Effects of an external electric field on one-phonon resonant and electron Raman scattering in a semiconductor quantum wire

In this work, the influence of an external electric field is studied in two cases: one-phonon resonant Raman scattering and one-phonon electron Raman scattering, processes that occur in a semiconductor quantum wire with cylindrical symmetry and finite potential barriers. Where we have considered that the electric field is homogeneous and transversal to the system axis. To carry out this study, we obtain a mathematical expression for the differential cross-section for both Raman processes, where for one-phonon resonant Raman scattering, intra-band and inter-band optical transitions are considered, while for one-phonon electron Raman scattering, only intra-band optical transitions are considered. Therefore, to determine the electronic states, we use a valid model when the electric field is weak with respect to confinement. In the case of the Fröhlich electron–phonon interaction, we use a model in which the oscillation modes are completely confined, a model that was developed within the framework of a macroscopic continuum model. Then, the singularities present in the Raman spectra and the effect of the electric field on their position and intensity are analyzed. Finally, how the electric field affects the electron–phonon interaction and the selection rules for optical transitions in a semiconductor quantum wire with cylindrical symmetry are shown.


Introduction
The properties of low-dimensionality systems, such as the semiconductor quantum wire (QW), allow their use in various applications, such as light sources for optoelectronic devices [1][2][3][4][5].Therefore, both theoretical and experimental characterizations of these systems are important.The presence of an external electric field in nanostructures has several effects on the structure of sub-bands depending on its symmetry.For instance, in the case of multiple quantum wells, the presence of an external electric homogeneous field or an internal electric field in a transversal direction to the barrier changes the energy of electronic states [6,7].It has also been reported [8,9] that polarizability and binding energy of a shallow donor in quantum dots change with variations in the electric field intensity.The use of electric fields has the possibility of allowing us to manipulate the energy of the electronic states of cylindrical systems such as the quantum wire, as well as the selection rules for optical transitions in this type of system.This effect or manipulation on electronic states and optical transitions is also possible using magnetic fields, however the use of the electric field is simpler and allows the use of a wide range of values.
On the other hand, in the case of semiconductor QW, the effects of an electric field are more varied than just changing the energy of electronic states [10][11][12][13], since it breaks the degeneracy of electronic states into 'even' states denoted by + ( ) m n , and 'odd' states denoted by - ( ) m n , ,and also change the selection rules of quantum transitions [10].In addition, the gap between these states increases with increasing electric field intensity, like what happens to a magnetic field.However, the electric field produces a change in the selection rules for optical transitions as well as for transitions due to an electron-phonon interaction this effect being different to the one produced by a magnetic field [14,15].
Raman scattering is a technique whose precision makes it an indispensable tool for the theoretical and experimental characterizations of low-dimensionality structures [16][17][18][19].An example of the use of Raman scattering as an instrument of study can be observed in reference [20], where it has been used core-shell nanostructures for surface-enhanced Raman scattering detection of pesticide residues.One of the conclusions of Raman studies is that confinement affects phonon oscillation modes.This is related to singularities in Raman spectra of semiconductor nanostructures related with longitudinal optical (LO) phonons and with transverse optical (TO) phonons, causing a violation to the selection rules in volumetric systems [21,22].The use of Raman scattering to study a low dimensional system from the theoretical point of view is based on obtaining the differential cross-section, allowing studying the structure of electronic and phononic bands by analyzing the singularities of the emission or excitation spectra.In addition, Raman scattering allows obtaining the selection rules for intra-band or inter-band optical transitions.In the case of semiconductor QW, the presence of an external homogeneous magnetic field breaks up the degeneration of electronic states producing an increment in the number of singularities in Raman spectra.However, in the case of homogeneous magnetic field, the same selection rules for systems where there are no electric field or magnetic field continue to apply [14,15].While, in the case of the external homogeneous electric field, the breakdown of the degeneracy of the electronic states also occurs, to which we must add that the selection rules are also broken, which produces an increase in the number of singularities in the Raman spectra [10].This means that the electric field intensity can change the energy and intensity of the singularities in a relatively controlled way, this being important for the design of sensors or light sources.
Therefore, the objective of this work is to study the effect of an external homogeneous electric field transversal to a semiconductor QW of cylindrical symmetry with finite potential barriers, considering two Raman processes.Firstly, one-phonon resonant Raman scattering (RRS) [14] where optical transitions are interband; and secondly, one-phonon electron Raman scattering (ERS) [15] where optical transitions are intra-band.For this purpose, we have considered a system grown in a GaAs AlAs matrix.In reference [10] the authors study the intra-band and inter-band optical transitions in a quantum wire in the presence of a homogeneous and transversal electric field.Meanwhile, in our work we also study the electron-photon interaction considering that the oscillation modes are completely confined, this developed within the framework of a macroscopic continuum model.Furthermore, we study the Raman spectra for two types of Raman scattering: the onephonon resonant Raman scattering where there are inter-band optical transitions and the one-phonon electron Raman scattering of a phonon where there are only intra-band optical transitions.Then, we will analyze the influence of the electric field in the singularities of Raman spectra and the selection rules.On the other hand, to determine the electronic states, the model proposed in reference [10] is considered.

