atom A at a site in the ith atomic plane within a period. We give belov an elementary molecular field treatment of the problem. Let m be the supercell label and-r, the lattice vector within an atomic plane, then the mean field experienced by the atom at the (m,i,r) site is vhere
&nti< $ > and
spin on any site of the itb plane vhen that site is occupied, separately, by an atom A or B. ‘Ibe effective hamiltonian is
= SAW+,
x;,,
-
SBB(SB, x;,
(7)
vhere x; = /sA?,
J<$+,>
+ (I-Ci_,)J’S;_,>
(‘-Ci)JB<$>J
Near the Curie pdifit, (3)
smell.
+ (‘-Ci+,)JBG’ft,>
TC
Writing
pf(B)s
i_,J& + (l-c~_,)J&_,>
zI being the number of nearest neighbors of an atomic eite on a ~ipgle plane, JA, JB and * Project supported by the Science Chinese Academy of Sciences.
x; = p&F
replaced
by
composition
fund of the
499
riA(B) become
- pA(B) i {S, A(B)),
sA+1,‘;:/3
given PyQ
ve have (9)
sA(B)(sA(B)t,)cA(B) i
iTA(B)Lbeing I
(8)
/3T&
(10)
by Eqs. (3) vith @A(B) i ’ (P: = 1). ml- any dpeedific
modulation,
(9) and (10) constitute
CURIE TEMPERATUREOF A HEISENBERGMAGNETICSUPERLATTICE
500
a closedset of equationsto fix the Curie temperature. We have solvednumericallyEqed(9)and (10) in tvo typicalca8e8: (1)Squarcwavemodulation, i=1,2,...,1/2 ci = i:, i=(1/2)+1,(1/2)+2,...,1 with1 even, Ld (2)Sinusoidal modulation + &Acos[2R(i-lj/l] 'i = 'A
I .I
1.0 -
-L
J,= I,J,-0.01,J = 0.1
0.9 -
CA= 0.4 , A=0.7
We have computedTo, in both cases,for differentvaluesof lt The resultsare illustrated by the ourveeshoun in Figs.1and 2. It can be .soenthat T, rices Piret steeplyvith 1, and then saturatesafter 1 exceeds,say, 10 to 20. Hhir is qualitatively the obeervedbehavior
’ 0.71 0 4
1.5
Fig. 2.
1.4-
Vol. 55, No. 6
’
8
1
I2
’
I6
1.
20
1
24
1
26
1
32
’
36
1
4C
Tc versus1 of a einueoidally modulated auperlattice with CA=0.4,A=O.7,
1.3-
JA=l, JB=O.Ol,J=O.l and SA=SB-,. JA=I,JB=0.2,J ~0.5
1.2-
of kk
I.1 -
IO
Fig.1.
0
I
I
I
I
4
6
12
16
I 20
I 24
I 26
32
Tc vereue1 of a square-vave modulated superlattice with JA=l, JB=O.2,J=O.5 end SA.SB+
vere~a
modulationvave-le.qth4.
For metallicsuperlattices, the exchange integralswill have valuesdifferentfrom that in pure metalsor uniformalloys.We must also considerthe contributions from Itinerant electronsto the superlattice magnetiem.Therefore, the presentsimpletheory16 not directly comparablewith measurements on, e.g., Ni/Cu superlattices. Nevertheless, the resultsare eatiefactory.
REFERENCES 1.
2.
3. 4. 5. 6.
E.M. Gyorgyet al, Phys. Rev. Lett.U 57 (1980). T. Jarlborgand A.J. Freeman,Phys. Rev. Lett. &, 653 (1980). J.Q. Zhen et al, Appl. Phya. Lett.,j& 424 (1981 k. J.Q. Zheng et al, J. Appl. Phys.22, 3150 (1982). N,K. Fleuariset al, J. APPL. Phys.2, 2493 (1982). E.M. Gyorgyet al, Phys. Rev. 2B, 6739 (1982).
7. 8.
9. 10. 11. 12.
R.E. Camleyet al, Phye. Rev. ZB, 26i (1983). P. Grunbergend K. Mika,khye. Rev. GB, 2955 (1983). M. Grim&itch et al, Phye. Rev. Lett. 51, 498 (1983). A. Kueny et al, Phya Rev. aB, 2879 (1984). S.V. Lyo, Solid State Cow il, 709 (1984). H.R. Ma and C.H. Teai,ChinesePhya. Lett.1, 92 (1984).