Spontaneous and piezoelectric polarization in the nitrides is analyzed. The slab model was designed and proved to be appropriate to obtain the spontaneous polarization in AlN, GaN and InN. The spontaneous polarization and polarization related electric fields in AlN, GaN and InN were determined using DFT slab calculations. The procedure generates single value of spontaneous polarization in the nitrides. It was shown that Berry phase polarization may be applied to determination of spontaneous polarization by appropriate addition of polarization induced electric fields. The electric fields obtained from slab model are consistent with the Berry phase results of Bernardini et al. The obtained spontaneous polarization values are: 8.69*10-3 C/m2, 1.88*10-3 C/m2, and 1.96*10-3 C/m2 for AlN, GaN and InN respectively. The related Berry phase polarization values are 8.69*10-2 C/m2, 1.92*10-2 C/m2, and 2.86*10-2 C/m2, for these three compounds, respectively. The GaN/AlN multiquantum wells (MQWs) were simulated using ab intio calculations. The obtained electric fields are in good agreement with those derived from bulk polarization values. GaN/AlN MQWs structures, obtained by MBE growth were characterized by TEM and X-ray measurements. Time dependent photoluminescence measurements were used to determine optical transition energies in these structures. The PL obtained energies are in good agreement with ab initio data confirming overall agreement between theoretical and experimental data.

Macroscopic polarization of crystalline materials is a physically important property of infinite systems such as solids1,2 and finite systems including molecules and nanoobjetcs.3 Polarization arises from luck of mirror symmetry in the crystal lattice, allowing for the existence of a material vector property. In the case of the solid it is therefore expected that microscopic definition of the property is formulated, in direct relation to the atomic lattice structure. Accordingly to the standard textbooks, the polarization is defined as dipole of a unit cell, divided by its volume.1,4 While in the case of finite systems so defined property is recognized as a well-defined quantity, in case of infinite systems, the polarization value depends on the choice of boundary conditions.5–7 It was noted that such quantity depends on the selection of the simulation volume so it could not be used to determine the physical property of the matter. The statement was connected to the surface term, which affects its value.8–12 Direct modification, such as using of large volume, does not remove this deficiency.2 It is not clear whether additional requirement of independency of the selection of simulation volume could be applied to determination of the polarization, based on its definition.13–15 

Macroscopic polarization is technically important physical property of pyro- and ferro-electrics, often used in many important technological applications.16 Natural consequence of polarization is an electric field in the crystal interior, affecting functionality of advanced electronic and optoelectronic devices. The built-in electric fields affect energies of quantum states that is known as Stark effect. The electric field in quantum structures changes energy of quantum states, giving rise to phenomenon known as Quantum Confined Stark Effect (QCSE).17 In addition, the strain induced fields may contribute to this effect significantly, especially in strained quantum low dimensional structures frequently used in modern devices that are built on polar nitrides (0001) surface.17–20 

A mere change of the energy of quantum states could be either beneficial or harmful; a really detrimental effect is spatial separation of electrons and holes that are shuffled to the opposite ends of quantum wells.21–23 Spatial separation reduces an overlap of the hole-electron wavefunctions, their radiative recombination rates and lowers efficiency of photonic devices.21,24–26 The negative influence of QCSE may be enhanced by Auger recombination or carrier leakage at high injection currents.27–31 That could lead to decrease of the device efficiency for higher injection currents, the phenomenon nicknamed as “efficiency droop”.32 A harmful influence of QCSE for optoelectronic devices is in turn beneficial contribution in electronic devices based on two-dimensional electron gas (2DEG), such as field effect transistors (FET’s) or molecular sensors. Electric field at AlN/GaN heterostructures stabilizes 2DEG leading to high carrier mobility which may be used for construction of fast electronic devices. The electric field, induced by dipoles of molecules attached to the surface, may contribute to the sensitivity of molecular sensors which opens new applications of such devices.

