THE EFFECT OF OPTICAL PHONONS ON THE ELECTRONIC STATES IN TWO-WELL RESONANT TUNNELING STRUCTURE DRIVEN BY ELECTRIC FIELD AT FINITE TEMPERATURE
Ju. SETI, М. ТKACH, М. PAN’KIV, О. VOITSEKHIVSKA Chernivtsi National University, 2 Chernivtsi, Kotsubynsky Str., Chernivtsi 58021, Ukraine, E-mail:
[email protected] Received November 12, 2015
The electron-phonon interaction in two-well resonant tunneling nanostructure driven by constant electric field at finite temperature is studied within the effective mass approximation for the electron and dielectric continuum model for the confined optical and interface phonons using the approach of temperature Green’s functions. The effect of electron-phonon interaction on electronic states and radiation band of two-well cascade of injectorless quantum cascade laser is observed. It is established that the frequency of laser generation weakly depends on the temperature while the width of radiation band increases. Key words: electron-phonon interaction, Green’s function, injectorless quantum cascade laser.
1. INTRODUCTION
Quantum cascade lasers (QCL) have been investigated and improved for over two decades. Their operating frequencies occupy the whole infrared and terahertz range of electromagnetic waves covering the transparency windows of the atmosphere and radiation frequencies of molecules. QCLs are still attractive both from scientific and practical point of view due to their perspectives in medicine, military, communication devices, environmental monitoring and so on. It is well known that the first QCLs [1, 2] were operating at low (cryogenic) temperatures, when the electron-phonon interaction almost does not influence on the characteristics of electronic states. Thus, the main attention was paid to study and improve the operating parameters of QCL varying their physical parameters and geometrical design. Rom. Journ. Phys., Vol. 61, Nos. 5–6, P. 980–991, Bucharest, 2016
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The temperature range of modern QCLs [3–5] is enlarged till the room ones. As far as the phonon occupation numbers are sensitive to the temperature, one can expect that electron-phonon interaction should renormalize the electron spectrum and, thus, effect on the characteristics of QCL radiation band. Now then, in the majority of papers [6–10], the electron-phonon interaction in resonant tunneling structures (RTS), being the basic functional elements of QCLs, was studied at finite temperatures. The theory was developed using the approximation of the effective mass for the electrons and dielectric continuum model for the confined optical and interface phonons [11, 12]. The electron-phonon Hamiltonian was obtained in the representation of phonon occupation numbers and in coordinate one over the electron variables. Using it, the probabilities of quantum transitions between electronic states were obtained within the Fermi golden rule. However, using this approach, it is impossible to calculate the shifts and decay rates of electronic states arising due to the electron-phonon interaction. In this study, the electron-phonon Hamiltonian is obtained in the representation of second quantization over all variables for the two-well RTS driven by constant electric field at finite temperature. The energy shifts and decay rates of the three operating electronic states in two-well cascade of injectorless QCL are calculated using the method of temperature Green’s functions. The effects of different mechanisms of electron-phonon interaction on the spectral parameters of electronic states are studied as functions of the intensity of electric field and temperature. 2. ELECTRON-PHONON INTERACTION IN TWO-WELL RTS DRIVEN BY ELECTRIC FIELD
The plane two-well nanostructure in constant electric field with intensity F (Fig. 1) is studied in Cartesian coordinate system with OZ axis perpendicular to the planes of RTS.
Fig. 1 – Potential energy profile for the two-well RTS driven by constant electric field.
