Impact of Relativistic Electron Beam on Hole Acoustic Instability in Quantum Semiconductor Plasmas

1 Introduction

Plasma waves are electrostatic and electromagnetic in linear and non-linear regimes. The electrostatic waves propagating with the dynamics of ions are low frequency waves such as lower hybrid, ion acoustic wave (IAW), hole acoustic wave (HAW) and ion cyclotron wave. The acoustic mode is one of the fundamental electrostatic modes that was extensively studied by the researchers for their vast applications in laboratory and space plasmas [1], [2], [3], [4], [5]. For instance, these waves play a significant role in the study of the naturally occurring plasmas in space such as corona, solar wind, comets, earth’s ionosphere and chromospheres etc. [6], [7], [8], [9], [10], [11], [12]. An example of electrostatic low-frequency compressional waves in electron-ion plasmas is the IAWs. In these waves ions play a similar role as that of the neutral atoms in the excitation of sound waves in neutral fluids [13]. The interaction among ions in plasmas extends over the large distances for their electric fields, due to which these waves can travel through the collisionless plasmas. The semiconductor materials consist of holes and electrons exhibiting the properties of solid-state plasmas. The oscillating electric field is established by the separation of charges due to mass difference between semiconductor plasma species [14]. This oscillatory electric field generates HAWs in semiconductor quantum plasmas. The experimental studies of the acoustic modes were carried out in a variety of materials like GaAs, MgO, Si, Al₂O₃ and SiO₂. These low-frequency modes can be studied in quantum semiconductor plasmas at a nanoscale by using the quantum-hydrodynamic model (QHD) of plasmas. The quantum effects become prominent and cannot be ignored when the average inter-particle distance is less than the de-Broglie wavelength associated with each of the plasma particle (holes and electrons). The quantum effects appear through Bohm potential, Fermi pressure, exchange and correlation potentials. The exchange and correlation potentials are due to the spin effects in the plasma that play a significant role in the ultra-small electronic devices made up of semiconductors [15], [16], [17], [18], [19]. The QHD for each of the plasma species provides complete description of the nano-sized semiconductor devices [20], [21], [22], [23], [24] and lasers [25], [26], [27], [28]. For example, nanowires, nanophotonics [29], [30], [31], resonant tunneling diodes [32], [33], [34], super-luminescence which occurs for picoseconds [35], [36], [37] and high gain photoconductive current filament [38]. Recently the excitation of low-frequency HAW was studied in quantum semiconductor plasmas [20].

In this manuscript we studied the excitation of low frequency, long wavelength electrostatic HAW in semiconductor plasmas. The HAWs are generated through the relativistic beam of electrons and then the effects of plasma parameters such as beam velocity, concentration of plasma species, thermal velocity of electrons and multiple quantum effects on them were investigated. There is an extensive amount of literature describing both the production and the application of the electron beam. Prono et al. discussed the production of intense relativistic electron beam (REB) with large power outputs (≥10¹² W) and high efficiency (>50%) with possible application of plasma heating in mirror machines, tokamaks and laser pumping. Several authors have studied the effects of REB interaction with plasmas. The REB can impart the strong electrostatic fields or hydromagnetic waves in plasmas [1]. Furthermore, the structure of the wake-field by using an ultra-REB pulse propagating through plasma, was studied extensively by Rosenzweig et al., Amatuni et al. and Ruth et al. The applications of the electron beam pumped into semiconductor quantum plasmas were elaborated in [20], [39]. On contrary to the optical pumping, the significant feature of the electron beam into the bulk plasmas is the independence of matching with the band gap energy. In our article we are studying the dispersion properties of the electrostatic waves exciting in semiconductor plasmas whenever the semiconductor is exposed to an REB. The point to be noted is that our electron beam is thermally relativistic whereas the electron beam speed or streaming is non-relativistic. Therefore, the macroscopic structure of the semiconductor might not be affected on exposing it to the REB. The study of electron beam effects on the electrostatic waves exciting in semiconductors is our quest. Recently, we investigated on the impacts of the non-REB on the HAWs, upper hybrid and lower hybrid waves exciting in the semiconductor plasmas. We found that the HAWs instability reduces on increasing the thermal temperature of beam electrons. In other words, the beam electron temperature tends to stabilise the HAWs. Therefore, the motivations behind the current study are:

1. Firstly to check if the thermally REB stabilises the HAWs or not.

2. The possible generation of electrons beam in laboratory persuades us to analyse the effects of the relativistic beam on the HAWs in semiconductor plasmas.

The present study provides information and feedback as to what can happen if a chip made of semiconductor or a simple semiconductor is exposed to the environment of relativistic beam in a lab or an astrophysical environment such as solar wind etc. On employing the QHD equations in Section 2, we have solved the fluid equations of semiconductor plasmas for the study of linear dispersion relation. The dispersion relation is used to obtain the growth rate of the electrostatic HAWs. A numerical analysis of the growth rate of the HAW instability is done with experimental parameters at nanoscale with a particular example of GaAs in Section 3. A summary of the results is presented in Section 4.