Model and applied theory
To achieve our objective we must determine the electronic states of a semiconductor QW with cylindrical symmetry, radius r 0 and length L long enough to be considered infinite.If we consider finite potential barriers, we have being V r the confinement potential and m ( ) 1 2 the effective mass of the electron in the conduction band or the effective mass of the hole in the valence band.In addition, the system is under the presence of an external homogeneous electric field applied transversal to the axis of the wire with intensity F, as shown in figure 1.Under these conditions and following the model proposed in reference [10], the energy and the wavefunction corresponding to the electronic states have the form: where e is the electron charge, 2 being á ñ r the average radius value and  r the part of the energy of the electronic states due to confinement.Then, are the even and odd Mathieu periodic functions with order n and eigenvalues n a and n b , respectively.These Mathieu periodic functions have a period π and defined parity where even states are those denoted by = + + ¼ m 0, 1, 2, , while odd states are those denoted by = --¼ m 1, 2, .With J v and K v we have represented the Bessel function of the first kind of order n and the modified Bessel function of the second kind of order n respectively.On the other hand, , and B m n , are constants obtained considering the normalization and Ben Daniel-Duke conditions, then showing the breaking off the degeneration of the electronic states.However, we must remember that these equations correspond to an approximate solution fitting well when the electric field is highly weak, or the system is highly confined [10].
Moreover, to study Raman scattering we must determine the differential cross-section [14,15], with the following form , , , 6 being c the speed of light, h w ( )the medium refraction index respect to the frequency, w l and w s are the frequencies of the incident and scattered photon, respectively with unitary polarization vectors e l and e .
s W is the transition probability per time unit calculated using Fermi's Golden Rule, then where E i and E f are the initial and final energies of the system, respectively.Then, E , i E , f and M fi depend on the type of Raman process being produced: (a) one-phonon resonant Raman scattering and (b) one-phonon electron Raman scattering.

Differential cross-section for a one-phonon resonant Raman scattering process
In the one-phonon RRS, transitions inter-band occur, for this reason, the conduction band is considered empty completely, while the valence band is completely filled.Furthermore, can be described the one-phonon RRS as a three-step process.Then, in the first step, a photon of incident radiation, with frequency w , l creates an electronhole pair (EHP) when an inter-band transition occurs.In the second step, an intra-band transition is produced due to an electron-phonon or hole-phonon interaction with the emission of a longitudinal optical phonon with frequency w .m n , Finally, the EHP recombines due to a new inter-band transition with the emission of a photon of secondary radiation with frequency w , s as shown in figure 2 [14].Therefore, is the confined part of the energy of the EHP, where  g is the gap energy between the valence and conduction bands.Furthermore, m Is the Hamiltonian of the electron-photon interaction [14,15].While Ĥph is the electron-phonon Hamiltonian in the framework of a macroscopic continuum model whose exact form can be consulted in references [23,24], then, where w m n , is the photon frequency and it is determined from the following transcendental equation [23]: e e e e e b b also, Being w LO and w TO are the frequencies of the oscillation modes of the longitudinal and transverse optical phonons, while b LO and b TO are the phenomenological parameters that describe the dispersion of LO-phonons and TO-phonons in the bulk.
To determine the matrix elements in equation (8) we must consider the wavefunction given by equations (2) and (3), and the Hamiltonians given by equations ( 13) and (14), therefore in the case of the electron-phonon interaction we have where m 0 is the electron free mass,

e e h h e e h h e h e h e e h h e h e h
While for the case of the electron-phonon interaction, we have , , , j j j j j j j j j j j j j j j j j j j j Being = j e is the electron contribution and = j his the hole contribution, while ( ) p 0 cv and ( ) p 0 vc correspond with the average value of the momentum between the conduction band (c) and the valence band (v).
To determine the differential cross-section, we first must substitute equations ( 9)-( 11), ( 17) and (21) in equation (8), and consider = E E i f and add for all values of k , z so that where

e e h h e e e e h h e e e e e e h h e e
å å  Finally, if we substitute equations (8), ( 9) and ( 24) in equations ( 6) and (7) considering that the delta function can be substituted by a Lorentzian, i.e.

