In this paper, we present the results of theoretical studies on the opticalelectric-magnetic properties of quantum wires in one-dirction semiconductors in two cases in the presence or absence of external magnetic fields.
Trang 1RESEARCH OF QUANTUM THEORY ON OF QUANTUM THEORY ON OF QUANTUM THEORY ON THE KINETIC EFFECTS IN QUANTUM WIRES UNDER
THE KINETIC EFFECTS IN QUANTUM WIRES UNDER
THE INFLUENCE OF AN EXTERNAL FIELD THE INFLUENCE OF AN EXTERNAL FIELD
Nguyen Thu Huong 1 , Nguyen Vu Nhan 2
1 Faculty of Basic Sciences, Air Force - Defense Force Academy 2
Science-Technology Center, Hanoi Metropolitan University
Abstract:
Abstract: In this paper, we present the results of theoretical studies on the
optical-electric-magnetic properties of quantum wires in one-dirction semiconductors in two cases in the presence or absence of external magnetic fields Physical problems are studied on the basis of quantum kinetic equations in two main directions: The theory of nonlinear absorption of electromagnetic waves by confined electrons in quantum wires (first problem) and quantum theory on a acoustoelectric (AE) field and a acoustomagnetoelectric (AME) field in a quantum wire under the influence of an external magnetic field (second problem) Analytical expressions for the absorption coefficient of electromagnetic wave in the first problem and the analytical expression of the AE field and AME field were obtained in the second problem The numerical theoretical results for the GaAs / GaAlAs quantum wires are cylindrical and rectangular
in size with different types of potential Calculated results are compared to the corresponding results, but in bulk semiconductors, quantum wells and semiconductor superlattices show differences such as the appearance of new resonance peaks in the absorption spectrum as well as in the graph of the AE field and AME field
Keywords:
Keywords: Quantum wires, one-direction semiconductor, 1D semiconductor
Email: huong146314@gmail.com
Received 15 November 2018
Accepted for publication 15 December 2018
1 INTRODUCTION
The advances in solid-state physics in recent decades have been characterized by the shift of major study subjects from three-dimensional structural crystals to low-dimensional structures In a low-dimensional structure, the free movement of conductor particles is limited to two-dimensional, one-dimensional or non-dimensional Most of the optical properties of electricity in these systems are changing Some new features, called the size effect appears [1-4] The transition from three-dimensional electronics to two-dimensional electronics (quantum loops, superlattices .) or one-dimensional (quantum wires .) has
Trang 2significantly changed both quantitatively and quantitatively the physical properties of which are the optical properties of the material Structural studies as well as physical phenomena in low dimensional materials show that the structure has dramatically changed many properties of the material and added many new superior properties not found in 3D electronics New materials with these semiconductor structures have helped to create components and devices based on completely new principles and revolutionary modern technologies in general engineering and engineering in the field of optics-electronics in particular
In this section we present the results of theoretical studies on some optical-electronic-electron properties in low-dimensional semiconductors such as quantum wells, quantum wires in two cases: in the presence or absence of external magnetic fields The physical problems mentioned in this report in two main directions are: The nonlinear electromagnetic absorption by confined electrons in low-dimensional semiconductors and
theory of AE field and AME field in low-dimensional semiconductors under the influence
of external wave field The results presented below are aggregate results from scientific works obtained in recent years
2 CONTENTS
2.1 Theory of absorbing weak electromagnetic waves in quantum wires in the presence of strong electromagnetic waves
The study of physical problems of the weak electromagnetic absorption coefficient in
low-dimensional semiconductors in the presence of magnetic field and no magnetic field has been studied by many scientists and our group in recent decades, and many scientifically significant results obtained Recently, we have expanded the problem by considering a second electromagnetic wave with strong amplitude and high frequency Below, we present two typical research results in this direction in the rectangular and rectangular quantum wiring of the one-dimensional semiconductor
2.1.1 The absorption of weak electromagnetic waves in cylindrical quantum wires in the presence of strong electromagnetic waves