Polarization related electric fields are therefore increasingly important in the device technology. The currently used approach divides polarization into the contribution from ionic and delocalized charge allowing to calculate the change of polarization only.9,10 The ionic part may be calculated directly, the delocalized contribution may be obtained using Berry phase formulation.5–10 

In the present paper the extensive investigation of the polarization is made, starting from analysis of the difference between polarization obtained by Berry phase approach and the polarization fulfilling boundary conditions at finite volume. The approach based on direct slab ab initio simulations is used to obtain spontaneous polarization for bulk wurtzite AlN, GaN and InN. The slab and Berry phase fields and polarizations are compared. The polarization related field in semiconductor superlattice is obtained from dielectric and minimal electric field energy formulations. The ab intio simulations are directly applied to GaN/AlN MQWs structure, and the derived electric fields are compared with those obtained from spontaneous polarization formalism. MBE grown GaN/AlN MQWs are characterized by TEM and X-ray measurements. Time dependent photoluminescence (PL) measurements were used to obtain energies of quantum states in MQWs structure. The results of theoretical calculations and experimental data are compared and the conclusions are drawn.

In the calculations reported below the two different ab initio packages were employed: Vienna Ab-initio Simulation Package (VASP),33–35 and Spanish Initiative for Electronic Simulations with Thousands of Atoms (SIESTA).36–38 The VASP basic code expressed many-body wavefunction as liner combination of plane waves which was terminated by the energy cutoff of 29.40 Ry (400.0 eV). Lepkowski and Majewski, has shown that the selection granted results sufficient for precise simulations of GaN elastic properties.39 The Monkhorst-Pack grid (7x7x1) was used for k-space integration.40 In order to speed-up calculations for Ga, Al, In and N atoms, the Projector-Augemented Wave (PAW) pseudopotentials for Perdew, Burke and Ernzerhof (PBE) exchange-correlation functional were used in Generalized Gradient Approximation (GGA).41–43 As standard for nitride systems, gallium 3d and indium 4d electrons were included in the valence band. The energy error criterion for the termination of electronic self-consistent (SCF) loop was set equal to 10-6. The ab-initio lattice constants: a = 3.195 Å and c = 5.206 Å for GaN, a = 3.112 Å and c = 4.983 Å for AlN, a = 3.563 Å and c = 5.756 Å for InN are in good agreement with x-ray data (a = 3.189 Å and c = 5.185 Å for GaN, a = 3.111 Å and c = 4.981 Å for AlN and a = 3.537 Å and c = 5.706 Å for InN).44–47 

The second, SIESTA code employs norm conserving pseudopotentials. The many electron wavefunction is expanded in a series of the numeric atomic orbitals, based an atoms that have finite size extent which is determined by the user.36–38 The pseudopotentials for Ga, Al, In and N atoms were obtained from all-electron calculations implemented in the ATOM program provided by the software authors.48 Gallium 3d and indium 4d electrons, included in the valence electron set, were represented by single zeta basis. For s and p type orbitals, more extended triple zeta basis set were used. The whole aluminum atom basis set was represented by a triple zeta function. Integrals in k-space for slabs were performed using (5x5x1) Monkhorst-Pack grid. The minimal equivalent of plane wave cutoff for the grid was set to 275 Ry. A convergence criterion for terminating SCF loop, the maximum difference between the output and the input of each element of the density matrix was equal or smaller than 10-4. VASP and SIESTA codes use the reverse charge convention in which the total electric potential is the electron energy.49 

PBE density functional needs refinement to obtain the correct band gap of nitride semiconductors. Optimal approach was proposed by Ferreira et al.,50 in two variants known as local-density approximation or generalized-gradient approximation half-electron technique (LDA-1/2 and GGA-1/2), which approximately includes self-energies of excitations in semiconductors, providing band gap energies, effective masses, and band structures in good agreement with experimental values.51 Present results were obtained by applying a modified GGA-1/2 correction to structures in which positions of atoms and a periodic cell were first relaxed to an equilibrium state by the use of the PBE approximation for exchange-correlation functional. In the geometry optimization, all atoms were relaxed until the force on each single atoms was lower than 0.005 eV/Å. Spin–orbit effects were neglected in these calculations, because high-lying valence band states lead to a small splitting (of the order of 10 meV). The DFT band gap energies of bulk AlN, GaN and InN were Eg(AlN) = 6.16 eV, Eg(GaN) = 3.47 eV, and Eg(InN) = 0.79 eV, respectively. Thus, a reasonably good agreement with low-temperature experimental data was obtained: Eg (AlN) = 6.09 eV52 Eg (GaN) = 3.47 eV53,54 and Eg(InN) = 0.70 eV.55 

