.lOURNALdl SCIENCE OF HNUE
M.alicni.iucal and Physical Sci., 2012. Vol. .i7. No. 7, pp. 106-112 This papci IS available online al lillp://sldb,hnuc.cdu vn
THK EOllAIIONS OK STA IK OF A.SYMMKTRIC M CLEAR MATTER
Le Vict Hoa and Lc Due Anh
Facalt}- ofl'ln.\ics, Hattoi National University of Education
Ahslract. The equations of stale (EOS) of asymmetric nuclear mailer lANMi in an extended Nanibu-Jona-Lasinio (ENJL) model was invesligaled by means of examining effective potential in one-loop approximation. Our numerical results show that isospin dependence of saturation density in our model is reasonably strong and critical temperature for liquid-gas phase Iransitijn decreases wilh increasing neutron excess.
Keywords: The equations of slate, asymmetric nuclear matter, isospin.
1. Introduction
The success of nuclear physics in satisfactorily explaining low and mediate energy nuclear phenomena leads to a strong belief that nucleons and mesons are appropriate degrees of freedom. At present, the relalivistic treatment of nuclear many-body systems introduced not long ago by Walecka [1-3| turned out to be a quite successful tool for the study of many nuclear properties: binding energies, effective nucleon mass, equation of state, liquid-gas phase transition, eel Along with the success of the Walecka's model, a four-nucleon model of nuclear matter (4-6| is introduced which consists of only nucleon degrees of freedom. In this article we will consider the isospin dependence of the energy of asymmetric nuclear matter in the extended Nambu-Jona-Lasinio (ENJL) model. The main goal of such studies is to probe the properties of nuclear matter in the region between symmetric nuclear matter and pure neutron matter. This information is important in understanding the explosion mechanism of supernova and the cooling rate of neuu-on
Received October 22,2012. Accepted November 6,2012.
Physics Subject Classification: 62 44 01 03.
Conlacl Le Viet Hoa, e-mail address; [email protected] 106
The equations of state of asymmetrit mu. leur mcniei
2 . C o n t e n t
2.1. T h e e q u a t i o n s of s t a t e of n u c l e a r m a t t e r
Let us consider the nuclear matter given by the Lagrangian density
£ = C{id- .\l)r-\- '^{li'tl')'- '; [r-i"I-)' -^{d^ryir)'-\-u'n"I'll: (2.i) where V' is nucleon field operator. M is the "bare" mass of the nucleon, (i = diag{pp.,(i„)\ fip^,, = {t[) ± (if/2 is chemical potential, f = a/2 with a are the i.sospin Pauli matrices, 7^ are Dirac matrices, and G'.,,., are coupling constants.
Bosonizing
In the mean-field approximation
(IT) = CT, (lip) = wi5o„, (K^) = bSsaSof, (2.3) Inserting (2.3) into (2.2) we obtain that
£„„=iji{ia-M'+~t°p-}i>-U(o,iii,b), (2.4) where
M- = M-G,a, (2.5) p' = P-G„U}-G,.T\ (2.6) U(o,i^,b) = i l G , C T " - ( ; „ a ) ^ - G , 6 ^ ] . (2.7)
The solution M " of Eq. (2.5) is the effective mass of the nucleon.
Starting from (2.4) we arrive at the inverse propagator
l(k,+p;)-M- -S.k 0
S-'(ifc;CT,w,6) = a.k -{ko+ii'^)-M' 0
0 0 {ko+p'„)-M'
\ 0 0 3.k
0 \ 0
—(T.k • ( (ko-t'lK,)-^' /
10 7
Thus
.irI.S- '(l,.a.uj,l,) = {k„-l /•;,')('.„ /•; )ik„ + l-:i)(l.u- !•:')• (2.9) in which
K; - K; I (//„ (.V'). f'^l ' ^ ' ± ( ' 2 '2"'''' ' ^ ' = /*•' + -""-(2.10) Based on 12.7) and (2.Si Ihc effective polcnlial is derived:
u =- ('(,Tu,vi) + ir.:.i-v '^^-[(:,a'-<:,.ic' -a,ii'\- ^Jj.-di,Un{\+,-'••-'•')
+ lii(l ) , ' • • " ' ) - H n ( l I , ' " ) + h l ( H r ' • • • ' • ' • ) | . (2.11)
The ground state of nuclear matter is determined by the minimum condition
^ = „, '^^0, ^ = 0 . 12.12,
do Uui 00
Inserting (2.11) into (2.12) we obtain the gap equations
^ = A / /.•-,/A-^{(n;+n;) + ( n - + 0 } = p ,
'^ = ;Ji/^"''-'*{K-0 + (",7-"n*)}=P.
