Computer simulation of high-field electron emission from crystalline and nanostructured silicon dioxide
V. Kortov, S. Zvonarev
Ural State Technical University, Ekaterinburg, Russia
Abstract.A physical model of the electron emission from charged dielectric films and near-surface layers of monocrystalline dielectrics is discussed.
The model assumes that free electrons are produced by thermal detrapping via multiphonon processes and tunneling. The free electrons moving to the surface in the field of the builit-in charge interact with optical and acoustic phonons. When the electric field strength is high an important role is played by inelastic electron scattering such as impact ionization and cascade processes. The model has been modified considering specific features of nanoscale materials. It takes into account processes involved in scattering of electrons on nanocrystal boundaries and the increase in the energy depth of surface trapping centers of charge carriers.
The physical model has been implemented in an algorithm, and a Delphi program has been developed for computing, electron emission. Emission from nanostructured and single-crystal silicon dioxides is simulated. It is shown that nanostructured SiO2 is characterized by a lower emissivity, a wider emitted-electron energy spectrum, and a higher dielectric strength, as compared to single crystals.
Introduction
Electron emission from crystalline and amorphous thin SiO2 films and near-surface layers exposed to an electron beam is the subject of extensivestudies. An electron beam charges the dielectric surfaceand generates a high-strength electric field in the near-surface layer. The motion of detrapped electrons to thesurface in the strong electric field leads to the hot-electronemission and, under certain conditions, to the electricalbreakdown[1]. Theoretical and experimental investigations show that the breakdown threshold field forthin SiO2 films is 10 to 16 MV/cm [2-3].
Nanostructured SiO2 is currently considered as amaterial with wide potential applications in micro- andoptoelectronics. Nanostructured SiO2 films are alsopromising as host materials for electroluminophoreswith a high quantum yield. To cause electroluminescenceof such films by impact ionization, an external electricfield of 6 to 7 MV/cm strong must be applied [4].
A physical model describing basic processes underlying the motion of an electron to the surface was modified in our previous studies dealing with simulation of the transport and the emission of electrons in irradiated crystalline dielectrics. Specifically, the rates of the electron scattering by optical and acoustic phonons were refined and processes of impact ionization and cascading were taken into account [5-7]. The present study deals with the substantiation of the electron transport model considering processes that take place in charged crystalline and nanostructured dielectrics. The use of the model for computer simulation of the electron emission in high-strength electric fields by the example of crystalline and nanostructured silica is discussed too.
Physicalmodel
The model has been developed for calculating the electron emission from charged near-surface layers in dielectrics exposed to electrons having the energy of (1-10) keV. The surface of dielectrics is charged positively under electron bombardment, while at some depth, which depends on the electron energy, trapping of electrons results in formation of a negative charge. A "plus"-"minus" structure and a strong electric field as high as a few megavolts [8] appear in the near-surface layer. Trapped electrons can be detrapped by the tunneling effect, thermo- and photoionization in the electric field. Their starting energy depends on the field intensity. Electrons drift to the surface and are scattered by acoustic or optical phonons depending on their energy; they also participate in processes of impact ionization and cascading. Some electrons can reach the surface, surmount the energy threshold, and be released to vacuum. The thermally stimulated electron emission current and the electron energy distribution are measured in experiments and the measured values can be compared with the corresponding calculation data.
Formation of free electrons
Electrons are detrapped via mechanisms of multiphonon ionization in an electric field [9]. In this case, an electron acquires an initial energy determined from the formula
,(1)
where is the electric field intensity, is the electron charge, is the effective mass of an electron, is the tunneling time of the nuclear subsystem, which is calculated from the formula [10]:
(2)
In the formula (2) the parameter means the electron-phonon coupling constant, which depends on the ratio between the optical and the thermal depth of the trap, is the Boltzmann constant, is the temperature, and is the local oscillation frequency of the center of the subsystem.
Scattering of electrons by longitudinal optical phonons
Electrons, which become detrapped by tunneling or ionization, move to the surface of dielectrics and interact with phonons. The scattering by longitudinal optical phonons influences the transport of electrons having the energy of fractions of an electron-volt to a few electron-volts. The rate of scattering of an electron having the energy E by optical phonons is calculated in terms of the Fröhlich theory [11]:
,(3)
where the signs (+) and () denote the generation and the annihilation of an optical phonon respectively; o, and are the absolute, optical and static permittivity respectively; ћLO is the energy of optical phonons; E is the electron energy; nLO is the Bose distribution of the number of phonons over modes ћLO at the crystal lattice temperature T, which is calculated from the formula [12]:
,(4)
Scattering of electrons by acoustic phonons
The scattering of electrons by acoustic phonons begins dominating when the electron energy is units to tens of electron-volts. The rate of scatteringby acoustic phonons depends on the energy at the edge of the Brillouin zone EBZ [13]:
when,(5)
when,(6)
where C1 is the deformation potential constant; is the dielectric density; CS is the sound velocity; M is the mass of the heaviest atom in a unit cell; nBZ is determined from the formula (4) at the acoustic phonon frequency BZ;N is the concentration of lattice atoms determined as:
,(7)
where is the coupling constant and is the phonon wave-vector.