Using the quantum-kinetic equation method, we construct quantum-kinetic equations for electrons in cylindrical quantum wires with an infinite potential From there, perform calculus and get the expressions of the current density vector and weak electromagnetic absorption coefficient by confined electrons in cylindrical quantum wires in the presence
of strong electromagnetic waves In calculations, we assumed that the weak electromagnetic wave is straight and linearly polarized, and satisfying high frequency
Trang 3conditions The energy spectrum of electrons in quantum wires is quantized according to the discontinuity levels assumed in the z axis direction As a result, we obtained the expressions for the z component of the line density vector (1) and the coefficient of weak electromagnetic wave absorption in presence of strong electromagnetic wave (2)
with
Formulas for A 1 , A 2, A 3 received due to the contribution of the absorption process and the radiation of a photon of weak electromagnetic waves
formulas for B 1, B 2 received due to the contribution in the absorption and radiation of a
photon of weak electromagnetic waves and strong electromagnetic waves B 3 received by contributions in the absorption and emission of a photon of weak electromagnetic waves
and two photons of strong electromagnetic waves Expressions defined F s, m , L s, m , M s, m is written in the following format:
The analytical expression (3) gives a weak electromagnetic absorption coefficient when there is a strong electromagnetic field in the cylindrical quantum wire with an infinite potential as a function, nonlinear dependence on the strong electromagnetic wave amplitude, and frequency of two waves as well as strong dependence on temperature T of the system and radius R of quantum wires These results are different in comparison with bulk semiconductors and quantum wells The results show that: In a quantum wells, the
1
>>
ωτ
Trang 4state of an electron is characterized by its quantum number n and wave vector , and in
the quantum wires this dependency is much more complex For example, in expression (3) there are four sums with four running indices, while in quantum wells there are only two sums with two running indices Differences are also expressed in numerical and in the graph of the absorption spectrum
The absorption coefficient (3) is calculated for the quantum wires GaAs / GaAsAl and
graphing The graph in Fig 1 is the dependence of the absorption coefficient α on the
temperature with the strong electromagnetic wave amplitude E 01 is different correspond to two frequency values of Ω1= 3.1013 Hz and Ω2= 1013 Hz with wire radius R =30 nm Found that, when the temperature T increasing from 20K to 400K, the absorption curve has
a maximum and a minimum In Figure 2 is the dependence of the absorption coefficient α
on the frequency of strong electromagnetic wave Ω1 with three different temperature values corresponding to Ω2= 1013 Hz, R =30 nm and E 01= 14.105V / m Absorption spectra show that in the investigated domain there is only one maximum and no minimum
Figure 1:
the temperature T
Figure 2:
Figure 3:
Figure 4:
The curve in Figure 3 shows the absorption spectrum α depending on the weak electromagnetic wave frequency Ω2 with three different temperature values corresponding
to Ω1= 3.1013 Hz, R =30 nm and E 01= 15.106 V / m Absorption spectra show a peak at
Trang 5Ω2= ω0 and another peak is lower when Ω2 ≠ ω0 Because optical phonons are strong oscillators, when the frequency of weak electromagnetic waves Ω2 equal ω0 of the phonon optical, the resonance peaks will appear and the absorption coefficient of the crystal is best The difference with the semiconductor semiconductor is the appearance of the second largest peak, which occurs due to a transition between the mini-regions of the electron in the quantum wires The graph in Figure 4 represents the dependence of the absorption
coefficient α on the amplitude of strong electromagnetic wave E 01 with Ω1= 6.1013Hz, Ω2= 3.1013Hz and R= 30nm The absorption curve has a resonant region and this area moves to
the right as the temperature decreases
The graph in Figure 5 shows the absorption spectrum of α on the amplitude of strong
electromagnetic wave E 01 with a different wire radius corresponding to Ω1= 6.1013Hz, Ω2= 3.1013Hz and T= 80K The absorption curve has a resonant region and this area moves to
the right in the direction of the radius of the wire In Figure 6, the absorption curve of the absorption coefficient α depends on the wire radius with three values of temperature corresponding to Ω1= 3.1013Hz, Ω2= 7.1013Hz, E 01= 15.106V / m Notice that apart from the main resonant peak, there are more sub-resonant on the left side in the direction of the reduction of the wire radius Obviously, the quantum size effect of the quantum wire has resulted in these sub-resonant
Figure 5:
Figure 5: The dependence of the absorption
Figure 6:
Figure 6: The dependence of the absorption
coefficient α on the radius of quantum wire
