Determined conditions of laser field on acoustic phonon increasing in semiconductor block

 Hong Duc University Journal of Science, E.4, Vol.9, P (33 - 37), 2017 
DETERMINED CONDITIONS OF LASER FIELD ON ACOUSTIC 
PHONON INCREASING IN SEMICONDUCTOR BLOCK 
Nguyen Tien Dung1 
Received: 14 September 2017 / Accepted: 10 October 2017 / Published: November 2017 
©Hong Duc University (HDU) and Hong Duc University Journal of Science 
Abstract: In this paper, I have been established the kinetic equation for phonons in 
semiconductor block under intense laser field. Using this equation, we find expression for the 
rate coefficient for the case degenerate electron gas. The condition of the acoustic phonon 
increasing in semiconductor blocks is discussed. 
Keywords: Acoustic phonon, semiconductor block, laser field. 
1. Introduction 
 Phonon amplification by absorption of laser field energy is a subject extensively 
investigated in different structures [1,2,3,4,8]. The main results of these papers are that by 
absorption of laser field energy, the interaction of the laser field with electron can lead to the 
excitation of higher harmonics and the amplification of phonon. With the development of 
modern experimental technology, the fabrications of low-dimensional structures are 
realizable. Naturally, phonon amplification by absorption of laser radiation in such confined 
structures should show the characterization of the electron-phonon interaction. 
 In this paper, we start from Hamiltonian of the electron-phonon system in Semiconductor 
Block (SB) under intense laser field; we derive a quantum kinetic equation for phonon in SB 
in the case of multiphoton absorption process. Then, we calculate the phonon excitation rate in 
the case of the electron gas is degenerative. Finally, we calculate the acoustic phonon 
excitation rate (APER) in a specific SB to illustrate the mechanism of the phonon amplification. 
2. Quantum kinetic equation for phonon in a Semiconductor Block 
 We use a simple model for a SB, in which an electron gas is confined by SB potential 
along the z direction and electrons are free on the x-y plane. It is well known that its energy 
spectrum is quantized into discrete levels in the z direction. A laser field irradiates which is 
normal to the x-y plane, its polarization is along the x axis, and its strength is expressed as a 
vector potential A(t) . The Hamiltonian for the system of the electrons and phonons in the 
case of the presence of the laser field is written as [8]: 
Nguyen Tien Dung 
Academic of Engineering and Technology, Vinh University 
Email: Tiendungunivinh@gmail.com ( ) 
 33 
 Hong Duc University Journal of Science, E.4, Vol.9, P (33 - 37), 2017 
 2
 1 e 
  
 H(t) p A(t)aapp  qqq bb Caa(b qpqpq b) q (1) 
    
 p2m c q p,q
 where ap and a p are the creation and annihilation operators of electron in SB, bq and 
bq are the creation and annihilation operators of phonon respectively, q   q is phonon 
energy for wave vector q . A(t) is the potential vector, depending on the external field. 
 A eAcosx 0  t,A 0 cE/ 0  (2) 
 Under intense laser field, the electron-phonon system is unbalanced, the phonon numbers 
change over time. The change over time of N (t) b b is described by the equation: 
 q q q t
 Nq (t) 
 i b b ,H t (3) 
 t q q t
 We obtain the quantum kinetic equation for phonons in SB: 
 Nq (t) 1 2   
 | Cq | J s J exp[i( s)  t]
 2   
 t p s.     
 t 
 dt ' [N (t ') 1]f (p q)[1 f (p)] N (t ')f (p)[1 f (p q)] 
 q q 
 i (4) 
 exp p q  p  q   (t ' t)
  
 [Nq (t') 1]f(p)[1 f(p q)] N q (t ')f (p q)[1 f (p)] 
 i 
 
 exp p  p q  q  (t ' t) 
   
