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Molecular mechanics of the ferroelectric to paraelectric phase transition in LiNbO3 via optical second harmonic generation
Volume 38. number 2
CIIEMICAL
1 March 1976
PHYSICSLETTERS
MOLECULAR MECHANICS OF THE FERROELECTRIC TO PARAELECTRIC IN LiNb03 VIA OPTICAL SECOND HARMONIC GENERATION
PHASE TRANSlTION
J.C. BERGMAN Bell Laboratories, Holmdcl, New Jcrscy 07733, USA Rcccived 13
November 1975
Tcmpcnturc dcpundcnt optical second harmonic generation (SIC) results xc used to monitor the 7O trigonai dcformation of the NW& uctahcdron in fcrroclcctric I.iNbO+ The StXG structural results compare favorably with those obtaincd from conventional (X-my) techniques. The tempcraturc dependence of the spontaneous polarization Ps, calculated vk~ the same trigonal deformation model is consistent with the cxperimcntal observations.
The rclationsbip between the induced polarization f and the optical field 6 acting on a given crystal is p=@;d&;y~3
+...
)
where x is the familiar linear polarizability nonlinear
terms
d and 7 xc
the coefficients
(1)
and the which
for second and third harmonic generation respcctivel y. A more rigorous formulation shows that the induced nonlinear (second harmonic) polarization (SW) is best expressed in terms of the Fourier components of the elcctlic field amplitude h-F I ’ are responsible
where G is the geometrical factor which rcliitcs the microscopic bond polarizability 13to the macroscopic crystal polarizability (I. We assume here that the tcmperaturc dependence of p is small compared to that of G i.c. aG/aT > ilp/aT. The rationale for choosing a crystd’s nonlinear polarizability (d) for studying a phase transition rather than the crystal’s linear polarizability x is that x(7’) seldom changes by more than a few percent, while d(T) often changes by over an order of magnitude. In this letter we apply the aforcmentioned technique ti, determine the molecular structure and mechanics of the fcrroclcctric to puraelectric phase transition in LiNb03 via the temperature dependence of d333_ WC relate the crystal SHG coefficient dss3 to the microscopic Nb +O bond polarizability fl via the general formalism
where d is a third rank (27 element) tensor whose elements are restricted by the symmetry of the mcdiurn and i,j,k refer to any of the three cartesian coordinates (:,2,3 ++ x,y,t). It 1~1srxcntly been shown [I] that by analyzing the temperature dcpcndcncc of optical second harmonic generation CSHG) coefficients (d’s) via a microscopic polarizability model [2] one can obtain detailed structuml information concerning the microscopic nature: of solid state phase transitions. The basis of the method stems from the simple relation
d333 = v-l
d=GP,
Assuming a bond of symmetry CNv, the form of P is reduced to the following two independent elements
230
(3)
(4)
where the matrix [G] represents the 3 X 3 of direction cosines wh.i& transforms our microscopic (bond) coordinate system into our macroscopic (crystal) coordinate system and v represents the volume of t.he unit ccli. Hcncc ~3lG3mG3rlh?tn
*
(5)
CIiEMlCAL I’IIYSICS LE’ITERS~
Volume 38, nu~nbcr2 8333 andP311
Substituting
=Plg3
=&31
36322
~0223
=8232e
these various elen~nts into eq. (4) and
collecting terms, we find d333 = v-’
w;,l3333
f3G33(G$
Since [G] is an orthonormal fG$ = !) therefore
fG;z)Pd-
ca
(7)
where the particular element G3, represents the cosine of the angle between the 3 axis of the bond and the 3 axis of the crystal. in this particular case, LiNb03 symmetry R3c, this corresponds to the cosine of the angle U shown in fig. 1. Since we hiive thirty six such Nb + 0 bonds per hexagonal cell [3j therefore 36 2
corresponds to the c~st~~logr~p~c C,(Z) axis found in LiNbO,. Since the six niobium oxygen octahedra are related by three translations and WC C, rotation therefore
mittrix i.c. (Gil + C&
d333=~L[G:3i3333 +3G&-G;3M3,*1,
iis33=Y-’
I rnrch 1976
b~cos3Bjt3btcoS0jsin2Bi
3
;tuis shown
2
(81
whcro fill S pJs3 :~ndfl-”-= ijsl I _ It should perhaps
noted thdt the three-fold rotation
be
in fig.