e e h h h h e e h h h h h h e e h h h h
Then, the differential cross-section for a one-phonon RRS process is obtained, which has the form where 2.2.Differential cross-section for a one-phonon electron Raman scattering process One-phonon ERS can be described as a process occurring in a single band, either in a valence band (by a hole) or in a conduction band (by electrons), i.e., all transitions are intra-band.In this case, we have and , ; 32 Then, we have several scattering mechanisms, all of three steps.The first process would be, first, an electron (hole) absorbs a photon of incident radiation; second, an intra-band transition occurs due to an electronphonon interaction (hole-phonon interaction) with the emission of a longitudinal optical phonon; third, the electron or the (hole) emits a photon of secondary radiation, as shown in figure 3. A second scattering mechanism would be, first, an electron (hole) absorbs a photon of incident radiation; second, the electron (hole) emits a photon of secondary radiation; third, due to an electron-phonon interaction (hole-phonon interaction) with the emission of a longitudinal optical phonon.Although this represents two of the possible combinations, the process can be produced in 6 different ways, thus [15] å å , , In the case of intra-band optical transitions, we must perform a change of reference system so that where , ; , To determine the differential cross-section equations, (21), (32) and (35) are substituted in equation (33),  where j j j j j j j j j j j j j j j j j j j j j j j w w w w w w j j j j j j j j j j j j j j j j j j j j j j j j w w w w w w j j j j j j j j j j j j j j j j j j j j j j j j Finally, substituting equations (32), ( 33) and (40) in equations ( 6) and (7), and the delta function by a Lorentzian (see equation (29 where

Discussion of results
To illustrate our results, a semiconductor QW grown in a GaAs AlAs matrix was selected so that the physical parameters used for the calculation of the electron/hole states [14,15] [14,15].On the other hand, we must remember that the solution obtained for the electronic states is just an approximation and its validity is in the subcritical region [10], where these states are described by Bessel functions of real order except for the states where = m 0 described by Bessel functions of imaginary order [10].Furthermore, in figures 5-10 the results are shown directly using the energy of the secondary radiation and not the Raman shift, since it is easier to compare the results.This is because we are studying two different Raman scattering processes, and their Raman spectra lie in different regions of the electromagnetic spectrum.
Figure 4 shows the energy of the confined part of the states of a semiconductor QW with a radius of 20 Å: (a) electrons and (b) holes.The red lines represent the energy of the odd states and the black lines the energy of the even states.In this figure, the unfolding of the electronic states due to the breaking of the circular symmetry caused by the electric field is observed.This unfolding, as can be seen, is different and non-linear for each electron or hole states [10].Moreover, due to a greater effective mass, the number of states of the hole is greater than the one for the electrons even if > V V .
e h In the figures from 5 to 7, the Raman spectra for an one-phonon RRS process are studied, which are obtained from the differential cross-section (see equation (30)).In this graph, a set of singularities related to the creation and recombination of EHP is shown (see equations ( 25) to (28)).Raman spectra of a one-phonon RRS show two types of singularities or peaks.Firstly, the peaks related to the first intermediate state ñ |a (called: incoming resonance), their position are given by In this case, the position of the outgoing resonance will depend only on the energy of the EHP. Figure 5 presents the differential cross-section for a one-phonon RRS for a semiconductor QW with a 20 Å radius considering an incident photon energy w =  3.0 eV.

