Analytical expressions for the rate of acoustic phonons excitations, the conditions for the amplification of sound and the m om entum conditions for electron to participate in the amplif
Trang 1VNU JOURNAL OF SCIENCE M athem atics - Physics T X V III, N()3 - 2002
O N T H E A M P L I F I C A T I O N O F S O U N D ( A C O U S T I C
P H O N O N S ) B Y A B S O R P T I O N O F L A S E R R A D I A T I O N I N
C Y L I N D R I C A L Q U A N T U M W I R E S W I T H P A R A B O L I C
P O T E N T I A L I N T H E P R E S E N C E O F M A G N E T I C F I E L D
N g u y e n Q u o c H u n g a n d N g u y e n Q u a n g B a u
D e p a r tm e n t o f P h ysics, C ollege o f S c ie n c e - V N U
A b s tr a c t Based on the quantum kinetic equation for phonon, the amplification
of sound (acoustic phonons) due to the absorption of laser rad iatio n in cylindrical quantum wires w ith parabolic potential in the presence of m agnetic field is studied Analytical expressions for the rate of acoustic phonons excitations, the conditions for the amplification of sound and the m om entum conditions for electron to participate
in the amplification of sound are obtained for the two cases: singlephoton absorption process and m ultiphoton absorption process T he differences betw een the two cases
of singlephoton absorption and m ultiphoton absorption are discussed; the numerical com putations and plots are carried o u t for a G aA s/G aA sA l q u an tu m wire T he results are com pared w ith bulk sem iconductors and q u an tu m wells
1 I n t r o d u c t io n
It is well known th a t the interaction of a laser rad iatio n w ith m aterials can lead to the excitations of higher harm onics arid the am plification of phonons T he problem has been widely investigated in the past in a num ber of p apers [1-6] In [1-3] the problem has been studied in bulk sem iconductors with singlephoton absorption process and m ultipho ton absorption process for degenerate and non-degenerate electron system In [4,5] the problem has been considered for quantum wells w ith singlephoton absorption process and
m ultiphoton absorption process and in the presence of ưiagnetic field All these authors shown th a t the phonon population grows w ith tim e u n d er some conditions
In tills paper, we stu d y the influence of m agnetic field on the am plification of sound (acoustic phonons) due to the absorption of laser rad iatio n in cylindrical quantum wires
w ith asym m etric parabolic potential Based on the q u an tu m kinetic equation for phonon,
we obtain analytical expressions for the rate of change of the population of the phonon states, the conditions for the amplification of sound, and m om entum condit ions for electron
to participate in the amplification of sound w ith th e presence of m agnetic field in th e two cases: singlephoton absorption process and m ultip h o to n absorption process All results are numerically com puted and plotted for a G aA s/G aA sA l q u an tu m wire
2 R a te s o f a c o u s tic p h o n o n s e x c ita tio n
Consider a quantum wire w ith elliptical cross section and asym m etric parabolic confining potential:
T y p e s e t by ẠẠ/I ì S-T^X
10
Trang 2O n th e a m p l i fi c a ti o n o f s o u n d ( a c o u s tic p h o n o n s ) by. 11
V { x , y ) = % - { t f x x 2 + t f y y 2).
Here f2Z)f1y are the effective frequencies of the potential and 771* is the effective electron inass
Assume the vector p o ten tial p of the titled m agnetic field B in the form p = (0, B z x } B x y — B yx ) ) electron wave function and energy can by w ritten as [7Ị:
$ -7>(x, y, z) — —= — ■ — c 71X H n ( y- I -= =
n’ ự l y/2*nllxy/ĩỉ V J y/2‘llly ự ĩ
h2k 2