b = ^ / ^ " A - ' d * - { K - - 0 - ( n ; - n : ) } = P „ I2.B) where
n~=n~_\ nl=nX\ n^ ^ n-^- n~ = nl; "^ =^ [,'•-••'' -\-i\~\
The pressure P is defined as
/ ' ^ - f i l t a k e n at minimum. (2.14)
Combining Eqs. (2.11), (2.13) and (2.14) together we get the following expression for the pressure
P = -YP? + 2ip^, + ^p;-f^jJ°°A-VA[ln(l-Fe-^--/'-)
+ ln(l + e-'^;/'') + ln{l + e"^-/^) + ln(l -t- e-'^t''^)].2 ' ' • ' 2 ''« ' T ' ' ' "
(2.15) 108
The equations of state ofa.iynimcliit mu Inn IIILIIICI
The energy density is obtained by the Legendre transform of /':
e = Sl(<T,w.6)-FTS-I-/I,,/),, -I ;;,/), c \ .. G„ , a, , 1 /•" ,
= V ' ' - '' V ' ' " ' " • ) ' ' ' '" ^ / * " ' * £ ! . ( " , ; + " , , I ii„ -I- ",;)• (2.16) with the entropy density defined by
+ ",7ln?i,7 + (1 - ",7)ln(l - 11,7)
-I- 11+In 17+ 4- (1 - ii + ) l n ( l - 11+) -I H,; h i n , | + (1 - ",',)hi(l - »,;) . (2.17)
Let us introduce the isospin asymmetry 11:
<^ = (l>n-l>,)IPH. (2.18) in which PB = Pn + Pp is the baryon density, and Pn, Pp are the neutron, proton densities,
respectively.
Taking into account (2.5), (2.13), and (2.18) together the Eqs. (2.15), (2.16) can be rewritten as
^ ^ _ ( . ! / - . I / - ) " , r67 , G,Q
-I- l n ( l - l - p - ^ * r ^ ) - H n ( l - t - e - ^ ^ ) - f l n ( H - e " ^ ' ^ ) ] . (2.19)
2G, [ ^ + ^ ] p 2 + 1 r edkE,(n; + < +n-+ n+). (2.i z a '^ Jo
Eqs. (2.19) and (2.20) constitute the equations of state (EOS) governing all thermodynamical processes of nuclear matter.
2.2. N u m e r i c a l s t u d y
In order to understand the role of isospin degree of freedom in nuclear matter, let us carry out the numerical study. First we follow the method developed by Walecka [1]
to determine the three parameters G^, G^, and Gy for symmetric nuclear matter based on the saturation condition: The saturation mechanism requires ± a t at normal density PB = pQ = Q.llfm"^ the binding energy eun = -M + e/ps attains its minimum value 109
[,c Vict Hoa and Lc Due Anh
{^i„n)o ^ -15, SA/.r, in Which e is given by (2,20). It is found that Gj = 13.62/m'and G, = 0 7,S(.\. As 10 fixing (!, let us employ the expansion of nuclear symmetry energy (NSE) around po
Fnum — " 1 -f I' t Pn - Po 3 I Po
i\ K,yin ( PB- PoY
• j 18 I PO I (2.21)
with (11 being the bulk symmetry parameter of the Weiszaecker mass formula, experimentally wc know ii, 30 - XiMiV'. I, and /<,„„ related respectively to slope and curvature of NSE at po
A,t,„. = 9pol 3 2 I
Then Gr is fitted to give a^ ~ 3,2A4cV. Its value is Cr = 0.1980,.
Thus, all of the model parameters are fixed, which are in good agreement with those widely expected in the literature [1]. Now we are ready to carry out the numerical computation.
In Figures 1 and 2 we plot the density dependence of E\„n{pB\ a) at several values of of temperature and isospin asymmetry a. From these Figures we deduce thai for comparison with the results of the chiral approach of nuclear mailer [7] the asymmetric nuclear matter in our model Is less stiff and the isospin dependence of saturation density is strong enough.