In summary, the numerical results show that, the absorption coefficient of weak electromagnetic wave in the presence of strong electromagnetic waves in the cylindrical
quantum wire GaAs / GaAsAl with infinite potential, complex dependencies on Ω1, Ω2, E 01,
R and T The absorption curves found several resonant peaks Under the influence of strong
electromagnetic waves, the absorption coefficient can be negative This means that under the influence of strong electromagnetic waves with condition is satisfied, the absorption coefficient transforms into a weak electromagnetic wave enhancement factor From (3) if
given E 01= 0 we will get back the weak electromagnetic absorption coefficient in the cylindrical quantum wire in the absence of strong electromagnetic waves The findings are
published in [5]
Trang 62.1.2 The absorption of weak electromagnetic waves in rectangular quantum wires in the presence of strong electromagnetic waves
Similar to the physical problem in part 2.1, based on the quantum-kinetic equation, we construct a quantum-kinetic equation for confined electrons in rectangular quantum wires with an infinite potentials in the presence of weak electromagnetic waves (weak amplitudes, high frequency) and strong electromagnetic waves (strong amplitude, high frequency) Calculations with the assumption that weak electromagnetic waves are linear with frequency satisfying high frequency conditions , the electron energy spectrum in quantum wires is quantized according to the degree of discontinuity along the
z axis From there, calculated the density vector of current and the absorption coefficient
of weak electromagnetic wave by the confined electronic in the rectangular quantum wires
in the presence of strong electromagnetic waves As a result, we obtain analytical
expressions for the absorption coefficients written as:
with
Trang 7The absorption coefficient of weak electromagnetic wavse (4) in the presence of strong electromagnetic waves in the rectangular quantum wire with infinite potential is highly complex dependence and depends only on the amplitude of strong electromagnetic
wave E 01 , the frequency of two electromagnetic waves, the temperature T and the size of the quantum wires (L x , L y) These results are different in comparison to normal semiconductors, quantum wells and cylindrical quantum wires From expression (4) when
given E 01 = 0 we will get back the absorption coefficient of weak electromagnetic waves in
the rectangular quantum wires with an infinite potential in the absence of strong electromagnetic waves
The absorption coefficient (4) is calculated and graphed for the rectangular quantum
wires GaAs / GaAsAl The graph in Figure 1 is the dependence of the absorption coefficient α on the temperature with the different amplitude of the wave E 01 with respect
to Ω1= 3.1013 Hz, Ω2= 1013 Hz and R =30 nm
Figure 7:
Figure 7: The dependence of the absorption
Figure 8: Figure 8: Figure 8: The dependence of the absorption
The graph in Figure 7 shows the dependence of the absorption coefficient of weak electromagnetic wave α on the frequency Ω2 of weak electromagnetic waves with different
temperatures correspond to E 01 and Ω1= 3.1013 Hz, L x = 24nm, L y = 26nm, E 01 =15.106Hz Notice that, on the absorption curve there is a main peak at Ω2= ω0 and several sub peak are smaller when Ω2≠ ω0 It is easy to see that the main peak changes insignificantly when the temperature changes Figure 8 shows the curve representing the absorption spectrum of
the coefficient α on the amplitude of strong electromagnetic wave E 01 with three different temperature values corresponding to Ω1= 6.1013 Hz, Ω2= 3.1013 Hz, L x = 24nm, L y= 26nm
On the absorption spectrum there is a maximum, the position of the peak is shifted to the right when the temperature decreases
Trang 8The graph in Figure 9 shows the dependence of
the absorption coefficient α on the size L x of
quantum wires with three different temperature
values corresponding to Ω1= 3.1013 Hz, Ω2= 7.1013
Hz, L y = 26nm, E 01 =15.106Hz On the absorption
curve, many peaks appear These peaks appear by
quantum size effects when the L x small, the
resonance peak in this area is clearer and sharper
than the remaining peaks
The numerical results also show another
important conclusion that, under the influence
of strong electromagnetic waves and under certain
Figure 9:
Figure 9: The dependence of absorbtion
conditions, the absorption coefficient of electromagnetic waves in the rectangular quantum wire can receive negative value That is, in this case the absorption coefficient transforms into a amplification coefficient of weak electromagnetic wave This is different than in normal semiconductors This difference is due to the quantum wire being one-dimensional semiconductors The findings are published in [6]
