Temperature-induced disruptive growth rate behavior due to streaming instability in semiconductor quantum plasma with nanoparticles

The nature of the growth rate due to streaming instability in a semiconductor quantum plasma implanted with nanoparticles has been analyzed using the quantum hydrodynamic model. In this study, the intriguing effect of temperature, beam electron speed, and electron-hole density on growth rate and frequency is investigated. The results show that the growth rate demonstrates a nonlinear behavior, strongly linked to the boron implantation, beam electron streaming speed and quantum correction factor. A noteworthy finding in this work is the discontinuous nature of the growth rate of streaming instability in boron implanted semiconducting plasma system. The implantation leads to a gap in the growth rate which further gets enhanced upon increase in concentration of implantation. This behavior is apparent only for a specific range of the ratio of thermal speed of the electrons to that of the holes.

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The growth in research in this direction is driven by unresolved problems in the transport mechanism of charge carriers due to the miniaturization of electronics propelled by the development of micro and nano-electronics based semiconductor devices [8][9][10].It has been reported long back that electrons and holes (e-h) in semiconductors can be considered as semiconductor plasma systems [11].In such systems, quantum corrections become relevant as the de-Broglie wavelength is comparable to the inter-particle distance.For a better understanding of physics, quantum corrections are being considered from a long time in the microscale and the nanoscale semiconductor-based devices [12][13][14][15][16][17][18].
The complexities in the flow dynamics of the charge carriers in semiconductors have a lot of open problems.One of the thrust areas in the perspective of flow dynamics relates to the study of instabilities of the waves that may be excited due to several reasons.It is an important area of investigation in different quantum systems due to their significance in understanding the various processes leading to the growth and decay of excitations of waves and noises under various circumstances [19].Different types of instabilities like modulational, parametric, and streaming have been studied extensively for a long time in the context of quantum plasma [20][21][22][23][24][25][26] in general.Recent studies have provided a thorough analysis of the many instabilities that might arise in semiconducting quantum plasma [27][28][29].The theoretical prediction and experimental verification of streaming instability have been successfully carried out in the twentieth century in classical plasma systems [30,31].Using the quantum hydrodynamic model (QHD), streaming instability has been studied for unmagnetized dusty plasma [32], semiconductor plasmas [33], electron-ion plasmas [34], and electron-positron quantum plasmas [35].
A respectable amount of work has been carried out to understand the two-stream instability in a quantum semiconductor plasma system.It is observed that in the presence of an external magnetic field, the dispersion relation in various types of semiconductor plasmas exhibits interesting features.The angle of the wave vector affects the branching of the modes into Langmuir and acoustic waves [36].Another work carried out to understand the propagation behavior of solitary waves in a semiconductor quantum plasma system in the presence of an external magnetic field depicted that the density ratios of the charge carriers, the type of the semiconductor, and the strength of the magnetic field can highly impact the wave parameters [37].The magnetic field strength and the plasma parameters also control the growth rate of instability.In one of the works, it is illustrated that the behavior of the phase speed and instability of the lower hybrid waves actuated by pumping an electron beam in a semiconductor plasma system is dependent on the number density of electrons, the streaming speed, and the external magnetic field [38].It is also revealed that by invoking quantum effects in case of twostream instability in n-type GaAs semiconductor plasma it can result in two new modes that are otherwise not present under classical consideration [6].
Yet another important phenomenon in the context of semiconductors is the doping of semiconductors which has revolutionized the performance of semiconductors and changed the facade of technology right from the discovery of diodes and transistors.Boron being one of the p-block elements has been extensively studied [39][40][41][42] as a dopant to explore its magnetic and optical properties for its potential explanation in spintronics applications in inorganic semiconductors in recent times.Electric-optical properties of boron doped C 3 N system showed a linear decrease in band gap with increasing applied electric field [43].Furthermore, the density functional theoretical result predicts that the boron-implanted system may provide a new platform leading to novel physical properties.Ion beam irradiation is a well-known technique to obtain desired atomic species and form nanostructures such as atomic vacancies [44,45], interstitial defects, and also to generate desired nanostructures inside semiconductors [46,47].The incorporation of nanoparticles inside semiconductors not only changes the chemical properties locally but also affects the band gap of materials due to quantum confinement generated by vacancies [48].During the process of ion beam implantation, nanoparticles of different sizes may form by the amalgamation of boron-boron or boron along with atomic vacancies.The generation of different-kinds of nanostructures depends on the energy and size of implanted/irradiated species [49].
The incorporation of charge carriers within nanoparticles (implanted ions) along with intrinsic carriers (holes and electrons) in semiconducting plasma is a relatively new concept.The addition of nanoparticles is effective in the crystal growth in organic semiconductors and thereby affects the mobility of the charge carriers [50,51].Sun, et al in their work have successfully demonstrated that the introduction of Au and Ag nanoparticles in the matrix of metal oxide semiconductors supplements its optical properties because of its surface plasmon resonance [52].
There has been a multitude of work regarding the fabrication and characterization of both inorganic and organic semiconductors.However, it is worth noting that the theoretical understanding and modeling by incorporating quantum effects in a semiconducting plasma system implanted with nanoparticles has not been comprehended as yet.
As mentioned earlier, the consideration of semiconductors as a quantum plasma system has proven to be an effective theoretical approach to explain the performance of semiconductors.New insights in physics were reported by considering various quantum effects [53][54][55] in semiconductor plasma systems.Highly doped semiconductors can be considered to be degenerate plasmas, in which the temperature is much smaller than the Fermi energy like that of the electron gas in an ordinary metal.
Understanding the various mechanisms for excitation of instabilities in semiconductor quantum plasmas holds the key to delve into the physics of wave propagation and thereby the flow dynamics of the charge carriers in semiconductors.In the case of two stream instabilities in semiconductor quantum plasma, it has been found that in most cases, electron beam speed plays a vital role as the free source of energy for the excitation of the instability [6,27,33].
To the best of our knowledge, the impact of temperature on instability in semiconductor quantum plasma in the presence of nanoparticles is not yet explored.The effect of temperature on the functioning of semiconductors is very demanding for situations where semiconductor-based devices in ultra-cold temperatures are used like that in lunar missions.
In this work, we propose to investigate the effect of electron and hole temperatures on the growth rate of streaming instability in semiconductor quantum plasma implanted with boron nanoparticles by injecting an external electron beam.The boron ion implantation changes the concentration of charge carriers in the system.Electrons and holes are considered to behave quantum mechanically and inertia less, while the nanoparticles were assumed to be immobile.However, the charge carriers within the nanoparticles have been considered mobile.The presence of nanoparticles is modeled in a similar way as that of quantum plasma with dust particles in which a generalized dielectric response for an unmagnetized quantum dusty plasma comprising electrons, ions, and dust particles with different quantum effects [32] was evaluated.The instability due to the streaming of electron beam has been analyzed, both theoretically and numerically.The detailed theoretical derivation, numerical analysis, results and discussions, and conclusions drawn for streaming instability considering the InSb semiconductor plasma system are presented in the sections below.