 Where Nq (t) b q b q t , the symbol X t means the usual thermodynamic 
average of operator X, J (z) is Bessel function, f() p ap a p t ,  e E0 q / m  . 
3. Phonon excitation rate in a SB 
 Above results [4] allow one to introduce the kinetic equation for phonon number of the 
q mode: 
 Nq (t)
 qN q (t) (5) 
 t 
 where q is the parameter that determines the evolution of the phonon number Nq (t) 
in time due to the interaction with the electrons. If q 0 the phonon population grows with 
time, whereas for q 0 we have damping. 
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 Hong Duc University Journal of Science, E.4, Vol.9, P (33 - 37), 2017 
 From (6), the phonon excitation rate becomes: 
 N q (t) 2 
 2 2 
 | Cq |  J   /  f (p q) f (p)(p q p q ) (6) 
 t  
 p  
 In the strong-field limit,    and the argument of the Bessel function in Eq. (6) is 
larger. For large values of argument, the Bessel function is small except when the order is 
equal to the argument. The sum over  in Eq. (7) may then be written approximately: 
 2  1
  JEEE        (7) 
   2
 Here E p q  p  q . The first Delta function corresponds to the absorption and the 
second one corresponds to the emission of /()  photons. In the strong-field limit only 
multiphoton processes are significant and the electron-phonon collision takes place with the 
emission and absorption of /()  photons. Substituting Eq. (7) into Eq. (6), the phonon 
 ()() 
excitation rate becomes q  q  q , where: 
 2 
 () 
  qC(q) f (p q) f (p)     p q p q (8) 
   
 p 
 In the following, we will calculate for the case in which the electron gas is 
degenerative. In this case, we may simplify the carrier distribution function by using the 
Boltzmann distribution function: 
 0 khi F  p
 f p  F  p . 
 1 khi F  p
 I calculate the rate of acoustic phonon excitation. For acoustic phonon, we have 
 2
 2 q
Cq here V, , va, and  are the volume, the density, the acoustic velocity and the 
 va V
deformation potential constant, respectively. 
 2 4
 () qm  eqE 
  0  (9) 
 q 16 2 v 2 V 2 m  q 
 a 
 Analyzing Eq. (10) we can obtain the conditions for the phonon amplification. From the 
 eE q
 () o 
condition q 0 , we obtain q 0 . The condition which the laser field must 
 m 
satisfy is: 
 eqE
 0 
   q (10) 
 m 
 in which: 
 35 
 Hong Duc University Journal of Science, E.4, Vol.9, P (33 - 37), 2017 
 2 2 2
 m  q  eEo 
 F  o qv ; v e x (11) 
 2q2 2m m
 The condition (10) simply means that if the drift velocity of electron q.E0 / m under 
the intense laser field, excesses the phonon phase-velocity, a deformation potential for 
multiphonon excitation can be generated in the SB. 
 In next to the condition (10), in the case of degenerate electron gas must also satisfy the 
condition (11), so the increase acoustic phonons are more difficult. Note that the condition 
(11) is not indicated by other authors when studying this effect [6,8]. 
4. Conclusions 
 I have analytically investigated the possibility of phonon amplification by absorption of 
laser field energy in a SB in the case of multiphoton absorption process with non-degenerative 
electron system. Starting from bulk phonon assumption and Hamiltonian of the electron-
phonon system in laser field we have derived a quantum kinetic equation for phonon in SB. 
However, an analytical solution to the equation can only be obtained within some limitations. 
 Using these limitations for simplicity, I have obtained expressions of the rate of acoustic 
phonon excitation in the case of multiphoton absorption process. Finally, the expressions are 
numerically calculated and plotted for a SB to show the mechanism of the phonon 
amplification. Similarly to the mechanism pointed out by several authors for deferent models, 
phonon amplification in a SB can occur under the conditions that the amplitude of the external 
laser field is higher than some threshold amplitude. This is the Cerenkov’s condition [8]. 
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 Hong Duc University Journal of Science, E.4, Vol.9, P (33 - 37), 2017 
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