T h at is why photoelectron spectra provide This work is the next, step Iiiforniation about olectroilic structure of solid states [1 Spec tioscorjN- LPS has been tiovelopecl.. Electron i
Trang 1A N G L E - R E SO L V E D L A SER - RAY PH O TO E L EC TR O N
S P E C T R O S C O P Y OF SIM PLE METALS
N g u y e n V a n H u n g
FiiCiihy of Physics, College of Natural Sciences - V N U
I INTROD UCTIO N
Photoelectron nu'asun'd in spectrometer have energy and momentum which are coniK'crod to th(' onos inside the crystal T h at is why photoelectron spectra provide
This work is the next, step Iiiforniation about olectroilic structure of solid states [1
Spec tioscorjN- (LPS) has been tiovelopecl According to this theory the intensity of LPS
l ụ f q Ĩ Ĩ E ,c ,) d E r ì n ,, ~ I r/3 p (T ,_ ,; + - E,)Ố{E - E, - nhw), (1)
here the liuoar contribution:
h ) ■ ■■U n IV /„ )
h J : y f„ (7Ỉ.V - E , - - E, - 2 huj') • ■ • ( % , - E, - Nhuj) (2)
f:
( ' I I V ' I } ! , ) u : , I ' i / ' ) ( / ; , I H " 1 / ^ )
Ef!^ - E, - 2huj){Ef> - E, - Ah^) ■ ■ {Ef;^ - E, - 2Nhuj) (3)
ha.sic \c< tor of K'ciproral vpc to! g, the height of potential barrier and surface - normal of
iosp(-cri\(>ly U’ w aio powf’rful radiations of laser ray Electron is emitted to the final
m atiix ('IciiK'nts are ronsideiecl to be coustant, the intensity of LPS spectra is characterized
by Plan(> Densitv of States (PDOS)
B Z
Trang 2whoro ly is band iiuiex k is wav(* voctor and A'o is distance of ( onsidoK'd ciOoS - section to
th(' center of Brillouin zone (BZ)
Our purpose ill this work is to calculato matrix elements ill (2) and (3), that nu'ans the iỉit(‘iisity (1) for siinpl(‘ metals and through tliat to ovaluatP the aiifth'-n^solvod LPS The numerical c-alrulation is carried out for Beryllium (Bo), a typical simple metal
The most important simple metals with configurations of the outer electronic shell
are presented in Tab 1 Th e positive ions in these siinpk' iiK'tals are small The con-
duetin^ electrons which are not strongly influenced by positive ions can move near froolv Thereforo these near free electrons can 1)0 (lpscrih(^d by Orthogonal Plane Wave (OPW )
Tab 1 Most important siinplo ni(' tals
with ( oiifiguiatioiis of outer ('Ifctionic shell in Iiputial atoms
~ | i/-'"'’" ' ) = / ■ < I I V’c K r r I ;,y = , N + i ■) I
whoH' tho core - fiinriioii is givi'ii by
()
and pseiuh'-fuuctioiis have th r form
ịh
111 th(\s(' íuiictiơus is \va\’0 \'(H’tor <‘01(' - stat(‘ and ÍÌ is Iioniializiii^ \'oluin<‘ Th(‘ 0 P \ \
(5) (lescriho th(' following electronic states
- Initial state, if j — 1(^1 = Ư.(J\ — g).
- Iiitriniediate states, if j — 2, 3, • ■ ■ , A\
- Final state, if j — N + I whicli is r r d u m l to one OPW , it means that
Trang 3Mu' function (G) is nonualiz(‘d acTonling to th r numl}rr Nc of basic rolls at R and cacli ( ('11 has Na atoms at ('// In the ahov(^ expiossions the inoniontuni conversation
A7 = k ÍOI transition from initial sta tr-to intprmodiato and filial ono is satisfied.
1 li(' (01 ('-iunctioii VV: ^ippf'ais o n l y in a c o m b i n a t i o n w h i c h is c a l c u l a t e d as f o ll o ws
Fioiu thí' oiilu)o(jualizatiou of O PW (5)
(9)
in uur ca.s(‘ \V(' liToivod
In tlie ('alculation \V(' ha\'e used the expressions of stnicturo factor
S { q - q ' ) = - ^ y
N o ^ I
(10)
( 11 )
(12)
(13)
(14)
and íli(' r('latioii
f’oi siniplr iiH'tals íh(' ort liogonalizin^ coiiiponiMit in O FW is small W(‘ can nogloct ihc second part in tli(' I inht of oqiiation(ll) that moans F'^ ^ 1 Mơie‘0V('i' f m ' oleríron (kn's I!0I ahsoiỉ) ('1(TÍloniai^iH'tic waves T h o n ’foio wr* locoivod
Usiui> thí' abo\'í' I(\sults we calculated th r inatiix elements for orio transition and
( ' I I / ) ~ - 'f - q + f / ) S { g - (fp - q).