£n, i (kz ) = — ~~ 4- ^ 1 ( n + 1/2) + àiL>2 (/ 4- 1 /2 ), (2)
1 + t e ) + t e ) 1 ~ l ' - \ JW Û 7 ’ 1 “ Wc - ^ is the cyclotron frequency, I I n ( x ) is H erm itte polynomial of order n, L is the length of the where M = m*
wire,
i m + fiỉ + w i + 2 ù , n J 1 + ( ^ ) 2+ ( ^ )
\ íìỈ + + 0,2 - 2Í2
i/2
W ith bulk phonon assum ption, the H am iltonian for the electron-phonon system of
a quantum wire in the external field can be w ritten as:
# ( 0 = X ] £" 'i ( k ~ +
n,i, 1c
(3)
where a and a / (fci and ò-ỹ ) are th e creation and annihilation operators of
electron (phonon); /c = (0,0, k z ) is the electron wave vector (along the wire’s axis: zaxis);
is the phonon wave vector; ~Ầ (t) = fịE o c o s ( íìt) is the potential vector, depends on the laser radiation; C n I n# l’C<t) = C - ệ ln I n ‘ I' is the electron-acoustic phonon interaction coefficient, where I C -ỹ |2= - ^ ự ĩ V is the principal volume of the crystal, £ is the deform ation acoustic potential, f) is the density of the m aterial, v s is the sound velocity in the m aterial, R is the wire’s radius, I n i n ’ I’ is the form factor
Prom Ham iltonian (3), we o b tain the q u an tu m kinetic equation for acoustic phonons
in quantum wires:
Trang 312 N g u y e n Q u o c H u n g, N g u y e n Q u a n g B a u
~ * 2 Ĩ 2 |C 'n,ỉ,n',i'("ỹ)|2 ^ ( n n , ỉ ( ^ ~ ~ t ) - " n ' l '( " ^ ) ) [ ( b t ) t i x
x ^ ^ ( « î ) ( / ế ì ) e x p ( ^ £ n '’r ^ ^ “ £n - ' ^ ~ - 0 - + í s í ì í ^ d t
W here the sym bol ( x ) t m eans th e usual therm odynam ic average of o p erato r x\ n n /( k ) = (a * ~>a l -j+)ti ^ — ~ ^ i i 2 ) *A/(æ) is the Bessel fonction of the first kind
Perform Fourier transform ation, from the dispersion equation for phonon, we o btain
th e rate of acoustic phonons excitations:
= E \ C n l , n > , l > ( t ) \2Y , \ n n ' , l ' ( t ) - n n ,l ( t - t ) } x
x ] C ^ ( « 2) ố {ff« ' ( ^ ) - £ n , i ( ~ k - ~ ỉ ) - - W i n } , (5)
u = —00
w ith ố(x) is th e D irac function
From (5), we process w ith non-degenerate electron system assum ption to obtain the rate of acoustic phonons excitations in the case of singlephoton absorption process and the case of m ultiphoton a b so rp tio n process
3 T h e a m p lif ic a tio n o f s o u n d in t h e c a s e o f s in g le p h o to n a b s o r p tio n p ro c e s s
In the case of singlephoton absorption process, assume th a t A <^c ÍỈ, from (5) we obtain the ra te of acoustic: phonons excitations in q u an tu m wires:
a { t ) = w ẩ E K W < * )I* *
9 n ,/,n \r
X e x p l ^ - 0 h u j i ( n + 1/2) + (3huj2{l + 1/2) - + ^2^ 2) + ~ ~ 2 ~ I x
(6)
w ith a = — n l ) — fox/2(ỉ — I ' ) + 0 = k n is B oltzm ann constant
Due to Ổ function in (5), only /c satisfies condition:
Trang 4contributing in the integral, or only electrons w ith m om entum satisfying condition (7) call dam p or amplify phonons
From (6), it is obvious when ÌAỈ-Ệ Q is negative, we have th e am plification of sound:
m * L \ 2 V " ị r , , - ^ , 2 _ ( h ịiu ì- ệ rn mp n '2 \
° < * > = t ] ị e x p ( 2 - “ V )
X exp f — (3huJi(n + 1/2) + /3hüJ2(l + 1/2)
« + 1/2> - f ặ ) x
(huj-Ệ + — u /) + ỉĩuJ2{ị 0 ) ^ •
4 T h e a m p lific a tio n o f s o u n d in t h e c a s e o f m u lt i p h o to n a b s o r p t i o n p r o c e s s
Use th e approxim ate form ula as in [8]:
fl(A2 - g ) 7T\/A2 — Z£2 >
with:
Ớ
Í 1,
I 0,
if X > 0
if X < 0 From (4), after several calculations, we o b tain the rate of acoustic phonons excita tions in the case of m ultiphoton absorption process:
i/=0
with:
X n i ( x ) - e xj){h(3u >x(n + 1/2) + h(3u>2(l + 1/2)) X
X exp ~ 2 fi2 ~ 2 ( ^ i ( n — n 0 + ^ 2 ^ - I') + X +
(n — n ') 4- hx)2{l — ư) + X + huJ-ệ )
X /„ {^ - Ị ạ - ặ ỌiMx(n - n ') + - 0 + x + »
/„(x ) is the complex Bessel function o f the order V.