2.2 The theory of AE field and AME field in cylindrical quantum wires with infinite potential
Studying the physical properties of low-dimensional semiconductor structures, scientists have paid much attention to the influence of sound wave to the properties of
low-dimensional materials such as AE effects and AME effects The propagation of external
sound waves into the semiconductor increases the transfer of energy and momentum of sound waves to conductive particles in the semiconductor and produces an electro-acoustic effect along the direction of sound waves If the material creates a closed circuit, it creates
an AE current that runs along the direction of the sound wave, if the open circuit produces
an AE field When there is an external magnetic field, in this semiconductor sample there is another effect called a AME effect and to appear a AME field Recently, this problem has
been studied by scientists in both theory and experiment in normal semiconductor, Kane semiconductor, two-dimension semiconductor However, in the one-dimension semiconductors (quantum wires) is left open Therefore, we are interested in studying the
theory of AE effects and AME effects in cylindrical quantum wires with infinite potential
Trang 92.2.1 The density of AE current in the cylindrical quantum wire when there is no external magnetic field
Proceed from the Hamiltonian operator describes the interaction of the sound wave-electron system and scattering between confined wave-electron with the acoustic phonon in the cylindrical quantum wire with infinite potential is written as follows:
(5)
(6)
For calculating the AE currents in a cylindrical quantum wire with infinite potential, it
is necessary to establish quantum-kinetic equations for confined electrons in quantum wires Performing the calculations with the assumption that the momentum recovery time
of electrons is approximately constant, we obtain the analytical expression for the density
of AE current in the cylindrical quantum wire with the infinite potential write in the form:
(7) here: ; The β = 1 /kBT ; εF is Fermi energy; Kn-the
Bessel function of type two From equation (7), found the nonlinear dependence of the AE
current on the temperature T, wave numbers, frequencies of external sound wave, and radius of quantum wires These results are completely different from the results in normal semiconductor and in the quantum wells of the same problem The findings are published
in [7]
2.2.2 The AME field in the rectangular quantum wires with infinite potential in the presence of external quantum field
Assume that external sound waves have frequencies ω q propagate along a cylindrical quantum wire with an infinite potential, considering the practical case from the
∑
∑
∑
∑
− +
+
+ +
=
+ + +
+
− +
+ +
q , ' n ,
' n l n q
k , ' n ,
n k n ' , p z k n ' , p ' z k k
' n l n z
p ,
n n , p z n , p z n , p z
) t i exp(
b a a U C b
b
b b a a C I a
a H
ε
−
k L
F
2
5
n,l,n',l'
s q 3 3
2m
e K ( ) 3K ( ) 3K ( ) K ( )
ξ
ξ
β
+
−
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F
2
6
n,l,n',l' s
e K ( ) 3K ( ) 3K ( ) K ( ) e K ( ) 3K ( ) 3K ( ) K ( )
βε
ℏ
±
−
=
l n ' l ' n
m R
) B B (
2 2 2
2 2
ℏ
ℏ
2
k
βω ξ
χ±= ±±ℏ
Trang 10experimental point at low temperature, when ωq/η=νS|q| /η << 1 and qd >> 1 (η is the frequency of oscillation of the electron, vS sound velocity, q is the number of sound waves outside and d is the average free path of electrons We consider external sound waves as
phonon streams Derivative of Hamiltonian describes the interaction between electrons and external phonon and electron scattering on the acoustic phonon in cylindrical quantum wire with the infinite potential in the presence of an external magnetic field in the second quantization writed in the form:
,
(8)
with
(9) here ; is the position of the electron in the cyclotron orbit, (x) is Laguerre conjugate polynomial Performing the calculations, we obtain the expression for
the density of AME currents in the cylindrical quantum wire with an infinite potential when
there is an external magnetic field is written in the form:
(10)
and obtained the general expression for the AME field in the cylindrical quantum wire with
infinite potential when there is an external magnetic field:
(11)
∑
∑
∑
∑
− +
+
+ +
=
+ + +
+
− +
+ +
q , ' n ,
' n l n q
k , N , ' n , N ,
' N N k ' n l n z
p , N ,
B z p , N , n
) t i exp(
b a a U C b
b
b b a a ) u ( J C I a
a H
ε
= ∞
∞
−
⊥
⊥
⊥ a ( p k )) e r a p ) dr r
)
u
(
z c
* ' N , ' n '
N
N
2
ψ
'
2
min( ', )
−
−
Sgn N N
N
2 /
2
⊥
=a q
2
ℏ
e m
,
=
∑
N
E
Y )]
sin (cos
A ) cos ( sin
[
ϕ ϕ
2 2
2 (1 sin )[ ci ( x ) sin( x ) si ( x ) cos( x )]
] cos Y )) x ( si ) x ( ci [(
sin ) sin ( )]
x sin(
) x ( si ) x cos(
) x (
ci
[
)) x ( sin ) x ( si ) x ( cos ) x ( ci ( cos sin ) sin (
Y
4 2
2
3 [ ci ( x ) si ( x )] 2cos Y 2M
4 [ ci ( x ) cos( x ) si ( x ) sin( x )]
)]
x ( cos ) x ( )[sin x ( si ) x ( ci ) x cos(
) x sin(
)]
x ( si ) x (
ci
[