l
To obtain this graph, only the process where intermediate states ñ |a and ñ |b correspond to the EHP designed as ( )  0, 1; 0, 1 , p were considered, see equations ( 12) and (30)).We must point out that this represents one of the cases allowed in the absence of an electric field [14].This is due to the creation and recombination of the EHP in case of the absence of an electric field requires = m m e h making the emission of a photon satisfy = m 0. Therefore, the presence of the electric field, which in this case is homogeneous and external, by breaking the symmetry of the system causes the appearance of the contributions where ¹ m 0. However, in references [21] and [22], it is demonstrated that in the case of a spherical quantum dot, it is possible to obtain contributions for ¹ m 0, if a different approximation other than dipolar is considered.Then, (a) = m 0 and In this case, being the two intermediate states equal, the contributions of the hole and the electron cannot be separated; this means that the incoming resonance, whose energy can be determined from equation (45)), and the outgoing resonance, whose energy can be determined from equation (46), are only separated by the energy of the phonon.If figures 5(a) and (b) are compared with figures 5(c) and (d) respectively, it is demonstrated that the position of the peaks of the differential cross-section does not significantly change as the electric field intensity increases.Then, the increment in the electric field intensity does produce an increase in the peak intensity, particularly for the peaks related with = m 1 where the increment is of 5 orders of magnitude.This shift is greater than the one reported in reference [25] for the quantum efficiency of a semiconductor QW with infinite potential barriers, while the increment for = m 0 is rather small but of a magnitude similar to the one reported in reference [25].This difference is because the effect of confinement is much weaker than the effect of the electric field.If figures 5(a) and (c) are compared with figures 5(b) and (d) respectively, it is noted that the position of singularities does not significantly change with the shift of = m 0 to = m 1.However, the shift of = m 0 to = m 1 makes the intensity of the Raman spectra decrease, but when the electric field increases, the intensity difference between the Raman spectra with = m 0 and = m 1 decreases.This is because in the absence of an electric field only phonon emissions with = m 0 are allowed according to the selection rules given in from here, it is concluded that the mode w  0,1 is the most important one, however, the intensity of each oscillation mode depends, partially, of the electric field intensity.Figures 5(b where the mode w  1,1 is much more intense than the other two.Figure 6 shows the differential cross-section for a one-phonon RRS process for a semiconductor QW of radius 20 Å considering the incident photon energy as w =  3.0 eV.In figure 6(a), the black line is multiplied by the factor indicated in the figure.Since the intermediate electron states are not the same, when considering the scattering mechanism for the one-phonon RRS selected, only an electron (not a hole) can necessarily be responsible for the emission of the phonon, so that only the contribution of the electrons is considered.This implies that the incoming resonance is  ( ) I 1, 1; 0, 1 , while the outgoing resonance is, ( ) O 0, 1; 0, 1 , and therefore both singularities are separated, in contrast with the previous figure, by an energy greater than the phonon energy.As in figure 5, the position of the peaks does not significantly change when the electric field increases.Besides, the shift from = m 0 to = m 1 does not significantly change the position of the singularities.On the other hand, the difference in intensity of the Raman spectra from the previous figure is considerably lower for all cases due to the way it was determined in references [10] and [14]   p This is especially true for = m 0 with a difference of more than two orders of magnitude, while in the case of = m 1 the difference is considerably small.However, when comparing figures 6(c) and (d), where = -F kV cm 60.0 , 1 it is the spectra denoted with black lines that present a greater intensity in both cases ( = m 0 and = m 1).Therefore, as the electric field intensity increases, the intensity of the spectra also increases for even states.In the case of the red lines, a combination of even and odd states occurs.In figures 6(a) and (c), the contributions of the phonon oscillation modes are shown, and we can conclude that the w  0,1 mode is the most important as it presents a higher intensity compared to the other phonon oscillation modes in comparison with the cases described in figure 5.