9
(16)
After lietailing c alculation of matrix elements (2) and (3) we received the intensity
ot au*2,le - irsolv(*(l LPS spectra (1) for simple metals in the following form
I ( w ẽ ; ĩ } , b , 0 :E,cĩl)dEdíìp ~ Ị d^k ^ { I M{ k , ư j , ẽ R u P j ) 1^
+ I M \ k , i/^,2w, p ;) 1'“^ ]9{E e ~ E{k, v) ) ỗ{E - E{k% u) ~ nfìuj), (17)
Trang 4where the m atrix elements for the linear contributions are given
latrix elements for the linear contributions are given
N
ư j ^ \ ) í P a { k 9 j + i )
E{k, ) - E{k, v) - u - ì )f ì ^ '
(18)
and for the unlinear contributionslinear c o n tnOUT,ions
N
M ' = Y^Cg>(K i ^ > a i P { ) n 2-'- P j ) S i P - 9 ] - Õ)
Cg'^^^{k,iyj + -í)ự’a{ĩ' + ểj + l)
(19)
' E{k Uj+i) - E{k% u) - 2(j - l)f^^
^j + ỉ
angle-resolved LPS spectra of simple metals according to equations (17-19) The linear contribution to LPS is dependent on polarisation of laser ray, but the unlinear onr is independent on it
III NUMERICAL RESULTS OF PDOS FOR Be
It was shown th at for evaluation of LPS spectra we can consider the matrix eleiiK'nts
to be constant and the intensity of angle-resolved LPS spectra is proportional and the to the PD O S [1-3] Beryllium (Be) is a typical simple metal, th at is wh>’ it is H(4('ctecl for
calculation of PDOS We investigate the valence band of Bf‘ and for laser ray Iih^' >
lOOOeF, In this case thp, intensity of angle- resolved LPS spectra of Be is given 1)V
Bp has hexagonal closed packing (hep) stnicture Its BZ is shown in Fig L riic energy band structure of Be has been calculated hefore [4,5] Using those riiorp,i('s \V(' calculated the contributions of a cross- section of BZ which is pei pi'iulictilai to tho outp,oiii”
direction Cs of photoelectron Figs 2 and 3 show PDOS for tlie d i m tioiis
(contribution from FAHK plane of Fig 1) and C s / / r K (ronti ibution from r ALM of
F ig l), respectively They contain the contributions of the 1st, the 2nd and the 3rd band Therefore, they contain the detailing information of elertioiiir stiu rtu io of tlio coiisidpK'd direction We can see in Figs 2 and 3 that from BZ ceiitor to the ('lU'igy of 0.4a.u tli(> electron behaves as free, that is why we receive the PDOS of free electron , They aio
more constant Therefore, the PDOS can characterize the aiiglo-iosolved LPS of Bo
Trang 5Fi(] Ỉ BZ of Be with hep structure
Fig 2 PDOS of Bo for F M direction
IV CONCLUSIONS
Trang 6E l e c t r o n s in t h i s cast' b e h a v e as IK'H! O I K ' S , t h a t is \vli\' tlu'ii s ta t f' s ai(' (I'SCIilx'd In'
O P W T h ( ' m a t r i x (‘lin iu n its Í la iis ir io u s th iu n i;h iu ti'i iiK ’d i a t c s t a t is au<l tlio n
to sp(‘cti‘oniot(‘i (hu* to Iiiulti])li(>tt)!i al»suiỊ)ti(Mi has IxM'ii i alculai(’<1 Í01 liui'ai au(l uuliiK'ai
c o n f r i b n t i o i i s c D n r a i u t h í ' f u i i T i i i ) u t i o u > i OÍ a í o i u i t ’ t u i u ’t i o n s s T u u t u K i a c t o i a n d
of })liutơ(*l(‘Ctioii is coataiui'd in Iiioiui'iitmu p These \alu('s ai(' iinpoitaut for
a u a h 's is ot ('Ip c tro n ic sT n u tu r o o f t h e c iw s ta l T h i ' liiK 'ai CU 11Ĩ 1 Tt) L P S i.'' tl('p (‘iu l('n t
on polarisation of laser ia\' but tho unlinoar OHO is in(l('poiHli‘iit on it PDOS of B<' tov
c litfe ie iii o n to o iu ft direc tio n s o f p h o t o o lo c t io n hav(' h o v u ( a k * u la r ( ‘(l.