Figure 1. The density dependence of binding energy at several values of temperature and isospin asymmetry n = 0 and o — 0.25
The equations of .stoic nj nsviiimetric nucleai imillcr
Figure 2. The density dependence of binding energy at several values of temperature and isospin asymtnetry i\ = 0.-5 and (\ = I
The EOS for several n slcp.s at some fixed temperatures LS presented in Figures 3 and 4 As we can see from the these figures, the crilical lemperalure for the liquid-gas phase transition decreases with increasing neutron excess.
Figure 3. The EOS for several n steps at tetnperatures T = QM('\' and T = \QMeV
" i ^ j j a ^ " - " _
/ • • ' •''/
•'i
1
'• • » . "
Figure 4. The EOS for several a steps at temperatures T = 15-\/( \" and T = 20.UcV'
L.e Viel H(Ki and Lc Due Anh
3. Conclusion
In this arlicic we have invesligaled Ihc isospin dependence of energy and pressure of llic .isyiiimelnc nuclear mailer on Ihe NJL-lype model. Based on the effective potential in [he one Itmp appio,\iiiialioii wc delcniiined Ihe expression of pressure by the effective liolenlial al the inininiuni. As a icsull. the lice energy has been obtained straightforwardly.
Tlic\ consliuile Ihe equaliims of stale (l-OS) of tbe asymmetric nuclear matter. It was indicated thai in llic asymiiiclric nuclear iimllcr. the isospin dependence of saturation density is reasonably strong and (lie critical tcnipcraturc for the liquid-gas phase transition decicascs wilh iiK leasing neulron excess. This is our major success. In order to understand heller the phase slructurc of Ihc asyiniiiclnc nuclear matter more detail study would be carried out by means of numerical computation. This is left for future study.
RKKKRKNCK.S III J. D. Walecka, 1974. Ann. Phys 83. 491.
| 2 | B. D. Scrot and J. D. Walecka, 1997. Phys. Lett. 8 8 7 . 172.
[3] B. D. Scrot and J. D. Walecka, 1986. Advances in Nuclear Physics, edited by J. W.
Negele and E. Vogt (Plenum Press, New York,), Vol. 16, p. 1
[4] Tran Huu Phat, Nguyen Tuan Anb and Le viel Hoa. 2003. Nuclear Physics. .A722, pp.
548c-552c.
[5] Tran Huu Phat, Nguyen Tuan Anh, Nguyen Van Long and Le \ i e t Hoa, 2007. Phys.
Rev. C76, 045202.
[6] Tran Huu Phat, Le viet Hoa, Nguyen Van Long, Nguyen Tuan Anh and Nguyen Van Thuan, 2011. Communications in Physics. Vol. 21, Number 2, pp. 117- 124.
[7] Tran Huu Phat, Nguyen Tuan Anh and Dmh Thanh Tam, 2011. Phase Structure in a Chiral Model of Nuclear Matter, Physical Review C84, ();4321
112
JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2012, Vol. 51, Nc. 7, pp. i 13-123 This paper is available online at http://stdb.hnue.edu.vti
ABILITY T O INCREASE A WEAK E L E C T R O M A C N E I I C WAVE BY CONFINED E L E C T R O N S IN DOPED SUPERLATTICES IN T H E PRESENCE
O F LASER RADIATION MODULATED BY AMPLITUDE
Nguyen Thi Thanh Nhan^ Nguyen Vu Nhan^ and Nguyen Quang Bau'
'Faculty of Physics. College of Natural Sciences. Hcmoi Natioiuil University -Faculty of Physics. Academy of Defence force - Air force Abstract. The analytic expressions for the absorption coefficient (ACF) of a wcdk; eleclromagnedc wave (EMW) caused by confined electrons in the presence of laser radiation modulated by amplitude in doped superiattices (DSL) are obtained by using the quantum kinetic equation for electrons in the case of electron-optical phonon scattering. The dependence of the ACF of a weak EMW on the temperature, frequency and superlatuce parameters is analyzed. The results are numerically calculated, plotted and discussed for n-GaAs/p-GaAs DSL. The numerical results show that ACF of a weak EMW in a DSL can get negative values.
So, by the presence of laser radiation modulated by amplitude, in some conditions, the weak EMW is increased. The results also show that in some conditions, the ability to increase a weak EMW can be enhanced in comparison with the use of non-modulated laser radiation. This is different from the case of the absence of laser radiation modulated by amplitude.