2 Derivation of Linear Dispersion Relation

A homogeneous dense semiconductor quantum plasma consisting of the electrons holes and beam electrons is considered. The REBs are externally pumped into the semiconductor plasma system as a source of energy with streaming velocity v_b0. The subscript notation “e,” “h” and “b” are used for electrons, holes and beam electrons, respectively. In the absence of any perturbation the plasma charge quasi-neutrality condition reads as, n_h0=n_e0+n_b0, where n_e0, n_h0 and n_b0 are the equilibrium number densities of semiconductor electrons, holes and beam electrons, respectively. In the study of the HAWs in semiconductor plasmas, the role of holes is considered similar to the ions in gaseous plasmas. The linearised set of equations for the semiconductor electrons, holes and the beam electrons with thermal effects as well as degenerate effects [arising through degenerate pressure due to the high number density of the electrons (holes), quantum recoil force due to the electrons (holes) tunneling through a potential barrier known as the Bohm potential, and electrons (hole) exchange and correlation effects due to their spins] are

where ℏ=h/2π (Planck’s constant), q_j, m(e,h)∗, and n_(e,h) are the charges, effective masses and number densities of semiconductor electrons and holes, respectively. The V_xcj is the exchange-correlation potential of semiconductor species and is defined as

Vxc(e,h) =[0.985e2εn(e,h)13(1+0.034aB(e,h)∗n(e,h)13ln(1+18.37aB(e,h)∗n(e,h)13))]

where aB(e,h)∗=εℏ2e2m(e,h)∗ is the Bohr atomic radius, ε is the relative dielectric constant of material and “e” is the magnitude of electronic charges [40], [41].

The equation of state for relativistic beam of electrons is:

This state equation is obtained from the well-known thermodynamic expression  PbV=−∂Eb∂V, where V and E_b are volume of the plasma system and total energy of the relativistic beam of electrons, respectively. The total energy E_b is obtained by solving the integral relation  Eb=nb a4π(m0c)3K2(a)V∫(ε−m0c2) ƒ(p)sinθp2dpdΦd, in spherical geometry where, ƒ(p) =exp(−c(p2−m02c2)1/2kBTb) is the modified Maxwellian distribution function describing the REBs. After performing the integration and some straightforward calculations, the total energy expression for the beam electrons appears to be as

Eb=nbVm0c2[K3(a)K2(a)−1a−1].

where, K_2(a) is the MacDonald function. The general expression for MacDonald function is ∫0∞coshnθ exp(−acoshθ)dθ =Kn(a) in which a is the relativistic factor that is the ratio of rest mass energy to thermal energy of beam electrons, i.e. a=m0c2kBTb. For very large value of relativistic parameter (a>>1), the state equation (4) of the REBs is converted to state equation (P_b=nk_BT_b) of non-REBs.

It is better to mention that thermal and streaming velocities of beam electrons make a different sense. Thermal velocity defining the thermal temperature of the beam electrons is associated with the random motion whereas streaming speed independent of temperature is concerned with the drifting of the electron beam. As far as the presence of relativistic factor in equation of continuity of beam is concerned, we ignore the inertial effects of electron beam on exciting the HAW effects. So, we do not use practically in our calculations the equation of continuity for beam electrons. As our equation of continuity is general and applicable to all the species of the plasma, we have hidden the relativistic factor in the definition of density and velocity for electron beam in equation of continuity. Our beam electrons are thermally relativistic whereas streaming speed is non-relativistic; therefore, we did not write the relativistic factor with beam streaming velocity. The degenerate Fermi semiconductor species (electrons, holes) obey statistical pressure law, P(e,h)= m(e,h)∗vF(e,h)∗2n(e,h)33n(e,h)02, for which vF(e, h)∗2=35vF(e,h)2[1+512π2{T(e,h)TF(e,h)}2] and vF(e,h)2=2kBTF(e,h)m(e,h)∗ where TF(e,h)=ℏ2(3π2n(e,h)0)2/32m(e,h)∗ is the Fermi temperature of semiconductor electrons and holes, T_(e,h) is the non-zero thermal temperature, n_(e,h)=n_(e,h)0+n_(e,h)1 number density with its equilibrium value n_(e,h)0 and perturbed number density n_(e,h)1, respectively. In (ω−k) space, the velocity components of the semiconductor electrons, holes, and the REBs can be obtained, from (2) and (3), respectively