Thesoundvelocityiscalculated considering three (two transverse and one longitudinal) branches of the acoustic phonon spectrum [13]:
,(8)
where CT and CL are the transverse and longitudinal velocities of the sound respectively.
Impact ionization
When moving to the surface, electrons acquire some energy on account of electron-phonon interactions and acceleration in the electric field. As a result, their energy can be as high as the ionization energy, which is found from the relationship [14]:
,(9)
In the formula (9) the effective mass of the valence band hole and the conduction band electron depend on the electron mass at rest : and respectively; is the forbidden gap width.
When the required energy is reached, electrons begin participating in the impact ionization. In this case their scattering rate is [14]:
when, (10)
where is the impact ionization coupling constant, is the impact screening parameter, and is a constant.
Cascading
When involved in the impact ionization, drifting electrons can generate new electrons by cascading. The probability that a secondary electron is formed depends on the electron drift length z and the electric field intensity F [15]:
, (11)
where and are constants, which depend on the material. In SiO21/cm and V/cm [15].
The electron drift length was determined as the projection on the normal to the surface of the mean free path l after each electron-phonon or electron-electron interaction taking into account the scattering angle. The length passed by an electron between interactions was calculated by the formula
, (12)
where is the time interval between interactions.
The initial energy of a cascading-generated electron depends on the energy loss by the primary electron during the impact ionization. The minimum energy loss by an electron required for generation of another electron during the impact ionization is determined by the forbidden gap width and equals eV for SiO2. If their initial energy is sufficiently high, secondary electrons can be accelerated in the electric field and generate so-called tertiary electrons, leading to the avalanche formation of electrons. Cascading of electrons is time-limited and lasts for about 10-14 s [16].
Energyandangularscatteringofelectrons
Electron trajectories were calculated by the Monte Carlo method with simulation of at least 10000 histories of free electrons detrapped by the action of different mechanisms.The type of the interactionwith phononswas determined using the Monte Carlo method with the random number generation.
Because of the scattering by phonons and the acceleration in the electric field, the electron energy E changes after each electron-phonon interaction according to the formula [17]:
, (13)
where lj is the free path; j is the scattering angle; is the phonon frequency. In this case, the electron energy increases and decreases due to the phonon annihilation and generation respectively. If the electron does not interact with a photon, its energy is influenced by the electric field only.
If the electron is scattered by phonons, not only the electron energy, but also the electron direction changes after each electron-phonon interaction. The angle between the electron directions before and after its interaction with a phonon is calculated from the formula [17]:
, (14)
, (15)
where E' is the electron energy after its interaction with a phonon and R is a random number from 0 to 1.
The angle between the electron direction and the normal to the surface is determined as
, (16)
where i is the interaction number and is the isotropic azimuthal scattering angle calculated from the formula.
Trapping of electrons by holes near the surface
It was already noted that the charge distribution in depth of the near-surface layer has a "+/-" structure. For this reason, an electron passes a region of a positive charge as it moves to the surface. Holes, which are located near the surface, can capture drifting electrons. The probability that electrons are captured by holes between electron-phonon collisions during their motion in the zone of the positive charge is defined by the relationship [18]:
, (17)
where , is the volume density of the positive charge, and is the trapping cross-section.
The probability that electrons are captured by holes in the entire zone of the positive charge should be taken into account by the following formula:
(18)
Release of electrons to vacuum
An electron, which reaches the surface, can escape from the surface if its energy is larger than the potential barrier taking into account the scattering angle :
, (19)
In this case, the energy of the electron release to vacuum is calculated from the formula:
, (20)
where Ei is the electron energy after the interaction.
Specific features of the electron transport in nanostructures
Processes of the electron transport in nanostructured dielectrics should be simulated taking into account the change of the phonon spectrum of nanoparticles, quantum limitations of the electron free path, porosity and other factors typical of the nanostructured state. However, of greatest importance in nanomaterials are processes involving the passage of electrons through numerous boundaries between nanoparticles. In this connection, the physical model used for calculation of basic processes of the electron transport in crystals was modified considering specific features of nanostructured dielectrics.
The objects of study were model nanostructures with crystals 5 to 20 nm in size. The structure of hexagonal crystalline nanoparticles with closely fitting faces was analyzed. Analogous opal-like thin-film structures were prepared in experiments (19). With the chosen model nanostructure, it was possible to disregard the porosity and use, as the first approximation, known parameters of SiO2 crystals necessary for calculation of electron-phonon interactions. Also, the model could be conveniently used for estimating the contribution from processes involved in crossing of nanocrystal boundaries by moving electrons.
The model and the calculation algorithm took into account that a free electron could be produced during photoionization either in the bulk or at the boundary of a nanocrystal located in the emission layer. As the electron was moving in this layer, its energy changed not only upon the interaction with phonons, but also upon crossing the nanocrystal boundaries. A potential barrier at the boundary of a nanoparticle can be surmounted if the electron energy Ei after the i-th electron-phonon interaction is larger than with account taken of the scattering angle (see formula (19)).