Keywords: Absorption coefficient, doped superiattices, weak electromagnetic wave, laser radiation.
1. Introduction
In recent times, there has been a growing interest in studying and discovering the behavior of low-dimensional systems, in particular, DSL. The confinement of electrons in these systems considerably enhances electron mobihty and leads to their unusual behaviors under external stimuli. As a result, the properties of low-dimensional systems, especially the optical properties, are very different in comparison with those of normal Received September 25, 2012. Accepted October 4, 2012.
Physics Subject Classification: 62 44 01 03.
Contact Nguyen Thi Thanh Nhan, e-mail address: [email protected]. vn
Nguyen Thi Thanh Nhun, Nguyen Vu Nhan and Nguyen Quang Bau bulk .sL-niiconduciois 11. -'| Ihc linear absorption of a weak EMW by confined electrons in low-dinicnsinnal syslcms has been invesligaled using the Kubo-Mori method [3, 4]
and the nonlineai .ihsorption ol a slrong HMW by confined electrons in low-dimensional syslcms has been siiuiii-d by using the quantum kinetic equation method 15. 6]. The inlkiciKc of lasL-r radiation on the absorption of a weak EMW by free electrons in normal bulk sLMiiicorKluciors has been investigated using llic quantum kinetic equation method [7. 81 and the presence of laser radiation and. in some conditions, the weak EMW is increased. The innuciicc of laser radiation (non-modulated and modulated by amplitude) on the absoiplion of a weak BMW in compositional superiattices has been investigated using Ihe Kubo-Mori mclhod |9l. The inlluencc of la.ser radiation on the absorption of a weak HMW in quantum wells has been investigated using the quantum kinetic equation method [10]. However, Ihc influence of laser radiation modulated by amplitude on the absorption nf a weak liMW in DSL is still open for study. Researching the influence of laser radiation on the absorption of a weak HMW plays an important role in experiments because it is difficult lo directly measure the ACF of strong EMW (laser radiation) by experimental means. Thcrclore, in this papci; wc study the ability to increase a weak EMW by confined electrons in DSL in the presence of laser radiation modulated by amplitude.
Tbe electron-optical phonon scattering mechanism is considered. The ACF of a weak EMW in the presence of a laser radiation field modulated by amplitude are obtained using quantum kinetic equation for electrons in a DSL. We then estimate numerical values for tbe specific n-GaAs/p-GaAs DSL to clarify our results.
2. Content
2.1. The absorption coeflicient of a weak EMW in the presence of a laser radiation field modulated by amplitude in a DSL
2.1.1. The laser radiation field modulated by amplitude
As in 19], here we also assume that the strong EMW (laser radiation) modulated by amplitude has the form:
F{t) = F,(0 ^ A ( 0 = F, sin {Q.t + a^) + Asin {iU + ^2) (2-1)
where, Fi and F2 has same direction, Hi and ih are a bit different from each other or After some transformations, we obtain:
F{t) = EQysm{nt-h^i) (2.2) with Foi = V ^ ' + ^2' + 2F,F2COs{AUt + Aa), AU = Qi - ^2, Aa = a, - Q^,
'—'-=^.'--|^*.(f..-).
114
Ability to increase a weak electromagnetic wave by confined electrons in doped .superiattices...
here, U is the reduced frequency (or the frequency of the laser radiation modulated by amplitude) and \AQ.\ is the modulated frequency. Foi is the intensity of the laser radiation modulated by amplitude.
- - F\ F> \ F In the case that Fi, F2, il\, ii.. i\ satisfy the conditions: ^ = ^ = , and AQ = 0, the above formulas can be approximated as in [9].
When AJi = 0, laser radiation modulated by amplitude becomes non-modulated laser radiation.
