= 30.7 nra .
3
* u n i t s : 1 » 6 m V s e c * '.
105 - 2nd C ond. 3rd Cond. V i b r a t i o n a l O v e r la p 0 .1 0 . 5 1 . 0 0 . 1 0 . 5 1 . 0 <vbvb>* 9 .5 4 6 22 9 . 3 0 0 1 7 2 .3 8 0 2 . 8 2 6 7 7 .5 0 6 6 1.4 0 0 < v b v b /v ( k ) > * * 1 .3 8 0 6 .9 3 8 9 .3 7 8 0 . 5 0 9 3 .0 6 7 2 . 5 5 1 ^aa 1 .1 5 0 Î .O6 7 0 .0 1 4 3 . 9 0 1 3 .2 4 9 1 . 3 5 0 ^bb (1.0 0 7) ( 0 . 9 1 2 ) (0.Ü1 3) (2.9 9 2) ( 2 . 5 5 5 ) ( 1 . 2 8 9 ) ^cc 0 .2 4 9 0 .2 1 0 0 . 0 1 0 1 . 0 3 8 0 .9 9 5 2 .2 0 1 (0.2 0 2) (0.1 6 6) (0.0 1 0) (0.6 5 6) ( 0 . 6 4 9 ) ( 1 . 7 0 4 ) ^ac -0 . 0 2 8 -0 . 0 1 8 -0 . 0 0 2 - 0 . 1 4 8 -0 . 1 6 1 - 0 .4 6 2 ( - 0 . 0 5 5 ) ( - 0 . 0 4 0 ) (-Ü .Ü0 3) ( - Ü .3 0 1) ( - 0 . 2 1 0 ) ( - 0 . 5 2 9 ) (a • ) 2nd Cond. 3rd C ond. V i b r a t i o n a l O v e r la p 0 . 1 0 . 5 1 .0 0 . 1 0 . 5 1.0 <vbvb> * 14.211 2 4 7 .1 8 0 2 8 1 .8 6 0 6 . 8 9 6 2 0 5 . 1 1 0 4 1 .1 7 2 < v b v b /v ( k ) > H** 1 .6 7 9 7 .2 8 0 1 1 . 0 7 8 0 . 9 3 0 4 .7 9 8 2 .1 9 3 ^aa 1 .2 9 5 1 . 1 6 0 0 .1 0 4 2 . 8 0 3 1 . 8 8 1 1 .3 4 4 ^bb ( 1 . 0 9 2 ) ( 0 . 8 9 8 ) (0. 1 0 9) ( 2 . 3 1 5 ) ( 1 . 7 0 7 ) (1 .2 1 1 ) ^cc 0 .1 9 6 0 .1 2 7 0 .0 1 2 0 . 6 7 5 0 .9 3 8 2 .7 3 4 ubb ( 0 . 1 5 4 ) ( 0 . 0 9 8 ) ( 0 . 1 4 8 ) ( 0 . 4 7 3 ) ( 0 . 7 7 6 ) ( 1 . 8 5 8 ) ^ac -0.Ü1 7 -0 . 0 0 7 - 0 .0 0 1 -0 . 0 9 8 - 0 . 2 1 3 -O. 5 3 6 ^ b 1 c . c -t r ro ( - 0 . 0 2 7 ) ( -0.0 0 3) ( - 0 . 1 5 4 ) ( - 0 . 3 2 4 ) ( -0.5 5 2) ( b ) T a b l e ( 4 . 1 2 ) C a l c u l a t e d m o b i l i t y r a t i o s o f e x c e s s e l e c t r o n s i n th e s e c o n d and t h i r d c o n d u c t i o n ban ds i n c r y s t a l l i n e a n th r a c e n e com p u ted i n th e mean f r e e tim e and f r e e p a t h ( l n p a r e n t h e s e s ) a p p r o x im a t io n s a t
9 5 K ( a ) and 290 K ( b ) u s i n g M a th u r-S in g h c o e f f i c i e n t s and s c r e e n i n g p a r a m e t e r 's = 3 0 . 7
* u n i t s : 1 » 6 m ^ /se c" . * * u n i t s : 1k>3 m /s e c .
106
an even d istrib u tio n o f carriers through the bands hence by using th is value e f f e c t (c) can be elim inated. As previously stated in se ctio n (A»4) the unit c e l l constants show no discontinuous changes with temperature and i t therefore seems reasonable to suppose that e f f e c t (b) leads to a temperature dependence o f the type T*"n .