Taking into account the three-fold tis shown in fig. 1, we can further simplify eq. (9) to
1
For an asymmetrically-trigonally dron, as in LiNbO,, the rciation tion cosines for the set of bonds end of the three-fold axis (co&) removed (cos 0 *) is
deformed octahebetween
the dircc-
nearest the
positive
and those farthest
COSUi=coso* - 3cos 5P44 _ Substituting
Cl11
this relation into eq. (IO) WCfind
d333 = 18V-’ ([(i.1546-_,}3-,3]p’l -3(0.3846-2rz
* 3.4638rz” - 3_n3)pl),
(12)
where PZG cos0. In the high temperature (T> 12OO'C) centric phase (lZ%) we find II = 0.5773 hence f3 = 54”44’ (fig. 1) i.e. we have an undistorted oct&edron and d333 = 0. In the row temperature phase (T= Z!?C} we find [3] II = 0.4736 and 0 = 6 1.7O. This corrcs-
ponds to
!i!bOs
&3
= 1W-’
(0.2096$~ - aao67d_)
l
(13)
Since for all altowed values of #z(0.4736 < IZ< 0.5’773) thi: coefficient of pii is at feast 30 times greater than that of 0’. and since we also expect [4] p” > PL, we thus neglect the 0’ term completely, giving us d3s3 = 18y-_rf(I_1546-~z)“-n3]lj’t
l
(14)
Recastirtg this in terms of thy deviittion of our octahedron from ideality i.e., # = 0 - 54.73” we find d 333-Fig. 1. N~obiuI~-oxygen octahedron showing the angle 0 ltetwfen the Ihrcc-fold and fuur-fotd axes. Tixc trigcxiai dw formation angle 0 is defined as the degree to which # deviates from normality (54”44’) i.e. Q = 0 - 54-44’.
l~~“~‘[sin~3.265cosQ,-4.898)
f O.S44sin39] ,
(1%
which for small @ i.e. (7O >, @b 0) can be simpIified
to 231
Volon~e 38,
CIIEhflCAL
number 2
PHYSICS
Using the room temperature values [5] of d, V and QSf 129x 10-%l,3.181 x 1(x2- cm3 and7” respectively we find via eq. (16) that /3” = 11.3 X 1Om30esu. This then gives LIS &=(105X
10-8esu)(3sin@-sin2@),
(17)
which describes the nonlinear
cocfficicnt d3j3 in terms of tilt dcgee of deformation # of the oxygen octahedron (fig. 1). We now use the temperature dependent SHC results of Miller and Savage [6] to predict the tcmperature dependence of @_This is shown in fig. 2. The rcla rive spread in the dss3 (obs) versus tempera-
ture and hence Q (talc) versus temperature is =6%. We see from fig. 2 that most of the “resymmctrizing” of the octahedron takes place above XOO°C. This type of behavior is quite similar to that found in the isomorphous ferroclcctric LiTaOj. In the cast of LiTnO3, the system has been accurately characterized by singIe crystal X-ray, neutron and optical SHG results, all three of which are in vs!ry close agrecrnent [7]. In the case of LiNbO3, howcvcr, the only iligh temperature structursl results are,from powder data [S]. These results (A’s) are rilso shown in fig. 2. The large error bars for these observed points allow us only to conclude that they agree with our SHG
SIiG
A61
-
X-Ray
A
I March 1976
LETTERS
results to within one standard deviation. It would bc interesting to compare our SHG structural results
with high temperature single crystal X-ray results which would of course be much more accurate than the points obtained from powder diffraction data. Since we now know the temperature dependence of our octahedral deformation, we can attempt to predict the temperature dependence of the spontaneous polarization P, via the same formalism as in cq. (4) i.e. e=rlGiipi,