2.1.2. The electron distribution function in a DSL
It is well known that the motion of an clcciron m a DSL is confined and that its energy spectrum is quanti/.ed into discrete levels. We assume that the quantization direction is the z direction. The Hainiltonian of the electron-optical phonon system in a DSL in an EMW field in the second quantization representation can be written as:
« = E =•• (p^ - h^(i'>) <f."".<>. + H ''M-''.-
+ Y. QAvfe)a:.,,-,„-,«„,P,(V+*;,-) (2-3)
where n denotes the quantization of the energy spectrum in the z direction (n = 0, 1,2,etc.); {n.p±) and {n\p± -\- qx) are electron states before andafter scattering, respectively;
P±{Q±) IS the in plane xOy wave vector of the electron (phonon); a^^ and On.pj^, {bt and bg-) are the creation and the annihilation operators of the electron (phonon), respectively;
? ~ {Q±!Qz)i •^{*) = H^oi cos(rit -j- ipi) H—Fo2cos(wt) is the vector potential of the EMW field (including two EMWs: a strong EMW with the intensity Foi and the frequency Q; a weak EMW with the intensity F02 and the irequency UJ); OJ^ ^ UIQ is the frequency of an optical phonon. C, is the electron-optical phonon interaction constant [7]:
2 ^ M _ l ) (2.4)
here V, e, £0 are the normaUzation volume, the electron charge and the electronic constant,Xo and Xoo are the static and the high-fi-equency dielectric constants, respectively. The electron form factor /„,n'(gs) is written as:
In^n'iQz) = X ! / ' ^ ' ' ' ' ' ^ " ( ^ - ^d)iJn-{z - ld)dz (2.5)
In a DSL, the electron energy takes the simple form:
Nguyen Thi Thanh Nhan, Nguyen Vu Nhan and Nguyen Quang Bau
er,{pi) 'l,'^!+H'(" + ^j (2.6)
here, in' is Ihc eflecdve mass nl eleclrtm, il'„{z) is the wave function of the n-th state for a single potential well which composes the DSL potential, d is the DSL period, .V^ is the
l;r('-'7T,/;>
numbcrof Ihe DSLperiod, Wp ( — I is the frequency plasma caused by donor doping concentration and .V,; is doped concentration.
In order lo establish the quantum kinetic equations for electrons in a DSL, we use the general quantum equation for statistical average value of the electron particle number operator (or electron distribution function) r/„,; (t) = ( « ^ ^ a^.p, ) [7]:
<>"u„ (/) / r , i \
'" ,; =([":,.•,"--.•'']>, (2.7)
where (V"), denotes a statistical average value at moment t and {^), = 'J'rlW'i-). with 11' being the density matrix operator. Starting from the Hamiltonian in Eq. (2.1) and using the commutative relations of the creation and the annihilation operators, we obtain die quantum kinetic equation for electrons in a DSL:
-~J, = ~ ^ ^ IQI^ \l„,«'{g,)\' YI M3iQL)Maigi)Jm{a2qx)Jfla2gi)
X exp {i {((s - l)n + (m - / ) . . • -tS]t + {.-. - 1)^-, }}
X / <'t2{\ii„.,y. {l2).\,r - nv,p,+,-,(«2)(.V„-+ 1)1
™P I ^ IE„'{P± +q±)- e „ ( p i ) - fiu,- - snn - mti...' + ihS] (t - (2)1 + {n„,fAt2){N,+ 1) - n„',f,+,-,(t2)-\i\
exp 1 ^ [e„.(pl + 5 l ) - e„{px) + huj,--shn - mhu + iM] {t - i j ) } - [""'.p-x-,-x((2)N,-- n„,j-Ji2)(iV,-+ 1)]
™P 1 ^ l^nipi) - £„.(pi - 5l) - hwf-shn - mikj + iM] {t - ij)}
- K',p^-4At2)(N,-+ 1) - n„,-^(i2)W,-]
Ability to increase a weak electromagnetic wave by confined electrons in doped superiattices...
exp •! T l^niPi) - €„'{p_L - 7i) -H /JWq- - shQ - mhw -|- ihS] {t - is) \ (2.i If we consider a similar problem in normal bulk semiconductors that the authors V.
L. Malevich and E. M. Epshiein published, we will see that Eq. (2.8) has a similarity to the quantum kinetic equation for electrons in the bulk semiconductor [8].