Figure 7 shows the differential cross-section for a one-phonon RRS process for a semiconductor QW with a radius of 20 Å, considering an incident photon energy of w =  3.0 eV. 1 In figure 7, unlike figure 6, it is the states of the hole and not those of the electron that are different.Therefore, when considering the scattering mechanism (one-phonon RRS) that we have chosen, the phonon emission can necessarily only be carried out by the hole and not by the electron.Then, only the gap contribution can be considered.This implies that the incoming resonance is ( ) I 0, 1; 1, 1 , while the outgoing resonance is ( ) O 0, 1; 0, 1 , and therefore both singularities are separated by an energy greater than the phonon energy, but less intense than the one observed in figure 7. Since the hole possesses an effective mass greater than that of the electron, the gap between these two states is smaller.In contrast with the results shown in figures 5 and 6, the increment in the electric field intensity produces a shift in the position of incoming resonances, this is because the confinement is smaller for the hole than for the electron.Moreover, the increment in the intensity of Raman spectra as the electric field intensity increases is only of two orders of magnitude for = m 0 and of one order of magnitude for = m 1, this is two orders of magnitude below the case described in figure 6.In all other respects, the Raman spectra in figures 6 and 7 are practically the same.In figures 8 to 10, the emission spectra for one-phonon ERS are studied.The set of singularities observed in these graphs was analyzed (see equations (41) to (43)).In this case, the emission spectra show several types of singularities or peaks: in the first place, the so-called resonant peaks, or singularities, characterized by an energy or position w ( )  s not depending on incident radiation w ( )  In the second place, we ding the so-called non-resonant peaks or singularities characterized by an energy or position depending on the incident radiation, then Equation (49) shows that as the initial state is the same as the final state, i.e.
Therefore, the electron or hole absorbed an incident photon, later emitting a photon of secondary radiation, returning to the initial state.It can be concluded from the equation (50 This implies that the electron or hole absorbed an incident photon, later emitting a phonon and a photon of secondary radiation, returning to the initial state.The singularities given by the equations (49) and (50) can be observed in any of the four polarizations s s s l The differential cross-section for a one-phonon ERS process for a semiconductor QW with a radius of 20 Å, polarization s s     4).Therefore, the emission of the phonon must be produced between the states + ( ) 1, 1 and - ( ) 1, 1 implying that this transition is impossible without the presence of the electric field, since these states are the product of the unfolding induced by the electric field.In the beginning, the intensity of this peak increases rapidly with the increment of the electric field intensity, but later decreases slowly.Furthermore, the position of the singularities is kept constant, demonstrating that the gap between the electronic states slightly varies with the variation of the electric field due to strong confinement.The energy structure of the phonon modes ( ) n 0, can be observed in the emission spectra, but this is not the case for the mode structures ( ) n 1, since the gap between them is considerably smaller and mode ( ) 1, 1 is remarkably predominant.To improve and complete the analysis, we are going to increase the radius of the semiconductor QW to 30 Å, this increases the number of electronic states associated with the electrons and holes.Therefore, in figure 9     an unfolding of the peaks is not produced.Also, the energy structure of the phonons is hardly observed as the increase of the gap radius between the different oscillation modes reduces, and the mode ( ) m, 1 predominates [23, 24].Another interesting aspect is that for the cases shown in figures 8 and 9, the peaks w ( )  m n m n , ; , but with a greater intensity since their energy or position is closer to the incident radiation energy (see equation (43)).
Finally, in figure 10 the differential cross-section for a one-phonon ERS process for a semiconductor QW with a radius of 20 Å, polarization s s The number of states corresponding to holes is greater than the number of states corresponding to electrons since their effective mass is heavier, though the potential barrier for electrons is considerably greater than the one for holes (see figure 4).In this case, the emission spectrum intensity decreases by 6 orders of magnitude in comparison with the spectra obtained considering only the contribution of electrons.If figure 10  Moreover, the behavior of the singularities is similar to the one described for the case of the electrons.Also, the variation of the intensity of the Raman spectra due to the shift in m or in the electric field intensity is similar to the ones obtained for the electron Raman scattering in reference [26].