P D O S o f B e fo r (liffpKM it out»,oiui*, ( l i i í T ĩ i o n s o f p h o t o ( 'l( ’c r io u lia v (' hoiui ( a l ( u la te d
Tlu'\' show tlio iK'ar fi(H* i)(‘havior of i'k'c-troiis ill simple iiK'tals and Tlii’ii cK'ai aiiisoĩro])>'
iu (liiftn c n r d iio c tio n s T l i ( n r f o n \ th e y c h a r a c ti'r iz i' t h ( ‘ aii;*li'-!<'S(jl\-(‘(l L P S i) io \i(lin < ; (lo ta iliu i^ i i i f o n i i a t i o n ot ('li'i t i o i i ic s t n K ĩ i i n ' o f B('.
A c k i i o w l e d g m e u t T lli^ w o r k M'as c a i'iii’d OIU u itili'i ĩ l u ’ p la n OỈ tlu ' F i i u l a i i i e n t a l
R ('S ('ar( li P r t ) ” r a ia \ o 4 3 l A
R EFEREN CES
[1.’ N V Huaj; l u n i f i dl o f Pl i Ị Ị s ỉ ỉ s. N,j3( 1981)()
[2.' X \ ' Fluij^ J ot i i ' S r i ( - f i c r Vol.8 .\'o l i lí)97)2õ.
' l i ị P R ( ' i i i i ( M t a n d X \ " H u m ; E f Ị > i i i h / ^ / / / / s / i , s 2 G ( H J 7 S ) 1 0 1
4.1 M- T a u t P i n / S S f u f S n L ( i) ) 5 4 ( li » 7 2 ; 119.
5.1 P R i 'i iu i T t a n d X V H u u ^ E r f ) - T c c h Fliij-S. 2 9 ( 1 9 8 1 ) 1.
T A P CHI K H O A HOC ĐHQGHN, K H T N t x v , n‘^2 - 1999
PHO QUAXC; ĐIẺX T l ” DUNG LASKH PIỈỤ THUỌC GOC' C TA KIM LOAI n O N CIA
N g u y ễ n V a n H ilu g
hài Iià\' cirừiii; <1() các Ịìluì qiiaiiiJ, (liíMi tư tlùii.i; ỉasí'1 (Í.Ỉ^Sì ì>liii ílnux UUI
c iia k iu i lo a i (lư u ^ iiin ( ! ã (h n r c t lá iili ^ iá C iic Uaii.u, Ita n ;^ lliiii c u a n r (Iưtrc l)i(Mỉ (liỗ ii !)ÍU 1^ MJiii; p lia u ;; t n r c ;;iao i O P W ’ ) C i \ ( \'(’U u> Iiia í r ạ n (líii \'ứ i CÌU ( iìii\'( 'H (lịc h f|u;t CiU' t ! ạ ii^ t liá i I Iiy í '11 t íiili \ ìi plii I Iiv r i ỉ C’liiui;; 1)ÍH> clnVa Iiliừ n ^ ílạ i h rự n ^ í|u a ii t rụ iii; (líì<
t n r i i ^ c h o C HU ĩ i i ' u ' (l i í Mi t ư c i ì a \ ‘à t t l u ' I i l i i r h à m n , ^ u \ ' ( ’n í i r lif' > ó c a n Í M K \ ' à ( ỈU- i l u M i ; ;
số \'ỈIU” Iiãii^ lưựii”,- Pli'au tính số luat (lu íiaii.i; Uỉai Ị)lian^ (PD O Si (lã tlnri liicii (lio
Bí' C h rm > ; (là t ln' lii r i i cííc (lã c I íiil! í ư hỉUi CIIÍI (lic ỉi t ư t n m ; ; k i m lo a i ( h n i ^ là ii \'à klioii;^
(ta n i; hux>u^ k lù Ciuauj; tư ra th i'o các liư ứ n ^ k liiu n h a u D o (ỉó I ^ l ^ o s CUII^ ( a]>
các rhou^ tiu chi lirt vV* cati tnic' (liọii tư cua B(' lilìạn tư các pho LPS I)lỉu tlnnK- oỏ(