(18)
whcrc P represents the spontaneous polarization fl represents the individual Nb + 0 bond dipole
moment. The 3 X 3 matrix G and the voIume Varc defined as in cq. (1). We assume here that the change in t!!c bond dipole versus temperature is small with respect to the change in G with temperature i.e. (ac;lar~a~~a7pn thiscase P=(18,u/3X
103V)(o.5773-cos0),
L
!;-:;,\, 2
1
1 0
200
SO0
-‘c
Fig. 2. Calculated (--)
*
a
600
,300
1000
1200
-
temperature depcndcnce of the
trigonal deformation an& I$as determined frem’SHG results. 7J1e observed X-ray powder remlts are given by S’S_
232
(1%
where P is in PC/cm’, p is in debye, V is in cm3 and 0 is given in fig. 1. Using the room temperature values [!&IO] ofP, Vnnd Q of 71&/cm*, 3.181 X IO-“” cm3 and 19= 6 I .7” respectively we find via eq. (19) that p = 18 d&ye, a value which is surprisingIy close to that found by simply assuming a charge transfer of 1.5 electrons from the Nb over to the oxygen i.e. 14 dehye. In fact, if WChad taken the bond length variations into account to eq. (19) we would have found exactly 14 debye. Ncvcrtheless, using the simpler model with g = 18 debye WC find Y = (687&/cm2)
I
and
(0.5773 - cosf?) .
W?
Using our temperature dependent deformation results (fig. 2) WCplot Pversus temperature in fig. 3. Except for the region between 1000°C and 1100°C the observed [9] results are in close agreement and the degree of misfit in this region (=lOSO”C) may not be statistically significant since both curves overlap at the i- 10% level. In summary we have shown that the degree of deformation of the oxygen octahedron in LiNb03 can be quantitatively described by observations of the temperature dependence of the nonlinear coefficient $33. The same trigonal deformation MOdeI aIs. has been successfully employed to characterize
Valume 38, number
2
CHEMICAL
CRIC.
e-
10%
Obs.
c-
IOX
tildi
-
interest and encoumgtment.
for his itltcrest .-A-
A
II
A
A
Refcseilces
A
PC&Z 40 I
A
I
I 4110
--“c
600
a00
1000
A-
1200
-_L_
Fig. 3. Calculated (---) and observed (a) tempcraturc: dependcncc of the spontaneous polarization PS.
the tcmpxxture
dependence of the spontaneous
polarization_ WC would
like to thank
our many
colfeagucs
J.G. Bergman aqd G.R. Cram!, Phys. Rev; Letters, to bc published. J.G. Dcrgman and G-R. Crane. J. Chem. Phys. 60 (1974) 2470. f31 S.C. Abrabams. J-M. Reddy and J.L. Bcrnstt’in, J. Phys. Chem. Solids 27 (I9S6) 997. and G.R. Crane. J. Solid State C&em. 12 I41 J.G. Rctpn (1975) 172. ES1 J.G. Berman and S.K. Kurtz, Mat. Sci. Eng. 5 (1970) 23.5. 161 K.C. Miller and A. Savage, Appf. Phys. Letters 9 (1966) 169. 171 J.C. Bergman, 3. Xm. Chcm. Sot., to be published. 181S.C. Abrahams, H.J. Ievinstein and J&l. Reddy, J. Phys. Cbern. Solids 27 (t966) 1019. 191 A.M. GIass, private co~nmun~c~tion. [IO] S.ft Wcmpte, &f. DiDomcnico Jr. and I. Camlibel. Appil. Phys. Lcttcrs I2 (196s) 209. [II
\
200
and comments.