Conclusions
In this work, we have developed a study on the influence of an external electric field over the processes of Raman scattering with inter-band and intra-band optical transitions: first, for one-phonon resonant Raman scattering, and second, for one-phonon electron Raman scattering.This study is conducted on a semiconductor quantum wire of cylindrical symmetry with finite potential barriers where the electric field is homogeneous and in a transversal direction of the wire axis.In addition, = T K 0 and parabolic electronic bands have been considered.For one-phonon RRS, the conduction band is completely empty while the valence band is completely occupied, and for one-phonon ERS, only intra-band transitions take place.To determine the electronic states, a model with the conditions imposed by the electric field is assumed as valid for systems where the electric field is weak compared to confinement.In this study, a system grown in a GaAs AlAs matrix is considered.
As a result of this study, the one-phonon RRS has been demonstrated that the electric field produces the occurrence of singularities in the Raman spectra associated with mixed phonon oscillation modes ¹ ( ) m 0 with incoming and outgoing resonances where ¹ m m .
e h On the other hand, in one-phonon ERS, unlike in onephonon RRS, the existence of resonant and non-resonant singularities is verified.As for one-phonon RRS, the electric field produced the occurrence of singularities in Raman spectra associated with mixed phonon oscillation modes.This is due to the presence of a homogeneous electric field in a transversal direction to the QW axis that breaks the selection rules for optical transitions that exist in a system of this type but without an electric field [14,15], making it unnecessary to consider other approximations dipolar-approximation for the electron-photon interaction [21,22].We must point out that the effect of the Coulomb different to the interaction (excitons) on a quantum dot with spherical symmetry was studied in references [21,22], obtaining as a result that the selection rules are not changed.In contrast, the presence of a magnetic field breaks the degeneration of states without changing selection rules [14,15], for optical and phonon transitions.
If we compare our results with those obtained in the reference [25] for the one-phonon RRS, we must point out that in our case we consider a system with finite barriers where the effect of the electric field is much more important since the effect of confinement in the electronic states is minor.Furthermore, the quantum wire is a more realistic model than the free-standing wire studied in reference [25].The foregoing is verified when comparing figures (5)-( 7) with figures 6 and 7 of the reference [25], where it is observed that the effect of the electric field on the position and intensity of the singularities in a QW is greater than on a free-standing wire.On the other hand, our results include the study of the one-phonon ERS, where the electron-phonon interaction is considered, unlike the study carried out in reference [26], where phonons are not included.So, considering the electron-phonon interaction allows us to study the effect of phonons on the Raman spectra and compare them with the one-phonon RRS.
On the other hand, in both Raman scattering processes described in this work, the electric field causes a displacement of the singularities.This displacement will depend on two factors: the electric field intensity and the strength of the confinement of the electrons and holes.We show that in both Raman scattering processes, going from = m 0 to = m 1 affects the peaks´intensity.In most cases, the electric field increases the intensity of singularities in the Raman spectra; however, in some cases, it is possible that the intensity of singularities increases rapidly with increasing F and then decreases slowly.Also, the electric field unfolds the singularities since it breaks the degeneracy of the electronic states.Furthermore, it was demonstrated that in the case of onephonon ERS, the radius of the semiconductor QW determines the effects observed in the Raman spectrum.The effects described in this work are because the electric field is homogeneous and transversal to the QW axis, since for the case of the electric field with cylindrical symmetry, the selection rules for the optical transitions do not change with respect to a system with no electric field [13].
Finally, in a system like this, when the temperature is different from zero, the importance of the electronphonon interaction increases.However, this effect depends on the type of transitions that occur, whether intraband or inter-band; therefore, the effect on one-phonon RRS is somewhat different from the effect on onephonon ERS.In the RRS, the importance of the electron-phonon interaction increases with increasing temperature especially anti-Stokes scattering, since there are more electrons at low energy levels than at high ones; so that more Stokes transitions will occur than anti-Stokes transitions, consequently the Stokes peaks will have a greater intensity, the distance between the Stokes and anti-Stokes peaks being the same with respect to the Rayleigh or incident line.In the case of the one-phonon ERS, we must consider that there are only intra-band transitions and that the gap between the electronic states is relatively small.Then, the increase in temperature will increase the importance of scattering mechanisms where as a first step a photon of secondary radiation is emitted or a phonon is emitted.These scattering mechanisms are related to the last four terms of equation (41).However, in a model like the one we have used, the intensities and position of the singularities in a Raman spectrum will not change with the change in temperature.Therefore, it is necessary to develop a multiparticle model in which a change in temperature and electron density would be reflected in a change in the position of the peaks and a variation in the intensity of the Raman spectrum.