The thick superlattice samples, consisting of 40 periods of GaN/AlN MQWs, were deposited by plasma-assisted molecular-beam epitaxy (PAMBE) on 1-μm-thick (0001)-oriented AlN-on-sapphire templates. In addition the GaN/AlN MQWs were capped by 30 nm thick AlN overlayer, which did not hamper optical measurements. The GaN QW width was systematically increased from 1 nm up to 6 nm, with the same the AlN quantum barriers (QBs) 4 nm thick.56 Deposition rate determined by the active nitrogen flux, was 0.32 monolayer per second (ML/s), at the substrate temperature was ∼720°C, as estimated from the Ga desorption time. The structure was grown uninterrupted under self-regulated Ga-rich conditions.57,58 To populate the ground state in the conduction band, each GaN layer was doped with Si to a concentration of 1.3 ×1019cm-3.

High Resolution X-Ray Diffraction (HRXRD) was used for characterization of the sample structure. Measurements used Philips X’Pert MRD X-ray diffractometer operating at the Cu_Kα1 wavelength, equipped with fourfold Ge(220) monochromator, a threefold Ge(220) analyzer and X-ray mirror. For each sample two types of measurements: a 2Θ/ω scan of 0002 symmetrical reflection and a Reciprocal Space Map (RSM) of the (-1-124) asymmetrical reflection, were obtained.

The 2Θ/ω scans of the (0002) reflection measured for each sample indicated that the designed AlN/GaN MQW thicknesses are consistent with the growth assumptions. The measured scans and simulation of the diffraction pattern by Epitaxy software provided an estimate of the quality of the sample.

The in-plane lattice parameter a of the GaN/AlN MQWs was extracted from the RSMs of the (-1-124) reflection. As shown in Fig. 1, with increasing GaN QW width the MQW in-plane lattice parameter a increases from values close to AlN to values close to GaN lattice parameter a. The trend roughly follows Vegard’s law, i.e. the a lattice parameter corresponds to that one of an AlGaN alloy with the average Al mole fraction derived from the relative widths of AlN and GaN layers in MQW structure.

FIG. 1.

In-plane lattice parameter a of the GaN/AlN MQWs as a function of the GaN well thickness. The thickness of the AlN barrier was 4 nm in all the cases. The blue circle and the red diamond represent the lattice parameter of bulk GaN 45 and bulk AlN,44 respectively. The line represents Vegard’s law obtained for average composition of the superstructure.

FIG. 1.

In-plane lattice parameter a of the GaN/AlN MQWs as a function of the GaN well thickness. The thickness of the AlN barrier was 4 nm in all the cases. The blue circle and the red diamond represent the lattice parameter of bulk GaN 45 and bulk AlN,44 respectively. The line represents Vegard’s law obtained for average composition of the superstructure.

Close modal

The XRD results prove that the structures are relaxed to the lattice parameter that corresponds to the average composition of the structure, in good agreement with Vegard’s law. The solid line in Figure 1 is a fit to the following equation:

a(zGaN)=3.111+0.0105zGaN5105zGaN2=3.111+2.6103(zGaNzAlN)1.25105(zGaNzAlN)2
(1)

where the latter dependence is formulated for molecular fraction assuming the thickness of the barrier zAlN=4nm. The relation follows the linear Vegard’s dependence, with nonlinearity ≤ 2%. Thus, it can be assumed that the thickness of the structure is sufficient to obtain the average lattice parameter confirming full relaxation of the structure, i.e. the influence of the substrate is removed almost entirely. The crystallographic quality may be assessed from TEM data which are presented in Figure 2 for the QW thickness equal to 4 (a, b) and 6 (c, d) nm.