A
60 -
G
Wc also thank
A.M. Ghss for allowing us to use his spontaneous polarization data prior to pubkation and G.K. Crane
b
4
1
PIIYSICS LETTERS
for
233
CIIEMICAL
1 March 1976
PHYSICSLETTERS
MOLECULAR MECHANICS OF THE FERROELECTRIC TO PARAELECTRIC IN LiNb03 VIA OPTICAL SECOND HARMONIC GENERATION
PHASE TRANSlTION
J.C. BERGMAN Bell Laboratories, Holmdcl, New Jcrscy 07733, USA Rcccived 13
November 1975
Tcmpcnturc dcpundcnt optical second harmonic generation (SIC) results xc used to monitor the 7O trigonai dcformation of the NW& uctahcdron in fcrroclcctric I.iNbO+ The StXG structural results compare favorably with those obtaincd from conventional (X-my) techniques. The tempcraturc dependence of the spontaneous polarization Ps, calculated vk~ the same trigonal deformation model is consistent with the cxperimcntal observations.
The rclationsbip between the induced polarization f and the optical field 6 acting on a given crystal is p=@;d&;y~3
+...
)
where x is the familiar linear polarizability nonlinear
terms
d and 7 xc
the coefficients
(1)
and the which
for second and third harmonic generation respcctivel y. A more rigorous formulation shows that the induced nonlinear (second harmonic) polarization (SW) is best expressed in terms of the Fourier components of the elcctlic field amplitude h-F I ’ are responsible
where G is the geometrical factor which rcliitcs the microscopic bond polarizability 13to the macroscopic crystal polarizability (I. We assume here that the tcmperaturc dependence of p is small compared to that of G i.c. aG/aT > ilp/aT. The rationale for choosing a crystd’s nonlinear polarizability (d) for studying a phase transition rather than the crystal’s linear polarizability x is that x(7’) seldom changes by more than a few percent, while d(T) often changes by over an order of magnitude. In this letter we apply the aforcmentioned technique ti, determine the molecular structure and mechanics of the fcrroclcctric to puraelectric phase transition in LiNb03 via the temperature dependence of d333_ WC relate the crystal SHG coefficient dss3 to the microscopic Nb +O bond polarizability fl via the general formalism
where d is a third rank (27 element) tensor whose elements are restricted by the symmetry of the mcdiurn and i,j,k refer to any of the three cartesian coordinates (:,2,3 ++ x,y,t). It 1~1srxcntly been shown [I] that by analyzing the temperature dcpcndcncc of optical second harmonic generation CSHG) coefficients (d’s) via a microscopic polarizability model [2] one can obtain detailed structuml information concerning the microscopic nature: of solid state phase transitions. The basis of the method stems from the simple relation
d333 = v-l
d=GP,
Assuming a bond of symmetry CNv, the form of P is reduced to the following two independent elements
230
(3)
(4)
where the matrix [G] represents the 3 X 3 of direction cosines wh.i& transforms our microscopic (bond) coordinate system into our macroscopic (crystal) coordinate system and v represents the volume of t.he unit ccli. Hcncc ~3lG3mG3rlh?tn
*
(5)
CIiEMlCAL I’IIYSICS LE’ITERS~
Volume 38, nu~nbcr2 8333 andP311
Substituting
=Plg3
=&31
36322
~0223
=8232e
these various elen~nts into eq. (4) and
collecting terms, we find d333 = v-’
w;,l3333
f3G33(G$
Since [G] is an orthonormal fG$ = !) therefore
fG;z)Pd-
ca
(7)
where the particular element G3, represents the cosine of the angle between the 3 axis of the bond and the 3 axis of the crystal. in this particular case, LiNb03 symmetry R3c, this corresponds to the cosine of the angle U shown in fig. 1. Since we hiive thirty six such Nb + 0 bonds per hexagonal cell [3j therefore 36 2
corresponds to the c~st~~logr~p~c C,(Z) axis found in LiNbO,. Since the six niobium oxygen octahedra are related by three translations and WC C, rotation therefore
mittrix i.c. (Gil + C&
d333=~L[G:3i3333 +3G&-G;3M3,*1,
iis33=Y-’
I rnrch 1976
b~cos3Bjt3btcoS0jsin2Bi
3
;tuis shown
2
(81
whcro fill S pJs3 :~ndfl-”-= ijsl I _ It should perhaps
noted thdt the three-fold rotation
be
in fig.