Mathematical modeling
We have considered a boron ion implanted semiconductor quantum plasma system consisting of holes (h), and electrons (e).A beam of electrons (b) has been considered as the source of free energy.The charge neutrality condition at equilibrium is j q j n j0 = 0 where q j is the charge and n j0 is the equilibrium number density of jth species (j is a common term representing holes (h), electrons (e), beam electrons (b) and charge carriers within nanoparticles (n)).
The QHD approach is used to solve the set of equations.The electrons, holes, charge carriers within implanted boron ions, and beam electrons have been considered to be independent fluids.In the present work for mathematical simplification, a one-dimensional physical system (along the x-axis) is considered such that x ≫ y, z.It can be considered as a wire where any kind of waves excited flows along the length of the wire (x-axis).Also, damping due to collisions is neglected.The basic governing equations are comprised of the continuity equation, the momentum equation, and Poisson's equation as given below [55].
The equation of continuity is, The equation of motion is, The Poisson's equation is, Here we are using the perturbations as Here, n j1 (n j0 ) is the perturbed (unperturbed) density, v j1 (v j0 ) is the perturbed (unperturbed) fluid speed, Φ 1 is the perturbed electrostatic potential, and p j = n j k B T j is the pressure term for jth species.T j is the temperature, k B is the Boltzmann constant, and m * j is the reduced mass of the jth species (m * e for electrons and m * h for holes).The last term of equation ( 2) represents Bohmian force due to Bohm potential (Quantum potential) [56][57][58][59].It was first proposed by Bohm in 1952 [56].
The linearized sets of equations are given by; By assuming a form of plane wave solution exp (ikx − iωt) for the perturbed quantities, where k (ω) is the wave number (angular frequency) of the propagating waves and using equations ( 7) and ( 8), we find n j1 and v j1 as, Substituting the value from equation (11) in equation ( 10), we will get Further simplification of equation ( 12) leads to, where, v tj = kBTj m * j is the thermal speed.From equations ( 13) and (9) we will get, where, ω pj = (4π q j 2 n j0 /m * j ) 1/2 is the plasma frequency.Equation ( 14) is the generalized form of dielectric response function in quantum plasma with nanoparticles which can be further expanded as where, where, v te , v th , v tb and v tn are the thermal speed of the electrons, holes, beam electrons, and nanoparticles respectively.Instability due to streaming of electron beam has been considered in the next section.Quantum effects for the holes and nanoparticles can be ignored as they have higher mass compared to that of electrons.