It is well known that to obtain explicit solutions from Eq. (2.8) is very difficult. In this paper, we use the first-order tautology approximation method to solve this equation [7, 8]. In detail, in Eq. (2.8), we choose the initial approximation of n„,^^ (t) as:
where fin,pj. is the balanced distribution function of electrons. We perform the integral with respect to /_>. and then we perform the integral with respect to t of Eq. (2.8). The expression for the unbalanced electron distribution function can be written as:
n'.q k,s,r,m^-oo
exp{-~i{[kQ, + nj -\-i6]t-h kipi}}
kQ + ru) + iS [ n„',p^-?^N^- nn,p^ [N^ + 1)
V^niVi.) — ^n'ivx — 9 I ) " ~ buj^- shO. — mhw -\-ihS
"n',p-x-q-x(-^q + ^) " ^r^.f^^g
SniPl.) — £71'(pl — g l ) + ^ q ~ ^^ ~ " * ^ + ^ ^
nn.p^N^ - " < p - x + g - x ( ^ q + 1)
Gn'iPL + gl) ~ £n(pl) — bw^— 5/ifl — mhu -\- ihS
nn,pA^Q Bn'{p± + Q±)-£n{p.
f+l)-nr^',f^+q^N^ 1 ^2.9) p i ) -I- hu;^ — shft — mhw -\-ihS }
where ai = ^ ° \ 02 = \,EQI and fi are the intensity and the frequency of a strong m'n^ rri*ur
EMW (laser radiation), EQ2 and w are the intensity and the frequency of a weak EMW, N^ is the balanced distribution function of phonons, ^1 is the phase difference between two electromagnetic waves and Jk{x) is the Bessel fimction.
117
Nguyen Thi Thanh Nhan, Nguyen Vu Nhan and Nguyen Quang Bau 2.1..1. C'aliMlaiioiisollhf al)s()rpli(>n cdciricienl of a weak EMW in Ihe presence of
laser radiation modulated l>v amplitude in a IXSL The cai ricr current density formula in a DSL takes the form:
, / ) ( ' ) - '--.Yil'< ,---''l'))",.,,5.(') (2.)0)
; / ) * • ^ ^ \ lie /
Because the molion of electrons is confined along the / direction in a DSL. we only consider the in plane \( )y current density vector of electrons ji (f).
The ACl' of a weak l;MW by emifiricd cleelnins in the presence of laser radiation modulated by amplitude in the DSL takes the simple form 17]:
n = , '',.„-(7i(/)Fn. Mil a;A (2.11) Because the strong EMW (laser radiation) is modulated by amplitude, according to
section (2.1.1), it is expressed by Eq. (2.2). According to tbe hypothesis, due to <SV.\ <£ IJ,
9 -
then m a small amount of lime there are about a few periods F = —, we can presume that {Aill -\- An) is changeless. Therefore, we let / get a certain specific value r in such a small amount of time. Then, we have:
£^01 = \fFf + F^ + 2/''i/-2<'os(A(?--i-A.O =r(>;(,s/,^-, = a-\-a = const. (2.12) From the Eqs. (2.9), (2.10). (2.11) and (2.12). we established the ACF of a weak EMW in the presence of laser radiation modulated by amplitude in DSL:
"2 (Ho.i - -tfii.-i) + 55 (Go.i - Go,-i) 1-^ (H-i,, - H_i,_, + ff,,
"16 ' ° - " " ' ' - • ' - ' + "^i-' - ^ i - i ) + ^ C^-^.! - G-2,-i + Gi,, - G2,-i)}
(2.13)
Ability to increase a weak electromagnetic wave by cimjiiied electrons in doped superlattice where:
n 1 •/4
7^)^%(l:^)k^<^^-^)-
. (" + 0 -^.' = /'-V ("' + ^) --v., = ^;r^.//„,,. = / \I..A^.)?dq-.
Fo, = ^ F j ^ -F /-V + 2 F | /-a c()H(Anr + A a )
^ eFpi __ rF,,,
G,7n = hup{n' — n)-\- hwo - shil - ruhuj. with s = - 2 . - 1 , 0 , l , 2 ; m =
•) is tbe angle between the two vectors FQI and F02. F and F^ are the intensities of two laser radiations that creates laser radiation modulated by amplitude (with the intensity Foi and the frequency Cl).
Equation (2.13) is the expression of the ACF of a weak EMW in the presence of external laser radiation modulated by amplitude in a DSL. From the expression of the ACF of a weak EMW, we see that ACF of a weak EMW is independent of E02 and dependent only on EQI , Q, w, T. d. N^. njy-
When AQ = 0, the above results will come back the case of absorption of a weak EMW in the presence of non-modulated laser radiation.
From epression (2.13), when we set FQI = 0, we will receive an expression of the ACF of a weak EMW in the absence of laser radiation in a DSL that has been investigated in [4] but by using the Kubo-Mori method.
