VNU. JOURNAL OF SCIENCE, Malhematics - Physics, T.xx. N03AP. 2004
CA LCU LA TIO N O F T H E ABSO RPTIO N C O E F F IC IE N T O F A W EAK ELE C T R O M A G N E T IC W AVE BY F R E E C A R R IE R S IN QUANTUM
W IR E S BY U SIN G T H E KUBO -M O RI M ETHO D
N g u y e n Q u a n g B a u , D i n h Q u o c V u o n g , N g u y e n B a L a m , L u u B i c h L in h D epartm ent o f Physics, College o f Science, V N U
A b s t r a c t : A n a ly tic e x p re s s io n s fo r th e h ig h -fre q u e n c y c o n d u c tiv ity te n s o r a n d th e a b s o rp tio n c o e ffic ie n t o f a w e a k e le c tro m a g n e tic w a v e (E M W ) d u e to fre e c a rrie rs fo r th e c a s e o f e le c tro n -a c o u s tic p h o n o n s c a tte rin g in Q u a n tu m W ire s a re c a lc u la te d b y u s in g th e K u b o -M o ri M e th o d in tw o c a s e s : th e a b s e n c e o f a m a g n e tic fie ld a n d th e p re s e n c e o f a m a g n e tic fie ld . In c o m p a ra tio rt w ith b u lk s e m ic o n d u c to rs , d iffe re n t d e p e n d e n c e o f th e h ig h - fre q u e n c y c o n d u c tiv ity te n s o r a n d th e a b s o rp tio n c o e ffic ie n ts o n th e e le c tro m a g n e tic w a v e f r e q u e n c y ® , th e te m p e ra tu re T o f th e s y s te m , th e c y c lo tro n fre q u e n c y Í Ì (w h e n a m a g n e tic fie ld is p re s e n t) a n d c h a ra c te ris tic p a ra m e te rs o f a Q u a n tu m W ire s is o b ta in e d . T h e a n a ly tic e x p re s s io n s are n u m e ric a lly e v a lu a te d , p lo tte d a n d d is c u s s e d fo r a s p e c ific Q u a n tu m W ire s o f th e G a A s / G a A sA I.
1. I n t r o d u c t io n
T h e c a p tio n o f e le c tr o n s in the lo w -d im e n s io n a l system s ig n if ic a n t ly re d uce d the v e r s a ltilit y o f th e electron s. T h is le a d s to s ev era l o p t ic a l a n d e le c tr ic a l p ro pe rties, d iffe r in g to tho se o f u s u a l sem icon d uctors[l-5]. T h e c a lc u la t io n s o f th e ab sorp tion c o e ffic ie n t o f w e a k e le c tr o m a g n e tic w a ve in q u a n tu m w e lls [6], a n d in th e doped s u p c r la tic c s [7] w e re stu d ied .
In th is p a p e r w e s tu d y th e h ig h -freq u en cy c o n d u c tiv ity te n so r a n d th e ab sorp tion c o e ffic ie n t o f a w e a k e lec tro m a g n e tic w a ve (E M W ) (w eak e le c tr o m a g n e tic w a ve h a s the form : E = Eoe'wt) d u e to fre e c a r rie r s c o n fin e d in q u a n tu m w ừ e s in th e cases o f the absence o f m a g n e tic fie ld a n d the p resen ce o f a m a g ne tic f ie ld a p p lie d p e rp e n d ic u la r to its b a rrie rs . T h e e le c tro n - a co u stic pho n on s ca tte rin g m e c h a n is m is a ssu m e d to be d om in a nt. T h e r e s u lts w e re c a lc u la te d n u m e ric a lly fo r a t y p ic a l q u a n tu m w ir e G a A s / G a A s A l.
2. H i g h - f r e q u e n c y c o n d u c t iv i t y t e n s o r a n d t h e a b s o r p t io n c o e f f i c i e n t o f w e a k e le c t r o m a g n e t ic w a v e b y f r e e c a r r i e r s i n q u a n t u m w i r e s i n th e ' c a s e o f th e a b s e n c e o f a m a g n e t ic f ie ld
A c y lin d r ic a l q u a n tu m w ire: th e ra d iu s R 0, the le n g th L , th e in f in it e con fin ed p ote n tia l: V ( r ) = 0 in s id e th e w ir e a n d V ( r ) =co elsew here.