Trang 514 N g u y en Quoc H u n g, N g u y e n Quang B a u
In a sim ilar way to (7), we have the m om entum condition for electron to participate
in the dam ping or am plification of sound:
N ote th a t if:
X n '1 V 2m* ) X n ,'v \ 2m* ) ’
then a (~ ỹ ) < 0, or the num ber of phonon grows w ith time
5 N u m e r ic a l r e s u l ts a n d d is c u s s io n
(10)
(11)
F igure 1. D ependence of the rate of phonons excitations on phonon wave vector in
th e case of singlephoton absorption process; R = 5nm , B = 5T, Í2 = 200T H z
Prom th e obtained results, we plot th e dependence of the rate of acoustic phonons excitations Q on phonon wave vector (F ig l), on laser frequency ÍĨ (Fig.2) and on tem p eratu re T (F ig.3) for the case of the singlephoton absorption process Param eters for num erical co m p u tatio n are m * = 0.067mo; B = 5Te s l a ] v s = 40877715” 1. All figures show
th a t the curves have peaks, which illu strate the m axim um of the am plification of sound,
a t q = 4.7106m “ l , ÍÌ = 130T H z , T = 100K Com pare these results w ith quantum wires
in the absence of m agnetic field, we realize th a t m agnetic field has certain influence on the am plification of sound However, the num erical results for bulk sem iconductor and
q u an tu m wells are different a t th e range of values of q an d Q for th e amplification to happen
N ote th a t (7) and (10) are th e conditions for electron to p articip ate in the dam ping
or the am pification of sound in th e wire, b u t not the conditions for the amplification to happen In the lim iting case w hen U-Ệ Í2 (for singlephoton absorption process) and (11) (for m ultip h o to n absorption process), a (~ ỹ ) < 0 and we have the amplification of sound, or the nu m b er of phonon increases w ith tim e as a direct result of th e presence of the laser radiation These form ulae are m ore com plicated th a n the corresponding formulae
in quantum wells [4-6] due to th e in tricate dependence of the electron wave function and energy sp ectru m on H erm itte polynom ial W ith bulk sem iconductors, because of the continuance of th e energy spectrum , th e dependence has a completely different form [1-3]
Trang 6O n the am plification o f so u n d (a co u stic p h o n o n s) by. 15
A J A T - 1 7 1 K , 1 / m , R « r n m , F N - r , T
F ig u re 2. T he am plification of sound as a function of laser frequency in th e case of singlephoton absorption process; T = — 1 0 0 °c, H = 15nm , t í = 5T
/ ■ 1 0 * l / m , u m c g a « i J O X H 3 , K « l i > n m , b - î ï T
F igure 3. Dependence of th e rate of phonons excitations on tem p e ra tu re in the case
of singlephoton absorption process; Í2 = 1307712, R ~ 15nm , q = 4.7106r a - 1 , B = 5T
In the case of m ultiphoton absorption process, th e ra te of acoustic phonons ex citations (9) and the m om entum condition (10) intricately d ep en d on th e variable (the exponent and the complex-Bessel function) T he dependence on th e complex-Bessel func tion /1/(2), z varies with th e intensity of the laser rad iatio n field (A = — ), shows th a t the am plification of sound depends on the intensity of the rad iatio n field w ith order greater than two, while the dependence in th e case of singlephoton ab so rp tio n process is of the order two (6) W hen condition (11) is satisfied, we also have th e am plification of sound Com pare these formulae w ith the expression in [1-3] we realize th e difference for b o th the dependence and the m om entim i conditions
Trang 716 N g u y en Quoc H u n g, N g u y e n Q uang B a u
6 C o n c lu s io n
In the conclusion, we want to emphasize th at:
1 The quantum kinetic equation for phonon in q u an tu m wires in th e presence of
m agnetic field was established, which has a sim ilar form w ith those in quantum wells and bulk wsemiconductors
2 Analytical expressions and conditions for the am plification coefficient of sound were obtained in the case of singlephoton absorption and m u ltip h o to n absorption
In proper conditions, the rate of acoustic-phonon excitations is negative and the phonon population grows w ith time
3 The num erical com putations and plots are carried o u t for a G aA s/G aA sA l quantum wire Prom the results, it is easy to see th a t there are th e ranges of value of phonon wave vector, laser frequency and tem p eratu re a t which the am plification of sound happens
4 M agnetic field has certain influences on the am plification of sound
A c k n o w le d g e m e n t: T his work is perform ed w ith financial su p p o rt from the N ational Program of Basic Research in N atu ral Science No 411301
R e fe re n c e s
1 Nguyen Q uang B au, Nguyen Vu Nhan, C hhoum m Navy, J o u r n a l o f S cien ce, N a t.S c i.,
15(1999),1
2 E.M Epstein, R ad io in P h ysics, 18(1975), 785
3 E M E p stein , Lett J E P T; 13(1971),511
4 Nguyen Q uang Bau, Vu T h an h Tam, Nguyen Vu N han, J S c ie n c e and T echnical
In v e stig a tio n s in A r m y , No 24, 3(1998),38
5 Nguyen Q uang Bail, Nguyen Vu N han, Nguyen M anh Trinh, P roceedings o f I W O M S
’99, Hanoi 1999, 869
6 Peiji Zhao, P h y s R e v , B 49(1994), 13589
7 V.A.Gey 1er, V A M a rg u U s ^ ^ /te v , B 61, 3(2000),1716
8 L.Sholiinal, T u n n e l e ffe cts in s e m ic o n d u c to rs a n d a p p lic a tio n s, Moscow, 1974