Streaming instability
To investigate the streaming instability, we assume that the phase speed of waves in plasma system is comparable to that of fluid speed of plasma species.We apply the approxim- which can be re-written as, This can be written again as, where, are the acoustic frequencies of holes, beam electrons, and charge carriers within the nanoparticles respectively.Since the waves in plasma are driven by an electron beam with velocity v b0 , we consider that the phase speed of waves in plasma is comparable to that of electron beam speed, that is ω = kv b0 + δ where δ << kv b0 and kv b0 ∼ Ω nA hence Ω nA = (ω − δ).Replacing Ω nA = (ω − δ) and δ = (ω − kv b0 ) in equation ( 17), we get, Now neglecting δ 4 term since δ is a very small quantity, we will get a cubic equation in δ as, where, Now solving equation ( 18) we will get three values of δ which are as follows, and where, .
The real and imaginary part of δ 2 after replacing ω = kv b0 (since δ << kv b0 ) is given by, Normalizing ω r and ω i in the scale of ω pb gives where, M ′ is the normalized M given by is the quantum correction for fluctuations in density.

Numerical analysis
The analytical results obtained for the real part of ω and the growth rate are evaluated numerically by solving equations ( 19) and ( 20) for streaming instability.The numerical evaluation of the growth rates and angular frequencies are carried out for some typical parameters of InSb semiconductors that have been considered in previous pieces of literature [60][61][62].The typical values of InSb semiconductor taken for the calculations are n 0 = 10 17 cm −3 , m * e me = 0.03, m * h me = 0.41, electron to hole density ratio (η) for our calculation is taken to be 0.92 and the electron to hole thermal speed ratio has been taken to 1.17.
To derive more insight into the behavior of the normalized Mach number we rewrite equation ( 21) as, Further, we rewrite equation ( 23) as β = αξ , where ξ is the multiplicative factor as mentioned in the equation.Using this renaming of the variable we write equation (24) in terms of α and ξ which will give.
From the general rule of the cubic root, the real root amongst the three roots will be positive or negative, depending on the sign of the radical.For practical feasibility, M ′ can have only real and positive values.This will lead to the condition Only the first condition will be applicable since our primary requirement is Using a little algebra and mathematical logic, we can establish that equation ( 26) is only valid when, (i) αξ The first condition (i) imposes a limitation on the values of ξ , which requires that Again, imposing the condition for positive values for the radicand in the square root term in equation (25) we will find that The term ξ is affected by the temperature of the electrons and holes.This is further explored numerically by plotting the normalized values of ω ′ i and M ′ .The effect of temperature is visible in figure 7 where the variation of the thermal speed of electrons and holes has been carried out which is indeed the variation of temperatures of the species.Similarly, the plot of the variation of M ′ with respect to k ′ is shown in figure 8. From both the plots it is plausible that there exists a value of normalized wave number where the real values of ω ′ i and M ′ are not possible.