Finally,
 r from the following transcendental equation is obtained, then

Figure 1 .
Figure 1.Schematic illustration of a semiconductor quantum wire with finite cylindrical potential barriers in the presence of a homogeneous electric field oriented in a transversal direction to the axis of the system.

h
To determine the oscillation modes of optical phonons, the following physical parameters were used[14,15]: w =  36.25 meV, For a better understanding and analysis of the results, only the contribution of the first three oscillation modes for each m is considered, i.e.: = n 1, = n 2 and = n 3[14,15].To obtain the Raman spectra, all equal lifetimes of the intermediate and final states were considered, this is

Figure 4 .
Figure 4. Energy of the confined part of electronic states for a radius of 20 Å in a semiconductor quantum wire with (a) electrons and (b) holes.
) and (d) show the contributions of the phonons oscillation modes: w =  34.89 meV,

1
prohibited in the absence of an electric field.Moreover, the shift from = m 0 to = m 1 makes the intensity of the Raman spectra increase maintaining a difference of around 4 orders of magnitude in the intensity of the spectra.This is because, in the absence of an electric field, selection rules require the emission of a phonon through intra-band transitions satisfying If figures 6(a) and (b) are observed, where = -Raman spectra with red lines related to the EHP labeled by -

Figure 5 .
Figure 5. Differential cross-section for a one-phonon resonant Raman scattering process for a semiconductor quantum wire with a radius of 20 Å and w =  3.0 eV, l considering an intermediate state ñ |a and as state ñ |b the electron-hole pair described by ( )  0, 1; 0, 1 .p

l
To obtain the black (red) lines in this graph, only the process where the intermediate state ñ|a is related to the EHP +

Figure 6 .p
Figure 6.Differential cross-section for a one-phonon resonant Raman scattering process for a semiconductor quantum wire with a radius of 20 Å and w =  3.0 eV, l

p
In figure6(a), the black line is multiplied by the factor indicated in the figure.
the case of an absent electric field, this singularity can be observed only for contributions s s ⋅ + -

l 1
and w =  0.6 eV, l considering only the electron contributions, is shown in figure 8. Then: (a) = m 0 and = -When comparing figure 8(a) with figure 8(b) and figure 8(c) with figure 8(d)

Figure 7 .p
Figure 7. Differential cross-section for a one-phonon resonant Raman scattering process for a semiconductor quantum wire with a radius of 20 Å and w =  3.0 eV, l

p
In figure4(a) the black line is multiplied by the factor indicated in the figure.
respectively, the shift from = m 0 to = m 1 produces a decrease of at least one order of magnitude in the peaksí ntensity; this decrease is smaller than the one shown for one-phonon RRS (see figures 5 and 6).On the other hand, when comparing figure 8(a) with figures 8(c) and (b) with figure 8(d), the increase in the electric field intensity produces a slight decrease in the peaks´intensity.This is because for a radius of 20 Å, there are only three quantum states associated with electrons ( ) 0, 1 , + ( ) 1, 1 and - ( ) 1, 1 (see figure

1 1 , 1 ,
the differential cross-section for a one-phonon ERS process for a semiconductor QW with a radius of 30 Å, polarization s Then, when comparing figure9(a) with figures 9(b) and (c) with figure 9(d), the shift from = m 0 to = m 1 produces a decrease by one order of magnitude in the intensity of singularities w  ( )  0, 1; 1, 1 , as in figure 8. Comparing figure 9(a) with figures 9(c) and (b) with figure 9(d) a slight decrease in the peaks´intensity is observed with increasing electric field intensity; this behavior is similar to that of the previous figure, where the radius of the QW was 20 Å.In contrast with figure 8, the w  ( ) peaks are split and can be observed individually.In addition, we can observe the non-resonant singularity w  ( )  0, 1; 1, 1 sl that unfolds with the increase of the electric field intensity, with this peak being more important in the case where = m 1.However, when comparing figure 9(a) with figures 9(b) and (c) with figure 9(d), the shift from = m 0 to

Figure 8 .l 1 ,
Figure 8. Differential cross-section for a one-phonon electron Raman scattering process for a semiconductor quantum wire with a radius of 20 Å, polarization s s -+ ( ) , s l and w =  0.6 eV.l

Figure 9 .
Figure 9. Differential cross-section for a one-phonon electron Raman scattering process for a semiconductor quantum wire with a radius of 30 Å with polarization s s -+ ( ) , s l and w =  0.6 eV.l Considering in this case only the contribution of electrons.In figure 9(c), the red line is multiplied by the factor indicated in the figure.

figures 9 (
figures 9(c) and (b) with figure 9(d), the increase in the electric field intensity produces two opposite effects.For = m 0, the peaks increase approximately one order of magnitude, and for = m 1 the peaks slightly decrease their intensity.For sigularities w  ( )  0, 2; 2, 1 ,s

Figure 10 .l
Figure 10.Differential cross-section for a one-phonon electron Raman scattering process for a semiconductor quantum wire with a radius of 20 Å, with polarization s s -+ ( ) , s l and w =  0.6 eV.l