The theory of electron-phonon interaction is developed using the model of closed RTS with infinitely wide outer potentials. In this model, the electron is
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characterized by the complete system of orthonormal wave functions, which allow us to use the method of temperature Green’s functions in order to study the spectral characteristics of electronic states renormalized due to the interaction with phonons. The model of effective mass is used for the electron, taking into account the non parabolic dispersion law for the conductive band [13] and rectangular potential profile, linearly decaying in each region of RTS due to the constant electric field directed along OZ axis: UE U, reg. j 0 ), reg. j 0,2,4 m b (1 E gb m ( z, E ) U F (z) U j eFz, reg. j 1, 2, 3 (1) E m w (1 E ), reg. j 1,3 U eFd, reg. j 4 gw
Here Е is the electron energy; mw, mb are the electron effective masses in wells and barriers of nanostructure without non parabolicity; Egw, Egb are the widths of energy gaps for the semiconductor crystals of wells and barriers; U2 = U is the potential barrier height; U1 = U3 = 0; d is the linear size of RTS (Fig. 1). Solving the stationary Schrödinger equation for the electron
e ik nk ( r ) n (z) , S
( r zn z )
(2)
where k and are its quasi-momentum and radius-vector in the plane xOy, S is the square of main region in this plane and n z is the unitary vector along OZ axis, we get the complete electron energy E nk E n
2k 2
,
(3)
2m n
as a sum of the energy of longitudinal movement (En) and kinetic energy in the plane xOy with effective mass correlated over RTS, [14] m n1
n z
2
/ mz, E dz .
(4)
Now then, the one-dimensional equation for the wave function Ψn(z) is obtained. Its solutions are known for the each region (j) of RTS
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The effect of optical phonons on the electronic states
n( j) z A (nj) en z , n z n( j) z A (nj) Ai((nj) (z)) B(nj) Bi((nj) (z)), ( j) ( j) (nj ) z , n z A n e ( j)
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j0 j 1,2,3
(5)
j4
where
(n0) 1 2m b (E n )(U E n ) ; (n4) 1 2m b (E n )(U E n V) ; V eFd ; (2m w (E n ) V d 2 2 )1/ 3 E n / V z / d , (nj) (z) 2 2 1/ 3 (2m b (E n ) V d ) [(E n U) / V z / d],
j 1, 3; j 2;
(6)
Ai(ξ), Bi(ξ) are the Airy functions of the first and second kind, respectively. The fitting conditions for the wave function and its density of current at all RTS interfaces and normality condition, [15], definitely fix the unknown coefficients A (nj) , B (nj) and En energy and, thus, the complete orthonormal set of wave functions ( r ) and electron energy spectrum E . Using the quantized nk
nk
wave function ( r ) ( r )a nk n, k
nk
(7)
the electron Hamiltonian is obtained in the representation of second quantization H e E a a nk , nk
nk nk
(8)
where a nk , a are the annihilation and creation Fermi operators. nk The spectra and polarization potentials of confined optical (L) and interface (І) phonons in the framework of dielectric continuum model, [11, 12], are defined from the equation
j ()2( r ) 0 ,
2 2Lj j () j 2 . 2 Tj
(9)
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Here εj∞ is the high-frequency dielectric constant, ωLj, ωTj are the frequencies of longitudinal and transversal vibrations of the bulk material creating j-th layer of nano-structure. The energy spectrum (ΩLj) and potential of polarization field (ФLj) for L phonons are obtained from eq. (9) at the condition 2 Lj ( r ) 0 , εj(ω)=0. Evidently, the energies of confined phonons are dispersionless and equal to the energies of longitudinal phonons of the respective bulk materials (ΩLj = ħωLj). The polarization potential of confined phonons for the j-th layer of RTS is written as an expansion into the two-dimensional Fourier range L (, z) Lj (q, z)e iq
(10)
jq
with coefficients (ФLjλ) containing all possible harmonics (λ = 1, 2, . . .∞). Here q is the two-dimensional quasi-momentum. The spectrum and potential of І-phonons polarization field is found from eq.(9) at the condition 2 I ( r ) 0 , εj(ω) ≠ 0, herein I (, z) C(q) j (q, z) eiq .