Results and discussions
A general dielectric response of the semiconductor quantum plasma has been presented using the QHD model and Poisson's equation.Interesting results are derived from the analytical work as seen from the graphical plots given in figures 1-8.From figure 1 we observe that the growth rate increases with the wave number and it depends upon the beam streaming speed significantly.The growth rate is nonlinear.It signifies that unstable waves with shorter wavelengths drive more energy due to streaming.As the streaming speed  increases the growth rate diverges as the wavelength becomes shorter.The increment in the growth rate is sharp initially but a transition in the growth rate is observed at around k ′ = 8 after which it starts going towards saturation.The pure classical case in the absence of quantum effect is plotted in figure 2. It can be seen that when H = 0 the growth rate is sensitive to the value of hole and electron thermal temperature and increases as the temperature ratio increases.It is evident that increasing thermal speed slows down the growth rate compared to the previous case.It may be because of the energy redistribution due to quantum effects.However, the growth rate converges similarly to the previous case when the temperatures of the holes are varied.As the cascading of energy is higher at shorter wavelengths, the graph shows that  the growth rate is significantly larger at shorter wavelengths making them more open to instability at higher values of k ′ .The nature of the pure quantum case is shown in figure 3, where for ultra-low electron-hole temperature, the growth rate increases significantly only when the value of the H parameter is very high.For the lower values of H, the effect is not prominent.This indicates the impact of temperature on growth rates.
Moreover, the nature of nonlinearity is also slightly different and varies with the strength of the quantum effect given by H.In figure 4, the comparison of the growth rate of the classical case with that of the quantum case is depicted.It is visible that with the inclusion of quantum effects, there is a significant change in the value of the growth rate.For the classical case, the growth rate appears to be linearly varying in magnitude when compared with the quantum case.The growth rate with quantum effect increases significantly in the initial phase when k ′ is less than 9 and then attains a linear trend similar to that of the classical case but with a higher magnitude.
The implantation of boron ions affects the stability of the semiconducting plasma system as it reduces the free energy of the system as reported for carbon implanted TiO 2 semiconducting systems [47].The growth rate of instability thus decreases as the boron ion is implanted in the system.This analogy is very well in agreement with the normalized growth rate versus the wave number plot in figure 5 where the growth rate is decreased for the boron ion implanted system.The growth rate further decreases as the concentration of implanted boron ions is increased from 4% to 6%.The effect of boron ion implantation is also visible in the normalized growth rate versus wave number plot in figure 6, where the gap in the growth rate can be seen only after the incorporation of boron ions in the system, which further increased as the concentration of boron is increased in the system.As the growth rate in the plasma system has been successfully measured by Zhang et al [63], experimental verification for measuring the growth rate may be realized for our predictions in the near future.
The striking result derived in this work is in the behavior of instability with respect to the temperatures of electrons and holes (figures 7 and 8).It is observed that for v ′ th 0.051 to 0.055 and k ′ nearly from 5 to 15 growth rate is disrupted.It may be conjectured that in this range of thermal speed, waves do not derive energy from the streaming electron beam and are not excited or waves may be damped.Calculations from the equation of M ′ show that the disruption in the growth rate and Mach number lies for ξ > 108 α 2 1/3 and for the value of k ′ nearly from 5 to 15.These ranges vary as we change the  thermal speed of the electrons and holes distribution which depends upon their respective temperatures.It is evident that the instability is excited, but forms a kind of break or a gap for a particular range of v ′ th = 0.051 to 0.055.It is observed that when v ′ te and v ′ th are varied, for a threshold value of v ′ th = 0.051, the disruption in the growth rate gets triggered.The gap starts shrinking as the value of v ′ th is increased.Such an effect is only visible when we vary the temperature distribution.

Conclusion
The effect of temperature on streaming instability in a semiconducting quantum plasma system implanted with nanoparticles and injected with a streaming beam of electrons has been studied.The nature and rate of instability vary by including the quantum effects.The novelty in this work lies in the analytical and numerical decryption of disruption in the instability related to the thermal speed ratio of the electronhole distribution.It is plausible that instability gets smeared in a certain range of temperatures such that energy transfer to the waves does not take place, thereby avoiding the condition of instability.Hence, it can be speculated that this particular range of temperature may act as a safe zone for the functioning of the semiconductors in ultra-cold temperatures.The implantation of nanoparticles modifies the magnitude of the growth rate.The gap in the growth rate was seen only after the implantation of nanoparticles and was found to be increasing with increasing implantation concentration.It will be interesting to see if such disruption is present in other types of instabilities such as modulational instability, parametric instability, lower hybrid instability, and upper hybrid instability with variable temperatures.This may be another piece of interesting work to be carried out in the future course of this work.
The linear analysis of the growth rate analysis may not be comprehensive enough to understand the mechanism of the cascading of energy from the longer waves to the shorter waves.More physics may be deciphered of the instability process by analyzing nonlinear mode analysis, the wavewave, and wave-particle interaction using the kinetic approach that can be studied as a continuation of this work in the future.
2A − 4α, which gives, √ B 2 − A 2 ⩾ B which leads to the conditions that B ̸ = 0 and (B + A) (B − A) > 0. From the second condition, we may get two more additional conditions as either B < −A, B < A or B > −A, B > A. From the first condition, B must be negative and |B| < |A|, and from the second condition B must be positive and |B| > |A|.

Figure 1 .
Figure 1.Variation of ω ′ i vs k ′ for different beam streaming speed v bo = 0.02ω pb , 0.03ω pb , and 0.04ω pb with v ′ th =0.3 and η = 0.92 for all.