where ω−k→ ⋅v→b0 is the Doppler shifted frequency, v_b0 is the streaming speed of beam electrons, and Ψb=[Φ−m0c2e[(K3(a)K2(a)−1a)−1]nb1nb0], is the effective beam electrons potential, and Ψ(e,h)=[Φ+m(e,h)∗q(e,h)[vxc(e,h)2+vF(e,h)∗2+k2H(e,h)2]n(e,h)1n(e,h)0] such that H(e,h)=ℏ2m(e,h)∗ is the effective potential of semiconductor species (electrons, holes) due to electrostatic potential Φ and quantum effects arising through Fermi pressure, Bohm potential, exchange and correlation potential of semiconductor species (electrons, holes) with square of exchange speed given as

vxc(e,h)2 ={0.328n(e,h)013e2m(e,h)∗ε+0.01116e2aB(e,h)∗m(e,h)∗ε(18.37aB(e,h)∗n(e,h)0131+18.37aB(e,h)∗n(e,h)013)}.

The simultaneous equations (1)–(6) provide us the perturbed number densities of the beam electrons and the semiconductor species (electrons and holes) as

The standard expression for perturbed number densities of semiconductor electrons, holes, and beam electrons can be given as,

Comparing the expressions of perturbed number densities given by (7)–(9) with the standard expression (10), we can represent the susceptibilities of semiconductor electrons, holes and beam electrons, respectively by

The dispersion relation for the HAW can be written as

or

Here, vs,(e,h)2={vxc(e,h)2+vF(e,h)∗2+H(e,h)2k2} is the effective speed of electrons/holes which is the sum of the exchange, Fermi and thermal speeds, and the speed due to tunneling potential. Taking the logical approximation of vb0≫ω/k⇒ω/kvb0≈0 for beam electrons and ignoring the inertia of negative carriers of semiconductor. The dispersion relation of the HAWs in the semiconductor quantum plasmas becomes

Equation (16) is the modified linear dispersion relation of HAWs in electron-hole-electron beam plasma including the effects of exchange-correlation, thermal temperature, quantum statistical pressure and quantum recoil effects of the degenerate electrons and holes. Here Cs,h2=ωPh2vs,e2ωPe2 is the square of sound speed containing the thermal effects and quantum characteristics such as degenerate pressure, Bohm potential due to tunneling and exchange-correlation potential, and the λs,De2=vs,e2ωPe2 is modified Debye length having all the quantum features of semiconductor plasmas. The absence of electron beam, ω_pb=0, drags the modified dispersion relation (16) to the standard dispersion relation of HAW propagating in semiconductor quantum plasmas.

Equation (17), elaborates the propagation of HAW in semiconductor quantum plasmas without relativistic beam of electrons. The first term on right hand side describes the contribution of holes, which comes from their thermal equivalent effects, whereas the second term indicates the contribution of the electrostatic field interactions exciting the acoustic waves in the semiconductor plasmas.

3 Numerical Solution and Graphical Description

The frequency of the wave is usually the superposition of real and imaginary frequencies of the waves, i.e. ω=ω_r+Im ω where ω_r is the real frequency and Im ω is the imaginary frequency whose sign indicates either the wave will grow or propagate through the plasma. The experimental observations of the acoustic modes were carried out for a number of materials and semiconductors such as Al₂O₃, Si, MgO, SiO₂, and GaAs. As our analytical plasma model is developed for semiconductor plasmas pumped by REB, considering one of the semiconductor plasma like GaAs. However, our model can also be applied equally to the other semiconductors such as GaSb, GaN and InP. It was observed in our previous article, that HAW instability arises in the semiconductor (GaAs) degenerate plasma when non-relativistic external beam of electrons is used as an energy source [41]. The normalisation of the imaginary part of (16) is given as

 Im ω2=Im ω2ωpe2, ωp(b,h)2=ωp(b,h)2ωpe2, ωpe2=ωpe2ωpe2=1, k2=k2vFe2ωpe2, T(e,h)=T(e,h)TFe,  vxe(e,h)2=vxe(e,h)2vFe2,  vb02=vb02vFe2.

The HAW spectrum is observed in Figures 1–4 that describe multiple features under different conditions and parameters like n=4.7×1020cm−3; me∗me=0.067,  mh∗me=0.5, ϵ=12.8.

Figure 1:

Relationship of (Im ω, k) at  a=mb0c2kBTb=229,000, 232,000, 235,000, Th=ThTFe=0,Te=TeTFe=0,v=vb0vFe=0.201,σ=ne0nh0=0.75 and θ=0.

Figure 2:

Relationship of (Im ω, k) at Te=TeTFe=0.1, 0.2, 0.3,a=mb0c2kBTb=235,000,Th=ThTFe=0.1,v=vb0vFe=0.201,σ=ne0nh0=0.75 and θ=0.