(11)
jq
From the fitting conditions for the polarization potential of I-phonons and induction at all RTS interfaces [11, 12], the system of equations is obtained for the functions j (q, z) jeqz jeqz : j (q, z j ) j1 (q, z j );
j ()
j (q, z) z
j1 () z z j
j1 (q, z) z
(12) z z j
Its non trivial solution, together with the condition that phonon field vanishes at z , gives the dispersion equation for the energies ( Isq ) of all modes (s) of I-phonons. Quantizing the potential of confined (ФL) and interface (ФI) phonons [12] we obtain the phonons Hamiltonian in the representation of second quantization 4 H ph H L H I
Lj(bLjq bLjq 1 j 0 q
2)
Isq (bIsq bIsq 1 sq
2) , (13)
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where b Ljq ( b Ljq ), b Isq ( b Isq ) are annihilation (creation) Bose operators of Land I-phonons. The electron-phonon Hamiltonian H eph e [ L I ] , after normalization and quantization of both phonon fields and transition from the coordinate representation to the second quantization one, finally is obtained in the form
Heph HeL HeI f nLj1n (, q)a n k q a nk (b Ljq b Lj q ) 1
jq n1nk
(14) f nI1n (s, q)a n k q a nk (b Isq b Isq ). 1
sq n1nk
Here ( ) 2 Lj 2 f n n (, q) (1) 1
f nI n (s, q) 1
() 8e j 2 2 2 2 S[ q (z j z j1 ) ] Lj
1 / 2
2
z z j1 ( j)* ( j) , dz ( z ) ( z ) sin n n1 z j z j1 z j1 zj
(15)
4 () 2qz 2qz j { [2j (s, q)(e j e j1 ) 2 4e j 0
qS
sq
2j (s, q)(e
2qz j
e
2qz j1
4
)]}1 / 2
zj
, ( j)*
dz n1
j 0 z j1
(16)
(z)n( j) (z)(q, z) .
are the binding functions for the electron in the state n1 , n with L- and I-phonons; θ(λ) = 1 at λ = 1, 3, 5... and θ(λ) = 0 at λ = 2, 4, 6... . The obtained Hamiltonian of electron-phonon system written in the representation of second quantization over all variables: H H e H L H I H eph , allows us to calculate the Fourier-image of electron Green’s functions at finite temperatures according to the rules of Feynman-Pines diagram technique [16]. At small concentrations of electrons and their weak binding with phonons, the Fourier-image of electron Green’s function is found from Dyson equation
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G n (k, ) [ E nk Mn (, k)] 1
7
(17)
with the mass operator M n (, k ) calculated in one-phonon approximation
M n (, k )
* (p, q)Fn1n (p, q) Fnn 1
n1pq
1 pq pq , ( 0). E n1 (k q) pq i E n1 (k q) pq i
(18)
The first term of mass operator describes the processes of electron-phonon interaction accompanied by the creation of phonons with average occupation
1)1 and second one – by the annihilation. Here we number pq (e pq introduced the generalized index for the phonon modes (р) which at p = j,λ numerates all L-phonons modes and at p = s – all I-phonon modes. Considering that the electrons move perpendicularly to the planes of RTS, with the energy close to the energy of the bottom of respective electronic band, we put k 0 in (18) and neglect the frequency dependence of mass operator in the vicinity of electron energies Еn taking into account a weak electron-phonon binding. Now, the real and imaginary parts of mass operator determine the shift ( n Re M n ( E n ) ) and decay rate ( n 2 Im Mn ( En ) ) of the n-th electronic state. / kT
3. ANALYSIS OF ELECTRONIC STATES RENORMALIZED DUE TO THE INTERACTION WITH OPTICAL PHONONS IN TWO-WELL CASCADE OF INJECTORLESS QCL
Using the developed theory, we investigate the effect of L- and I-phonons on the spectral characteristics of electronic states and radiation band of two-well cascade of injectorless QCL, observed in [17]. Geometrical parameters of RTS with GaAs wells and Al0.15Ga0.85As barriers are taken the same as in the above cited experiment [17]: а1=7.1 nm, а2 = 16.7 nm, b = 3.1 nm. Physical parameters are the following: ε∞w=10.89, ε∞b = 10.48, ħωLw = 36.25 meV, ħωLb = 35.31 meV, ħωTw = = 33.29 meV, ħωTb = 33.17 meV, mw = 0.067 me, mb = 0.080 me, E Tgw 0 K = = 1520 meV E Tgb 0 K = 1626.5 meV, E Tgw121 K = 1481.3 meV, E Tgb121K = 1589 meV, UT=0 K = 130 meV, UT=121 K = 125 meV. According to the general idea of experiment, [17], the QCL is functioning when the electrons get from the state |3 > into the state |2 > radiating the
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electromagnetic wave with the frequency ω32 = 4.6 ТHz and, further, perform the radiationless phonon-assisted quantum transition from the state |2 > into the ground one |1 >, with the difference of energies corresponding to the energy of optical phonon ( ω21 ≈ 36 meV). The coordinated operation of QCL cascades is provided by the applied constant electric field with the intensity F = 17 kV/cm. The calculated energy spectra of L- and I-phonons are presented in Fig. 2. It is clear that the energies of L-phonons are dispersionless and eight modes of I-phonons are characterized by weak dispersion.