Expression (2.13) is similar to the expression of the ACF of a weak EMW in the presence of laser radiation in a quantum well that has been investigated in [10], but different from [10] in wave function, the energy spectrum and the electron form factor
^n,n'iQz), in addition to the laser radiation which in this case is modulated by amplitude.
Here it is very difficult to calculate//n,n' = / 1/Tiy(q'z)|^rfg2 by hand as in [10], so we have to program the calculation to be done on a computer.
119
Nguyen Thi Thanh Nhan, Nguyen Vu Nhan and Nguyen Quang Bau 2.2. Nunieiical results and discussion
In order to clarily the mechanism for the absorption of a weak EMW in a DSL in llic presence of laser radiation modulated by amplitude, in this section we will evaluate, plot and discuss the cxpicssion of the ACF for the case of a DSL with equal thickness ,/„ (/,, of the n-dopcd and p-doped layers, equal and constant doped concentration HD = ",i: n-GaAs/p-GaAs. The parameters used in the calculations are as follows [4, 7]:
\,^, = ]().9, \ii ^ 12.9, ;» - 0.067m(i, "111 being the mass of free electron, d = 80 nm, nil = 10'"nr', fiw„ ^ •.W2r,ii„V.j = I.
Tm.wiiu'e Tl'., '
Figure I. The dependence of a on T Figure 2. The dependence ofn on Sil Figure 1 describes the dependence of Q on temperature T, with .V,^ - 15, TID = 10"m-^ n, = 3.10'^ Hz. n2 gets five different values: 2.6.10" Hz, 3.10" Hz, 3.4.10"
Hz, 3.8.10" Hz, 4.10" Hz; ii, =- f,Q2 = f, F, = lO.lOT/m. F2 = 15.10«V/m, u) = 10"//c. The five different values of ^2 correspond to the five different values of Afi: 0.4.10" Hz, 0 Hz, -0.4.10" Hz, -0.8.10" Hz, -10" Hz. Figure 1 shows that when the temperature T of the system rises from 30K to 400K, its ACF decreases, and Uien gradually increases to 0. From Figure 1 we also see that when T gets a value which is under 80K, the ACF of a weak EMW in the presence of non-modulated laser radiation is greater than one in the presence of laser radiation modulated by amplitude. This means that the absorption of a weak EMW is reduced when a strong EMW is modulated by amplitude and, when T gets a value which is over lOOK, the ACF of a weak EMW gets values greater than one for the case of a non-modulated strong EMW. In addition, the ACF also gets negative values, i.e. the ACF of a weak EMW becomes increased coefficient of a weak EMW. So, when T gets a value which is over lOOK, the ability to increase a weak EMW in the presence of laser radiation modulated by amplitude is decreased in comparison with that in the presence of non-modulated laser radiation; and when T gets the value about from 80K to lOOK, the ACF of a weak EMW in the presence of laser radiation modulated by amplitude can be greater or smaller than one in the presence of non-modulated laser radiation.
Ability to increa.se a weak electromagnetic wave hy confined electrons in doped superiattices...
Figure 2 describes the dependence of Q on An (|Ar!| is the modulated frequency), and also with the above conditions and seven different values of T. From Figure 2 we see that the curves can have a maximum or a minimum in the investigative interval.
Both Figures 1 and 2 show that in the high temperature region, the ACF is almost independent of Afi, i.e. the amplitude modulation of laser radiation hardly affects the ability to increase a weak EMW in the presence of laser radiation.
\y
Figure 3. The dependence of<\ on il Figure 4. The dependence ofn on ^ Figure 3 describes the dependence of a on H (reduced frequency), with Nd ~ 15, no - 1023m-\ Qi = 2.5.10^^Hz -E- 8 WHz, AQ = 0.5 IQ^'^Hz , oi = ^ , as = - , Fl = S.lO^V/m, F2 = lO.lO^K/m , w = lO'^/fz and five different values of T The curves in this figure can have a minimum or no minimum in the investigative interval.
Figure 4 describes the dependence of a on the frequency ui of the weak EMW, with Nj = 15, TiD = 102^m"^ fii = 3.10^^Hz, ai = - , a2 = 7- . and five different values of
3 6
Jia corresponding to the five different values of Afi, w ^ 0.5.10^^Hz -E- 20.10^^Hz. This figure includes two subplots: the first subplot with Fi = 4.10^V/m, F2 = 8.10^V/m, T = 30A*, the second subplot with Fj = lO.lOV/m, F^ = l5.lO^V/m, T = 90K.