Taking into account the three-fold tis shown in fig. 1, we can further simplify eq. (9) to
1
For an asymmetrically-trigonally dron, as in LiNbO,, the rciation tion cosines for the set of bonds end of the three-fold axis (co&) removed (cos 0 *) is
deformed octahebetween
the dircc-
nearest the
positive
and those farthest
COSUi=coso* - 3cos 5P44 _ Substituting
Cl11
this relation into eq. (IO) WCfind
d333 = 18V-’ ([(i.1546-_,}3-,3]p’l -3(0.3846-2rz
* 3.4638rz” - 3_n3)pl),
(12)
where PZG cos0. In the high temperature (T> 12OO'C) centric phase (lZ%) we find II = 0.5773 hence f3 = 54”44’ (fig. 1) i.e. we have an undistorted oct&edron and d333 = 0. In the row temperature phase (T= Z!?C} we find [3] II = 0.4736 and 0 = 6 1.7O. This corrcs-
ponds to
!i!bOs
&3
= 1W-’
(0.2096$~ - aao67d_)
l
(13)
Since for all altowed values of #z(0.4736 < IZ< 0.5’773) thi: coefficient of pii is at feast 30 times greater than that of 0’. and since we also expect [4] p” > PL, we thus neglect the 0’ term completely, giving us d3s3 = 18y-_rf(I_1546-~z)“-n3]lj’t
l
(14)
Recastirtg this in terms of thy deviittion of our octahedron from ideality i.e., # = 0 - 54.73” we find d 333-Fig. 1. N~obiuI~-oxygen octahedron showing the angle 0 ltetwfen the Ihrcc-fold and fuur-fotd axes. Tixc trigcxiai dw formation angle 0 is defined as the degree to which # deviates from normality (54”44’) i.e. Q = 0 - 54-44’.
l~~“~‘[sin~3.265cosQ,-4.898)
f O.S44sin39] ,
(1%
which for small @ i.e. (7O >, @b 0) can be simpIified
to 231
Volon~e 38,
CIIEhflCAL
number 2
PHYSICS
Using the room temperature values [5] of d, V and QSf 129x 10-%l,3.181 x 1(x2- cm3 and7” respectively we find via eq. (16) that /3” = 11.3 X 1Om30esu. This then gives LIS &=(105X
10-8esu)(3sin@-sin2@),
(17)
which describes the nonlinear
cocfficicnt d3j3 in terms of tilt dcgee of deformation # of the oxygen octahedron (fig. 1). We now use the temperature dependent SHC results of Miller and Savage [6] to predict the tcmperature dependence of @_This is shown in fig. 2. The rcla rive spread in the dss3 (obs) versus tempera-
ture and hence Q (talc) versus temperature is =6%. We see from fig. 2 that most of the “resymmctrizing” of the octahedron takes place above XOO°C. This type of behavior is quite similar to that found in the isomorphous ferroclcctric LiTaOj. In the cast of LiTnO3, the system has been accurately characterized by singIe crystal X-ray, neutron and optical SHG results, all three of which are in vs!ry close agrecrnent [7]. In the case of LiNbO3, howcvcr, the only iligh temperature structursl results are,from powder data [S]. These results (A’s) are rilso shown in fig. 2. The large error bars for these observed points allow us only to conclude that they agree with our SHG
SIiG
A61
-
X-Ray
A
I March 1976
LETTERS
results to within one standard deviation. It would bc interesting to compare our SHG structural results
with high temperature single crystal X-ray results which would of course be much more accurate than the points obtained from powder diffraction data. Since we now know the temperature dependence of our octahedral deformation, we can attempt to predict the temperature dependence of the spontaneous polarization P, via the same formalism as in cq. (4) i.e. e=rlGiipi,