Assuming fo r the moment that t is constant over the temperature range 95°K to 290°K then fo r an excess electron in the f i r s t
conduction band the contribution to the to ta l temperature dependence o f e f f e c t (b) is o f the order T” 1*1 fo r excess electrons in the ab plan e»
S im plified calcu lation s by Friedman (7 0 ), assuming acoustic phonon sca tterin g , led to a temperature dependence fo r tq o f the type T~ 1,0 . Thus the predicted to ta l temperature dependence should be approximately T- 2 **» A sim ilar analysis using the elements o f
the m obility tensor fo r vib ra tion a l overlap fa cto r unity lead to a temperature dependence in the _ab plane o f the type T“ 1»8 . The la t t e r value is in fa ir agreement with the observed value o f
Kepler (11) o f T“ 1»5 .
Using sim ilar arguments i t can be shown that fo r excess holes e f f e c t (b) is highly an isotrop ic leading to to ta l temperature
dependence along the £ £ and _£* axes o f T” 2*5 , T" 2«0 and T” 2 » 3 , re sp e ctiv e ly , fo r v ib ra tion a l overlap fa cto r 0 .1 , and T“ 2« 3 ,
X“ 1.2 and fo r vib ra tion a l overlap fa cto r u n ity. The pred icted temperature dependence fo r hole conduction along the c ’ axis is within the extremes o f the reported values o f T“ 1«1* to TT2» 3 (14, 15, 16), however, Kepler (11) noted no observable anisotropy in the .ab plane. I t is in te re stin g to note that the anisotropy o f the temperature dependence r e fle c ts the anisotropy o f the thermal comparison c o e ffic ie n t s , a , v i z . o > a , > c l .
107 -
Similar trends are observed fo r the temperature va ria tion computed using screening parameter ç • 24,6 nm 1 where the respective
dependences along the ^a _b and c* axes are !
T“2 *1* , T~1,e and T“ 2» 1 , fo r v ib ra tion a l overlap fa cto r 0 .1 , and T "2» 1 » T“ 0*6 and T“ 2*1 fo r v ib ra tion a l overlap fa cto r unity»
For values calculated using the Mathur-Singh c o e ffic ie n ts the pred icted temperature dependences o f both electron s and holes in the ab plane are sim ila r to the values quoted above. However, fo r hole
conduction along the jc* axis the value - T-1* is much higher than observed experim entally.
In the foregoing discussion the e le c tro n ic conduction along
the <c* axis and the role o f higher conduction bands in determining the temperature dependence has been om itted. The predicted temperature dependence o f the m obility along the c ' axis f o r the f i r s t conduction band is o f the order T~ 3 whereas fo r the second and th ird conduction bands x t xs o f the type T 2 » however, due to the re la tiv e ly large band widths along the _c' axis coupled with the close proximity o f
the two energy bands one would expect e f f e c t (a) to be o f greater importance in the higher energy bands. Since, in general, e f f e c t (c) leads to a lowering o f the o v e ra ll temperature dependence this may be s u f f i c i e n t to reverse the sign o f the temperature dependence fo r
ca rrie rs in these bands. 4 .5 ( v i ) Hall e f f e c t
The r a tio o f the Hall to d r i f t m obility has been calcu lated in the manner outlined by Le Blanc (1 00).
Consider a one ca rrie r zero transverse current Hall experiment in which the current Hall fi'a ld and magnetic f i e l d vectors are
p a r a lle l to the orthogonal axes _a, t> and c* re sp e ctiv e ly . Let the energy o f the ca rrie r o f wave vector k be E(k) , it s group
- 108 -
v e lo c it y v(Jc) and inverse e ffe c t iv e mass M- 1(k) and assume that sca tterin g can be accounted fo r with a relaxation time function x(k )« I t can be shown that, fo r Boltzmann s t a t i s t i c s , the ra tio o f Hall to d r i f t m ob ilities along a is a function only o f the d ire ctio n o f application, o f the applied magnetic f i e l d , B .
H/
y /jij,
k0T « x 2 (va2 - 2vavb M ^ -l ♦ vb2 M -1aa >> (4 olO)B
<<T V 2>> <<T V, 2>>
The angular brackets in d ica te a s t a t i s t i c a l average over the Boltzmann d is t r ib u t io n in k space«
For sim p licity an abbreviated form o f the anthracene band structure i s considered which retains only those terms corresponding to in te m o le cu la r in teractions between neighbours in the ab plane« Thus
E+ (k ) - 2E3 ± 4E9 c o s ( i ko a) co s(J kola) ,
where the synbols have th eir usual meanings«
Assuming the mean free time approximation to be v a lid then fo r k 0T > band width the above r a tio reduces to
, ' - ? k0T * V ) (4.11)
B 11 _C
from which i t can be seen that the sign o f the Hall e f f e c t is
determined by the sign o f the resonance in teg ra l between the molecule at the o rig in and that at p osition (0, b_, 0 ) . A more general
ca lcu la tion has been done by Hermann (104). His th e o re tica l argument assumes the energy E(jc) to be a cosine function o f k and fo r the extreme case o f u « k 0T the ra tio becomes -2 .1 5 k0T/band width which again predicts an anomalous Hall e f f e c t fo r narrow bands«
The ra tios o f the H all to d r i f t m ob ilities calculated using
equation (4.10) are lis t e d in table (4 .1 3 ). The agreement between theory and experiment fo r both sets o f molecular o rb ita ls with 5 - 3 0 . 7 nm" 1 is good, however, the values calculated with the m odified screening parameter are rather low. Le Blanc (100) has shown that the r a tio y /y D fo r b) |j x b is re la tiv e ly in sen sitive to the band assymetry ( i . e . the ra tio E^/E^ but i s very sen sitiv e to the band widths. I t therefore appears that the band widths calcu lated using the modified S la ter function and S.C.F. atomic functions are too high, in dicatin g that these wave functions over estimate the true wave function at large distan ces.