From Figure 4 we see that the curves in the first subplot have a maximum where OJ = CJQ while the curves in the second subplot have no maximum and can have a minimum or no minimum in the investigative interval.
Figure 5 shows the ACF as a function of the number of DSL period Nd, with Dl = 3.10^^7?^, no = 10'^m-^ ai = ^,03 = ? , Fj = 6.10^V/m, F2 = 10.10«V/m,
3 0
T= 10QK,uj = lO^^/f^ and five different values of ^2 corresponding to the five different values of AQ. From this figure, we see that when Afi = OHz, the ACF gets a negative value and smaller than one for the cases of Afi = 0.4.10"H2, -O.S.IO^^H^, -lO^^Hz, and greater than one for the case of AQ = -O.A.lO^^Hz. So, the ability to increase a weak EMW is enhanced when laser radiation is modulated by amplitude with An = -0.4.10^^7/2; and when laser radiation is modulated by amplitude with AQ = 121
Nguyen Thi Thanh Nhan, Nguyen Vu Nhan and Nguyen Quang Bau
0 . 4 . 1 0 " / / z , - 0 . 8 1 0 ' ' ' / / 2 , - 1 0 ' ' ' / / j , the ability to increase a weak E M W is not enhanced in comparison with the case of non-modulated laser radiation (Afi = OHz), but is dccrea.scd.
- lO-OHl lQ.fl41Q"Hl
— ttQ-flflin"ni ta' 10" Hi
1 f
- ^
1 K T, •" r>
^ ^ _ -
W « 4 t 0 " K I
— — » n - - o « i o " w no- 10" w
" ^ \ "
Figure 5. The dependence ofn on N.i Figure 6. The dependence ofn on np Figure 6 describes the dependence of Q on the Nj, with Nj = 15. fii = S.W^Hz , a^ = l^a.^ 1^ /Tj = 6 . I 0 ' ' W n i , F 2 = lO.lO'nym . T = lOOAT, u = IQ^^Hz
3 6
and five different values of Q2 corresponding to the five different values of AH. From this figure, we also see that the ability to increase a weak E M W Is enhanced when laser radiation is modulated by amplitude with AQ — - 0 . 4 . 1 0 ' ^ / / : : , and when laser radiation is modulated by amplitude with Afi = 0 4 WHlz, -O.S.lO^^Hz, - 1 0 ' ^ / / 2 , the ability to increase a weak EMW is not enhanced in comparison with the case of non-modulated laser radiation (Afi = QHz), but is decreased. From dependence of the ACF of a weak EMW on the doped concentration no of DSL, if we measure the value of the ACF of a weak EMW by experimental means, we can infer a value of doped concentration no (of DSL) which corresponds to that value of the ACF.
These figures show that under the infiuence of laser radiation, tbe ACF of a weak EMW in a DSL can get negative values, and it means that the ACF becomes an increased coefficient. So, due to the presence of a strong EMW, in some condifions the weak EMW is increased. This is different from a situation In which there is an absence of laser radiation. In addition, if a strong EMW is modulated by amplitude, in some conditions the ability to increase a weak EMW can be enhanced.
3. Conclusion
In this paper, we investigated the ability to increase a weak E M W by confined electrons in a DSL in the presence of laser radiafion modulated amplitude. We obtained an analytical expression of the ACF of a weak EMW in the DSL. This expression shows that the ACF of a weak EMW is independent of E02 and dependent otily on Eoi, fi, w,
Ability to increase a weak electromagnetic wave by confined electrons in doped superiattices...
T,d, Ni,nD. We numerically calculated and graphed the ACF for n-GaAs/p-GaAs DSL to clarify. These results showed that due lo the presence of a slrong EMW modulated by amplitude, in some conditions, the weak EMW is increased and this ability to increase a weak EMW can also be enhanced in some conditions in comparison with the case of non-modulated laser radiation.
Acknowlettgements. This research which has been completed thanks to the financial support of the Program of Basic Research in Natural Science of VNU and NAFOSTED of Vietnam (Number 103.01 -2011.18), TN-12-07.
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