B a sed on th e K u b o - M o r i m eth od w it h th e fra c tio n s c u t-o ff w it h in second order a p p ro x im a tio n o f th e in te ra c tio n , w e o b ta in e d fo llo w in g fo r m u la fo r th e a b sorp tion c o e ffic ie n t o f w e a k e le c tr o m a g n e tic w a ve in th e q u a n tu m w ir e s w it h e lectron -acou stic phonon s c a tte rin g m e c h a n is m [8-9]:
C a lc u la tio n o f t h e A b s o r p tio n C o e ffic ie n t o f .. 9
(1)
H e re , N* is th e in d e x o f re fra c tio n and c is the lig h t ve lo city , ƠXX(<I>) is the p e r p e n d ic u la r co m p o n e n t o f th e h igh - frequen cy c o n d u c tiv ity te n so r in th e absebce o f m a g n e tic file d w h ic h is a fu n c tio n o f fre q ue n cy Ü) o f th e e le c tr o m a g n e tic field , te m p e r a tu re T a n d p a ra m e te rs o f q u a n tu m w ires, w h ic h is n o n -lin ea r.
3. H ig h f r e q u e n c y c o n d u c t iv i t y t e n s o r a n d t h e a b s o r p t io n c o e f f i c i e n t o f w e a k e le c t r o m a g n e t ic w a v e b y f r e e e le c tr o n s in q u a n t u m w ir e s in p r e s e n c e o f m a g n e t ic f i e l d
In the case o f p resence o f m agnetic fie ld (m agnetic fie ld B //O Z ), u sin g K u b o-M o ri method, th e c om pon ents o f th e absorption coefficient o f electrom agnetic w a ve in qua ntum w ires w ith electron-acoustic phonon s catterin g m echanism in presence o f externa] m agnetic field:
o f the h ig h frequen cy con d u ctiv ity tensor in presence o f e xtern al m agnetic fie ld w h ich is a fu nction o f frequen cy (Ứ o f the electrom agnetic field, c yclotron frequen cy Q o f external m agnetic field, tem p era tu re T an d p aram eters o f q ua ntum w ires, w h ic h is non-linear.
4. N u m e r ic a l c a l c u l a t io n a n d d is c u s s io n a xx((ũ,Q)= - ^ - R e ơ xx(ci),Q),
cN*
(2)
W h ere Q = -^- is the frequency cyclotron, o^m.q) is th e p e rp e n d icu la r com ponents
F ig u r e 1. Dependence o f the absorption coefficient of E M W on the energy o f E M W and
temperature in the case of the Absence o f a M a g n e tic F ield
F ig u r e 2. Dependence o f the absorption coefficient o f E M W on the energy o f E M W and
radius in the case o f the presence o f a M agnetic Fie ld
10
Nguyen Quctng Bau, Dinh Quoc Vuong...
B a s e d o n t h e a n a ly t ic a l e x p re ss io n s fo r th e p e rp e n d ic u la r co m p o n e n ts o f th e h ig h fre q u e n cy c o n d u c t iv ity te n so r, th e a b s o rp tio n c o e ffic ie n t o f w e a k e le c tr o m a g n e tic wave b y fre e e le c tr o n s in q u a n t u m w ir e s w it h e le c tro n -a c o u s tic p h o n on s c a t te r in g m ech anism in th e c a se s o f p re se n ce (2) a n d a b sen ce (1) - (3) o f e x te r n a l m a g n e tic fie ld fo r a ty p ic a l q u a n tu m w ir e G a A s / G a A s A l w it h values: n = 0.01m eV;m = 0.067m 0 ,e = 2.07e0 ; r = 5 0 A ° a n d th e sy s te m is a s su m e d to be a s ro om te m p e r a tu re (T = 300°K).
A c k n o w le d g m e n t s :
T h is w o rk is c om p le te d w it h fin a n c ia l su p p o rt from the P ro g ra m o f B a sic Research in N a tu r a l S cie n ce 411204.
R e f e r e n c e s
1. N. M o r i a n d T .A n d o , Phys. Rev., B40(1989) 6175.
2. H .R u c k e r,E -M o lin a r i and P .L u g li, Phys.ReV., B45(1992) 6747.
3. . J.P o z e la a n d V .Ju cie n e ,Sov.Tech.sem icond, 29(1995) 459.
4. A .S u z u k i, Phys. R ev., B 4 5 ( 1992) 6731.
5. G .M . Shm elev, I.A. C h a ik o v s k i a n d N guyen Q u ang B a il, Sov.Phys. Tech. Semicon., 12(1978) 1932.
6. N g uy en Q u a n g B a il, T r a n C ong Phong, J. Phys. Soc. Japan., 67(1998) 3875.
7. N g uy en Q u a n g Bau, N g uy en V u N h a n and T r a n C ong Phong, Korean Phys. Soc, 41(2002) 149.
8. R. Kubo, J . Phys. Soc. Ja p a n , 12(1957) 570.
9. H . M o ri, Progr. Thcor. Phys, 34(1960) 399.
