W hen th e conditions of p aram etric resonance (PR) are satisfied, p aram etric interactions and transform ations (PIT) of sam e kin d s of excitations, such a s phonon-ph[r]
Trang 1P A R A M E T R I C R E S O N A N C E O F A C O U S T I C A N D O P T I C A L
P H O N O N S I N D O P E D S U P E R L A T T I C E S
L u o n g V a n T u n g , T r a n C o n g P h o n g , P h a m T h u y V in h , N g u y e n Q u a n g B au
D epartm ent o f Physics, College o f Science, V N U
A b s tr a c t: T h e p a ra m e tric re so n a n ce o f a c o u s tic an d o p tic a l p h o n o n s in D oped
pre dicte d u sin g a s e t o f qu an tu m k in e tic e q u a tio n s fo r t h e p h o n o n s D is p e rs io n s
o f th e re s o n a n t ph o n o n fre q u e n c y an d th e th re s h o ld a m p litu d e o f th e fie ld fo r
p a ra m e tric a m p lific a tio n o f th e a co u stic ph o n o n s in D o p e d S u p e rtattic e s a re
in a D op ed S u p e rla ttic e s n-i-p-i o f G a A s:S i/G a A s:B e
1 I n t r o d u c t io n
I t is well known t h a t in th e presence of an ex te rn al electrom agnetic field (EEF), an electron gas becomes non-stationary W hen th e conditions of p aram etric resonance (PR) are satisfied, p aram etric interactions and transform ations (PIT) of sam e kin d s of excitations, such a s phonon-phonon, plasm on-plasm on, or of different kinds of excitations, such as plasm on-phonon will arise; i.e., energy exchange process betw een th ese excitations will occur [1] The PIT of acoustic an d optical phonon h a s been consider in bulk sem iconductors [2] The re s u lt of th e stu d y show th a t the P IT can speed u p th e dam ping process for one excitation an d th e am plification process for an o th e r excitation F or low-dim ensional sem iconductors, th ere have been serveral w orks on th e gen e ratio n an d am plification of acoustic phonons [3] However, in o ur opinion, th e energy exchange processs betw een two different kinds of phonons in low-dim ensional system s, w hich a r e driven by a PR of a two- phonon kind, have not yet been reported It should be noted th a t th e m echanism for P IT is different from th a t for phonon am plification u n d e r a laser field [4Ị an d from PR of a defect mode[5]
In [6] we have studied th e P IT in a quantum well w ith non-degenerative electron gas
In order to continue th e ideas of [2, 6], th e purpose of th is p ap e r is also study the param etric resonance of acoustic an d optical phonons, b u t in a doped sup erla ttic e (DSL) in which the electron gas is non-degenerative
2 Q u a n tu m e q u a t io n
We consider a DSL,a vector potential à (t)= Ã() cos(Qt) I f the F rohlich electron- acoustic an d optical phonon in tera ction potential is used, th e H am iltonian for th e system of the electrons and th e acoustic a n d optical phonons in th e la s e r field is H(t)[7]
U sing H am iltonian H (t)a n d realizing op erato r algebraic caculations a s in [6.7], we obtain a set of couped qu an tu m tra n s p o rt equations for th e acoustic phonons:
Trang 2i|(b5>t=(N-Hw|l (V
3 A c o u s tic p h o n o n s d is p e r s i o n a n d c o n d i t i o n f o r p a r a m e t r i c a m p lif ic a tio n
We lim it o u r calculation to th e case o f th e first o rd e r resonance (h =1), in which
Uq ± = n We also assu m e th a t th e electron-phonon in tera ctio n s satisfy th e condition
and D Ị • f (q ) is the electron-acoustic phonon interaction
coefficient an d electron optical phonon interaction coeffcient)