Fig. 2 – The energies of L- and І-phonons as functions of quasi-momentum q, ag is the lattice constant of GaAs well.
In order to investigate the effect of electron-phonon interaction on the renormalized parameters of electronic states in two-well RTS, we calculated the electron spectrum (En), complete shift of electron energy levels ( n Ln In ), their decay rates ( n nL nI ) and the partial contributions ( Ln , In , nL , nI ) caused by the interaction of electrons with L- and І-phonons, respectively. In Fig. 3 all these parameters are presented as functions of the temperature (Т) at the fixed intensity of electric field F = 17 kV/cm (Fig. 3а) and of the intensity at the fixed temperature Т = 121 К (Fig. 3b). From Fig. 3а it is clear that the temperature, varying from cryogenic (Т ≈ ≈ 0 К) till room one (Т = 300 К), almost does not effect on the electron energy spectrum. The small shift of Еn into the low-energy region is caused by the fact that the electron effective mass in n-th state depends on energy gaps widths, formula (1), which decrease when the temperature increases.
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The electron-phonon interaction causes the negative shifts (Δn) and decay rates (Гn). The absolute values of the shifts and decay rates of presented three states increase, Fig. 3а. Herein, nL1, 2 nI 1, 2 in the whole range of temperatures while 3L 3I in the range 0 К – 175 К and 3I 3L in the range 175 К – 300 К. As for the shifts, the partial contributions due to the interaction with L- and I-phonons are of the same order. Herein, | L1 || I1 | and | In 2,3 || Ln 2,3 | . (a)
80
E4
80
60
60
40
E3
40
20
E2
20
0
E1
250 T[K]300
150
200
250T[K] 300 1,0
L
3 I
0,8
3
3
0,6
5
10F[kV/cm]15
0
5
10 F[kV/cm]15
0
L
3
-3 L
-2
150
200
L
2
-4
250T[K] 300 0,8
2
I
2
0,6
0
L, I
200
10 F[kV/cm]15
0
5
10 F[kV/cm]15
-1
0,2
-3
-4
0
50
100
T
exp
150
200
0,0 250T[K] 300
-4
Fig. 3
0
5
0,0 20
20 0,08
0,06
2
2
exp
2
-2
I
F
L
0,4
2 2
0,2
5
L
-3
I
0
2, 2 [meV]
-1
150
0,4
3 0,0 250T[K] 300
L, I
100
exp
3
0,2
2 2 [meV]
50
2 2 [meV]
0
T
0,8
L
3 100
3
1,0
0,4
-3
50
3
0,6
I
3
0
1,2
I
-1
20
20 1,4
3
-2
-4
exp
F
L, I
3
0
L, I
200
3,3 [meV]
150
2, 2 [meV]
100
exp
L, I
50
T
3, 3 [meV]
100
L, I
L, I
-2
50
3 3 [meV]
3 3 [meV]
0
-1
L, I
E3 E2
-20
0
0
E4
0
E1
-20
0
(b)
100
En[meV]
En[meV]
100
L
2
0,04 I
2
10 F[kV/cm] 15
0,02
F
exp
0,00 20
The effect of optical phonons on the electronic states (a)
(b)
-1
0,8
L
1
0,6
L
1
-2
1
[meV]
1,0
1
I
1
0
0
-1
1
0
50
100
T
exp
150
200
0,0 250T[K] 300
L
1
0,06 0,04
I
1
-3
0,2
-4
0,10
0,08
L
0,4
-3
20
1
-2
I
1
10 F[kV/cm] 15
5
L, I
250T[K] 300
L, I
200
1, 1
150
L,I
L, I
100
1, 1 [meV]
50
1, 1 [meV]
0
0
989
1, 1 [meV]
10
0,02 -4
0
5
10 F[kV/cm]15
F
exp
0,00 20
Fig. 3 (continued) – Dependences of electron energies (Еn), their shifts (Δn, nL, I ) and decay rates (Гn, nL, I ) on the temperature (Т) at F = 17 kV/cm (a) and on the electric field intensity (F) at T = 121 K (b).