Because o f the e ffe c t s o f band mixing i t is not possib le to
determine the values o f the Hall to d r i f t m o b ilitie s by equation (4,11) in the higher bands. However, the energy bands along the a ax is, fig u re (4 .5 ), show eith er an increase in the second conduction band or a s lig h t decrease in the third conduction band with increasing k. The bands along the £ axis show a very sharp decrease with in creasin g k in d ica tin g that Eg is p o sitiv e and E3 more stron gly negative,
thus giving ris e to an anomalous Hall e f f e c t in these bands. In je ctio n o f electron s in to the second and third conduction bands w il l not a lte r the sign o f the Hall e f f e c t , but i f the electron s are in je cte d from the f i r s t conduction band in the manner described by Sano (103), leaving behind an excess h o le , the magnitude and p ossib le sign o f the Hall e f f e c t fo r excess holes w il l vary since holes in this band w ill have the opposite sign to those in the valence band. Such a change o f sign has been observed fo r one o f the carriers in cry sta llin e
anthracene, however, the sign o f the ca rrie r could not be determined ( 99) , The situ ation in the case o f electron bands calcu lated using the
Mathur-Singh molecular o r b it a ls , figure ( 4 .6 ) , is not quite so stra igh t forward. Using arguments sim ila r to those above
H u e ck e l = 3 0 .7 nm H u eck el - 2 4 .6 nm M-S = 3 0 . 7 nm I I I
Temp. V i b r a t i o n a l E l e c t r o n H o le E l e c t r o n H o le E l e c t r o n H ole H ole E le c t r o n H ole
O v e r la p 0 .1 2 . 1 8 - 6 . 4 5 0 .9 1 - 1 . 9 6 1 .6 3 - 1 3 . 9 4 95 K 1 .0 0 .2 2 - 0 . 6 5 0 . 0 9 - 0 . 2 0 0 . l 6 - 1 . 3 9 - - 0 .1 9 .7 7 - 2 4 . 2 5 3 .1 2 - 6 . 9 7 1 0 .1 7 -3 9 .9 1 290 K 0 . 9 8 - 2 . 4 3 - 2 5 + 1 0 1 3 .6 - 3 5 . 7 1 .0 0 .3 1 - 0 . 7 0 1 .0 2 - 3 - 9 9 T a b l e ( 4 . 1 3 ) R a t i o o f th e H a l l t o D r i f t m o b i l i t i e s i n c r y s t a l l i n e a n t h r a c e n e . I F ig u r e ta k e n fr o m R e f ( 2 2 ) . I I F ig u r e s c a l c u l a t e d fr o m d a t a i n R e f ( 9 9 ) a ssu m in g t h e m o b i l i t y o f e x c e s s e l e c t r o n s and h o l e s i n th e ab p la n e t o be 1 .4 ]o -4 m / v o l t - s e c ( 11 ) and th e s i g n o f th e H a ll e f f e c t t o be a n o m a lo u s( 2 2 ) .
110 -
conclusion that the magnitude o f the Hall e f f e c t in the two hands is sim ila r but o f opposite sign» The separation o f the energy bands, with v ib ra tio n a l overlap fa cto r 0 .1 , is o f the order k0T at room
temperature therefore both bands w il l be populated giving ris e to a low value o f the Hall constant.
4.6 Conclusion
The energy band structure and ca rrier m ob ilities in cry sta llin e anthracene have been calculated in the tigh t binding approximation in which the wave function fo r a cry sta l containing an excess electron or h ole is constructed using both Hueckel and Mathur - Sin^a molecular o r b it a ls as a basis in constructing the Bloch sum. The wave functions constructed using Hueckel molecular orb ita ls give b e tte r agreement w ith experiment than th eir Mathur - Singh counterparts, although both
p r e d ic t a m obility along the c ' axis several orders o f magnitude lower than observed experim entally. Inclusion o f higher energy bands in the transport mechanism can, under certain conditions, remove this apparent discrepancy.