In th ese lim itations, if we w rite the dispersion re lations for acoustic an d optical
phonons as (0ac(g) = 0 )a + ixQ and (0Op(ộ) = tú0 + ix„, we obtain th e re s o n a n t acoustic phonon
modes:
< % * > ■ * » « , + i f i v ± v „ )A ( q ) - i<T + T „ ) ± V [(v„ ± v „ ) A ( q ) - « T - t „ ) f ± A2 j (2)
In eq (2), th e signs ( ± ) in th e subscript of correspond to th e signs ( ± ) in front
of th e root an d th e signs ( ± ) in th e subscript of co±^ correspond to th e o th e r sign pairs
These signs depend on th e resonance condition (Oj + V- = (Ì For instance, th e existence of
a positive im aginary p a r t of (ûj~ im plies a param etric am plification of th e acoustic phonon In such cases th a t x « 1, th e m axim al resonance, an d q= <7x ( q z = 0), w here
V ” " ý 2 | C » , n ' ( q f ĩ K ) ; to " - ■ ÿ Z K n ' f i f l ' i V i j ) (35
From eq (4), th e condition for th e re sonant acoustic phonon m odes to have a positive
im aginary p a r t lea ds to: |a| > 4t0t0 U sing th e s e condition an d eqs (3),(4), we obtained the threshold am p litu d e for th e E E F for the degenerative electron gas:
727Tpm3/2n en n fa>qMen,,n(a>« ) +
E o > E th = fi2q2e I _ e x p [-P (e n - En- )]
X ex p f [(en- n ((flq ))2 + (En n(Vq))2 ]ỳj[l - expip/kOq )] [1 - exp(pfiViị )] (5)
w here p = — — , k g is th e Boltzm ann co n stan t a n d T is th e te m p e ra tu re of the
k BT
Trang 3E quation (5) m eans th a t th e param etric am plification of th e acoustic phonons is achieved w hen th e am p litu d e of th e EE F is higher th an the th resh o ld am plitude To
num erically estim ate th e th reshold am plitude Eth for th e p aram etric am plification of
th e acoustic phonons, we use th e superlattice n-i-p-i of GaAs: Si/GaAs: Be w ith th e
p aram eters a s folows [6,7]: ị = 13.5eV , p = 5.32g/cm'3, v„ = 5370ms \ =50 meV, s 0 =100, d=40 nm, n n =1023 1/m3, x»= 10.9, Xo = 12.9, m = 0.067mo, m„ is th e m ass of free electron, an d
t i v * h v () =36.25 meV In th is case, th e threshold am plitude m ust be from lOkV cm '1 to
25kV cm 1 for th e wave nu m b er of phonon from 10s m '1 to 109 m
4 C o n c lu s io n
In this paper, we obtained a general dispersion eq uation for p aram etric
am plification an d tran sfo rm atio n of phonons However, a n an alytical solution to the equation can only be obtained w ithin some lim itations Using th e s e lim itations for simplicity, we obtained dispersions of th e re so n an t acoustic phonon modes an d the threshold am plitude of th e field for acoustic phonon p aram etric am plification S im ilarly to the m echanism pointed o u t by several authors for bulk sem iconductors a n d q u an rtu m wells, param etric am plification for acoustic phonons in a doped s u p erla ttic e can occur
u n d er the condition th a t th e am plitude of the external electrom agnetic field is h ig h er th an some threshold am plitude N um erical results for GaAs/GaAsAl q u an tu m w ell clearly show the predicted m echanism P aram etric am plification for acoustic phonons an d th e threshold
am plitude depend on th e physical param eters of th e system an d a re sensitive to the tem perature
Acknow ledgments: T his work is com pleted with financial support from the Program of Basic
Research in N a tu ra l Science 411204
R e fe re n c e s
2 E M Epstein, Sou Phvs Semicond 10(1976), 1164.
4 P Zhao Phys Rev B 49(1994), 13589; Feng Peng, J Phys.: Condens Matter 11(1999)
4039
6 T c Phong, N Q Bau, J Kor Phys Soc 422003) 647.
7 N Q Bau N V Nhan, T c Phong J Kor Phys Soc 41(2002), 149.