Fig. 3b proves that at the fixed temperature (Т = 121 К), the increasing intensity of electric field (F) brings to the almost linear decay of all energies E n (F) with weak anti-crossing between E 3 and E 2 in the vicinity of F = = 8 kV/cm. The growing intensity differently effects on the shifts and decay rates of electronic states: i) The shift of the ground state (Δ1) almost does not depend on field intensity while the decay rate Г1 increases as function of іt. Herein, in the whole range of F one can see that | L1 || I1 | , 1L 1I . ii) The absolute value of the shift Δ2 increases and | I2 || L2 | in the whole range of F. The decay rate 2 decreases, herein 2I 2L in the range F = = 0 – 8.5 kV/cm and 2L 2I at F = 8.5 kV/cm – 20 kV/cm. iii) The absolute value of the shift Δ3 decreases and | I3 || L3 |, in the whole range of F. The decay rate 3 decreases, herein 3I 3L at F = 0 – 5 kV/cm and 3L 3I at F = 5 kV/cm – 20 kV/cm.
Such behavior of Δn and Гn as functions of F is caused by the peculiarities of contributions performed by intra-band (Δnn) and inter-band ( nn1 n ) interactions producing the complete shifts n nn nn Lnn Inn (Lnn Inn ) and 1 1 1 n1 n
decay rates n nn nn 1 n1 n
L nn
I nn
L (nn 1 n1 n
n1 n
I nn ). 1
It is also clear from
Table 1, where En Δn, Гn, Δnn, and Гnn (n = 1, 2, 3) are calculated for the two-well RTS driven by electric field with intensity F = 17 kV/cm at Т = 121К.
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Table 1 Energies, shifts and decay rates of electronic states n 3 2 1
En [meV] 34.43 15.48 –19.06
Δnn [meV] –1.75 –1.40 –1.75
Δn [meV] –2.13 –3.27 –2.27
Γnn [meV] 0.06 0.04 0.06
Γn [meV] 0.20 0.06 0.11
These results show that although the electron-phonon interaction brings to the decay rates and negative shifts of electronic states, but taking into account that the magnitudes Δ1, Δ2 ,Δ3 differ not so much, this interaction weakly effects on the T 0 K energy of QCL radiation ( E32 = 20.3 meV), making it smaller at 0.2 meV and causing the broadening of radiation band at 0.3 meV when the temperature increases till Т = 121К. 4. CONCLUSIONS
1. The theory of electron-phonon interaction in two-well plane RTS driven by constant electric field at finite temperature is developed within the effective mass and rectangular potentials approximation for the electrons and dielectric continuum model for the confined optical and interface phonons. 2. Using the approach of temperature Green’s functions, the contributions of electron-L- and I-phonons interactions into the complete shifts and decay rates of the three operating states of two-well RTS, being the cascade of injectorless QCL, are calculated depending on the temperature and electric field intensity. It is shown that the main contribution into the shift and decay of the ground electron state is performed by confined optical phonons and into the shift of the excited states – by L I interface phonons. The hierarchy of decay rates ( n 2 , n2 ) is determined by the intensity of electric field and temperature. 3. It is shown that the interaction between electrons and optical phonons in the wide range of temperatures weakly effects on the frequency of laser generation and causes a broadening of radiation band.
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