Having crossed the boundary, an electron can continue moving, with the scattering by phonons and penetration through boundaries of other nanocrystals. The motion stops when the electron becomes thermalized (E = 3/2kT) and is captured in a trap. If the energy is insufficient for surmounting the barrier at the nanocrystal boundary, the electron reflects from the boundary, is scattered by phonons, and is thermalized.
Resultsofsimulationanddiscussion
An algorithm was constructed and a program in the Delphi language for calculating the electron emission was written on the basis of the physical model described above.
The concentration of detrapped electrons diffusing to the surface is controlled by thermal ionization and tunnelingprocesses. Figure 1 shows free-electron concentrationsas a function of the electric field strength for single-crystaland nanostructured SiO2. The contributions of the electron detrapping mechanisms mentioned above to thefunction corresponding to nanostructured silicon dioxide are also shown as an illustration. In relatively weakfields (less than 2 MV/cm), free electronsare produced mainly by thermal ionization, and theirconcentration gradually increases with the electric field.This result is explained by the Poole–Frenkel theory,which predicts that a strong electric field reduces theenergy depth of a trap. As the field strengthexceeds 2.5 MV/cm, tunneling begins to contribute toelectron detrapping. The concentration of tunnelingelectrons increases exponentially with the field strength and thermal ionization can be neglected at 5 MV/cm.
Fig. 1. Concentration of free electrons generated in the near-surface layer vs. the electric field strength: ( 1 ) single-crystalSiO2 ; ( 2 ) nanostructured SiO2 . Curves for the nanostructured material correspond to different mechanisms: ( 3 ) tunneling detrapping; ( 4 ) thermal ionization.
The transport of electrons in bulk crystalline and nanostructured samples of SiO2 was studied by the Monte Carlo simulation of N = 10000 histories of free electrons, which had the initial energy of 4 eV after photoionization of traps. The parameters necessary for calculation of electron-phonon interactions at T = 300 K were assigned universally adopted values [11, 14]. The potential barrier at the boundaries between nanocrystallites was taken equal to =0.1 eV.
Electron trajectories in crystalline and nanostructured SiO2 with particles 20 nm in size were calculated. It was assumed that free electrons started at a depth of 50 nm. Some typical electron trajectories are shown in fig. 2. It is seen that the electron trajectories in the crystal and the nanostructured sample are considerably different. As electrons are moving in the crystal, they go through numerous interactions with phonons before their thermalization and move to a distance of more than 100 nm from the start point. The electron trajectories in the nanostructured sample are shorter. The electrons get thermalized after they have crossed 3-5 boundaries between nanocrystals. Therefore, the distance to their start point is not over 30 nm.
Fig.2. Trajectories (1, 2, 3) of electrons in nanostructured (a) and crystalline (b) silica.
The energy spectrum of electrons emitted from crystalline silicon dioxide was calculated taking the following parameters: the electron start depth of 30 nm; the electric field intensity of 0.6 MV/cm; the electron affinity on the surface of the sample equal to 0.5 eV; the initial energy of electrons at the start equal to kT; and the temperature of 300 K (Fig. 3a). In experiments the electron energy was measured by the retarding potential method. It is seen that the calculated and experimental spectra of the emitted electrons are in satisfactory agreement, pointing to the validity of the constructed physical model.
The energy spectrum of the released electrons was also calculated for nanostructured silicon dioxide with particles 20 nm in size (Fig. 3b). The calculation parameters were as follows: the electric field intensity of 1.9 MV/cm; the electron affinity of 0.1 eV; the potential barrier at the nanoparticle boundary taken equal to 0.1 eV; the initial energy of electrons at the start equal to kT; and the temperature of 300 K. Contrary to the crystalline sample, the energy distribution in nanostructured SiO2 had a pronounced "tail" of the high-energy component. This result can be explained by specific features of the transport and the emission of electrons in these samples, which are disregarded in the proposed model. The calculated energy spectrum is similar in its shape to the Maxwell distribution. The calculated values of the mean and the most probable energy of the released electrons are in good agreement with the experimental data.
Fig.3.Energydistribution of electrons emitted fromcrystalline (a) and nanostructured (b) silicon dioxide.
Cascading processes significantly contribute to the electron emission from near-surface layers of dielectric materials in high electric fields. Figure 4 shows the concentration () of free electrons produced by cascading processes as a function of the field strength computed for the single-crystal and nanostructured samples. The electron concentration in the avalanche region produced by cascading processes is higher for the single crystal as compared to the nanostructured material, even though the cascading thresholds are similar for both structures. This is primarily due to the higher free-electron concentration at the same field strength for crystal as compared to the nanostructure. The results obtained in this study demonstrate that the dielectric strength of the nanostructured material is higher than that of the crystal. According to our simulations, the breakdown threshold field for single-crystal SiO2 isapproximately 10 MV/cm, which agrees with the literature data [2,3]. For nanostructured SiO2, the breakdown threshold field is estimated for thefirst time and is found to be approximately 12 MV/cm.Note also that the slope of the electron concentration inthe avalanche region as a function of the field strength islower in the latter case as compared to a single crystal.