Figure 3:

(a) Pressure variations of beam electrons with respect to scaled beam electron density  σ=nb0nh0 for a fixed value of relativistic factor a=229,000. (b) Pressure variations of beam electrons with respect to scaled beam electron density  σ=nb0nh0 for a fixed value of relativistic factor, a=232,000. (c) Pressure variations of beam electrons with respect to scaled beam electron density  σ=nb0nh0 for a fixed value of relativistic factor a=235,000.

Figure 4:

(a) Pressure variations of beam electrons with respect to scaled beam electron density  σ=nb0nh0 for a fixed value of relativistic factor a=0.1. (b) Pressure variations of beam electrons with respect to scaled beam electron density  σ=nb0nh0 for a fixed value of relativistic factor a=1.

Figure 1 shows the effect of relativistic factor associated with externally injected electron beam. The negligible relativistic effects come into play when the relativistic parameter “a” has higher value. It is noted from different curves shown in Figure 1 that the spectrum of the HAWs becomes unstable at very large values of relativistic factor. It is worth mentioning that for all values of a, the semiconductor plasma species are taken non-relativistic, however quantised. The smaller values of relativistic parameter that impart comparable thermal energy with the rest mass energy of beam electrons reduce the spectrum of acoustic modes as well as the growth rate. It is confirmed that at values of a which corresponds to REB, there is no longer instability in the acoustic waves for all the possible values of plasma parameters. Such trend was also observed in our previous article, that is, the temperature of non-REB tends to stabilise the HAW instability in semiconductor quantum plasmas. So in the current study the observed stabilising character of REB to the HAW instability arising in semiconductor dense plasmas is as expected in the previous study of these waves when energy source electron beam was non-relativistic. It is clear from Figure 2, an increment in the value of thermal energy of external beam electrons tends to less resonate with the field energy of acoustic modes and henceforth, reduces the growth rate.

It is also clear from Figures 1 and 2 that the instability curves gradually become narrow with increasing temperature of semiconductor electrons and external relativistic beam of electrons. It is also clear from the mathematical relation, as temperature of beam electrons increases the value of relativistic factor decreases. When thermal energy of beam electrons is compared to the rest mass energy of electrons, the beam becomes relativistic and relativistic factor approaches to unity. Hence it is confirmed that the instability exists only when the beam electrons are non-relativistic whereas instability disappears when electron beam becomes relativistic. To find out what is happening here we plotted the state equation of beam electrons in relativistic and non-relativistic regimes. Shortly different values of the relativistic parameter a define either the plasma is relativistic, ultra relativistic or non-relativistic. Figure 1 elaborates the instability of the HAWs for three different values of relativistic parameter a=235,000, 232,000 and 229,000. It is noticed that HAW instability reduces on reducing the relativistic parameter and leads to no instability at even smaller values of relativistic factor (the small values of a that correspond to relativistic beam). Figure 2 shows the graphical retrieval of HAW instability due to non-relativistic electron beam at larger values of relativistic factor. The plot (2) authenticates the literature that the larger values of a=229,000 direct the electron beam to non-relativistic. These results once again confirm our previous work with non-relativistic electron beam. The plot (2), therefore, is nothing except only to verify that our beam becomes non-relativistic for large values of relativistic factor.

In Figure 3a–c, we have plotted the normalised state equations of both the relativistic and non-relativistic beam electrons vs. beam electron density divided by the hole number density σ at equilibrium in the semi-conductor plasma. It can be noted that at large value of relativistic factor a the pressure values of relativistic and non-relativistic beam electrons are very close to each other. In other words the value of relativistic factor a in the order of 3×10⁵ make the beam electrons to be non-relativistic which shows that increase in the value of a reduces the relativistic effect of beam electrons. In Figure 4a and b we have plotted the normalised state equations for both relativistic and non-relativistic beam electrons. It is found that at lower value of relativistic factor the pressure described by relativistic state equation is much larger than the pressure obtained from non-relativistic equation. Thus we conclude that the REB stabilises the HAWs exciting in quantum semiconductor plasmas.

4 Summary

We derived the dispersion relation for the HAWs in semiconductor quantum plasmas under the influence of external relativistic beam of electrons. We found that there is no instability arising in the HAW profile under the influence of relativistic beam of electrons i.e. relativistic factor (a=1). The variation of different parameters does not play any role to cause the instability such as, electrons to hole number density ratio, propagation angle, streaming speed of beam electrons and thermal effects of semiconductor species. The instability appears on the other hand when the external beam of electrons is non-relativistic. We also checked that the growth rate increases on increasing the streaming speed of beam electrons, electrons to hole number density ratio and relativistic factor (having more non-relativistic plasma). The growth rate decreases with increasing normalised semiconductor species temperature. This shows that our dispersion relation is truly valid for the case when REB with relativistic factor a=1 suppressed the instability in semiconductor quantum plasmas. It is showing full agreement of qualitative and quantitative form of equation of state for REBs.