(18)
whcrc P represents the spontaneous polarization fl represents the individual Nb + 0 bond dipole
moment. The 3 X 3 matrix G and the voIume Varc defined as in cq. (1). We assume here that the change in t!!c bond dipole versus temperature is small with respect to the change in G with temperature i.e. (ac;lar~a~~a7pn thiscase P=(18,u/3X
103V)(o.5773-cos0),
L
!;-:;,\, 2
1
1 0
200
SO0
-‘c
Fig. 2. Calculated (--)
*
a
600
,300
1000
1200
-
temperature depcndcnce of the
trigonal deformation an& I$as determined frem’SHG results. 7J1e observed X-ray powder remlts are given by S’S_
232
(1%
where P is in PC/cm’, p is in debye, V is in cm3 and 0 is given in fig. 1. Using the room temperature values [!&IO] ofP, Vnnd Q of 71&/cm*, 3.181 X IO-“” cm3 and 19= 6 I .7” respectively we find via eq. (19) that p = 18 d&ye, a value which is surprisingIy close to that found by simply assuming a charge transfer of 1.5 electrons from the Nb over to the oxygen i.e. 14 dehye. In fact, if WChad taken the bond length variations into account to eq. (19) we would have found exactly 14 debye. Ncvcrtheless, using the simpler model with g = 18 debye WC find Y = (687&/cm2)
I
and
(0.5773 - cosf?) .
W?
Using our temperature dependent deformation results (fig. 2) WCplot Pversus temperature in fig. 3. Except for the region between 1000°C and 1100°C the observed [9] results are in close agreement and the degree of misfit in this region (=lOSO”C) may not be statistically significant since both curves overlap at the i- 10% level. In summary we have shown that the degree of deformation of the oxygen octahedron in LiNb03 can be quantitatively described by observations of the temperature dependence of the nonlinear coefficient $33. The same trigonal deformation MOdeI aIs. has been successfully employed to characterize
Valume 38, number
2
CHEMICAL
CRIC.
e-
10%
Obs.
c-
IOX
tildi
-
interest and encoumgtment.
for his itltcrest .-A-
A
II
A
A
Refcseilces
A
PC&Z 40 I
A
I
I 4110
--“c
600
a00
1000
A-
1200
-_L_
Fig. 3. Calculated (---) and observed (a) tempcraturc: dependcncc of the spontaneous polarization PS.
the tcmpxxture
dependence of the spontaneous
polarization_ WC would
like to thank
our many
colfeagucs
J.G. Bergman aqd G.R. Cram!, Phys. Rev; Letters, to bc published. J.G. Dcrgman and G-R. Crane. J. Chem. Phys. 60 (1974) 2470. f31 S.C. Abrabams. J-M. Reddy and J.L. Bcrnstt’in, J. Phys. Chem. Solids 27 (I9S6) 997. and G.R. Crane. J. Solid State C&em. 12 I41 J.G. Rctpn (1975) 172. ES1 J.G. Berman and S.K. Kurtz, Mat. Sci. Eng. 5 (1970) 23.5. 161 K.C. Miller and A. Savage, Appf. Phys. Letters 9 (1966) 169. 171 J.C. Bergman, 3. Xm. Chcm. Sot., to be published. 181S.C. Abrahams, H.J. Ievinstein and J&l. Reddy, J. Phys. Cbern. Solids 27 (t966) 1019. 191 A.M. GIass, private co~nmun~c~tion. [IO] S.ft Wcmpte, &f. DiDomcnico Jr. and I. Camlibel. Appil. Phys. Lcttcrs I2 (196s) 209. [II
\
200
and comments.
A
60 -
G
Wc also thank
A.M. Ghss for allowing us to use his spontaneous polarization data prior to pubkation and G.K. Crane
b
4
1
PIIYSICS LETTERS
for
233