FIG. 2.

Cross-sectional HAADF STEM images of AlN/GaN MQWs, showing the arrangement of GaN QWs with various thicknesses – (a,b) 4 nm and (c,d) 6 nm.

FIG. 2.

Cross-sectional HAADF STEM images of AlN/GaN MQWs, showing the arrangement of GaN QWs with various thicknesses – (a,b) 4 nm and (c,d) 6 nm.

Close modal

As shown in Figure 2, the structures have very good crystallographic quality, however some dislocations and thickness changes of one monolayer can be observed.

Polarization in a finite size system was determined using its definition, which related this quantity to electric dipole.1 An extensive review of polarization in finite size systems was presented by Noguera and Goniakowski.3 Nevertheless, such approach could not be used to infinite system directly, as it was pointed out, that the magnitude and the direction of the polarization may be changed by mere change of the simulated volume.11–14 This deficiency was resolved by Resta who has used periodic system to obtain the boundary independent expression based on geometric phase.5,6 The obtained result, was determined modulo preselected value, thus the polarization was defined not as a single vector, but as claimed by Spaldin, a lattice.60 Thus an infinite number of the values could be associated with the polarization in strictly infinite system.

In the following we define spontaneous polarization in finite system following Landau and Lifsic definition as an electric dipole density, i.e. electric dipole divided by the volume.1 The infinite system in condensed matter physics is always considered in the thermodynamic limit, i.e. the finite system with its size going to infinity. The spontaneous polarization is defined as the polarization in the system in absence of any external electric field in the thermodynamic limit, i.e. of the infinite large size. Nevertheless, the vacuum outside exists as it is always present in the limit procedure and that factor has to be accounted explicitly in the determination of the spontaneous polarization and the related field values.

Resta approach was subsequently used by Bernardini et al. in their ab initio determination of polarization of nitrides.19 Using the zinc blende phase as a reference zero polarization system they obtained two slightly different values of polarization of the nitrides. Potentially, ab initio calculations may be used in direct determination of the electric fields in bulk nitrides. Nevertheless as argued by Vanderbilt and King-Smith, the field contribution arising from polarization and surface charge could not be distinguished using geometric analysis only.9 Thus, the surface charge directly contributes to the electric field affecting its value. In the following we will show how to remove the surface charge contribution to obtain polarization in the bulk from surface slab calculations.

Determination of electric field from Berry phase polarization is not straightforward because Berry phase procedure requires electric field to vanish.5 Therefore the following two relations have to be fulfilled:

{Pi(0)0Ei(0)=0
(2)

In the above relations and in the following we use scalar notation showing the vector components along the polarization axis only, whereas the perpendicular components are all zero. By superscript ‘(0)’ we denote the fields corresponding to Berry phase state. Any physically sound condensed matter system is finite, thus the system boundaries have to be accounted for. Thus it has to be assumed that the polar medium is surrounded by vacuum. In absence of the external electric fields and of the free charge, the electric displacement vector D vanishes in the whole space, i.e. in the slab and in the vacuum (i = slab, vac)1,2 i.e.

Di(pol)=εoEi(pol)+Pi(pol)=0
(3)

which is in contradiction with Eq. 2. Therefore, the Berry phase state is efficient procedure which could be used for calculations but the resulting polarization values cannot be applied directly because it could not be realized physically. It remains in contradiction with the continuity of electric displacement vector at the vacuum-slab interface. In order of attain the physically realized state, the supplemental electric field in the slab has to be added. Berry state corresponds to spontaneously polarized medium with the external field such that the field inside the medium vanishes. As in Berry state the field is zero the correction field, emerging due to the external field, is equal and it has the opposite direction to the spontaneous polarization induced field Eslab(pol). The field additionally changes the polarization of the matter according to material susceptibility χ, therefore the spontaneous polarization Pslab(pol) is equal to

Pslab(pol)=Pslab(0)+εoχEslab(pol)=Pslab(0)+εo(ε1)Eslab(pol)
(4)

Combining Eqs. 3 and 4, the value of spontaneous polarization electric field may be obtained as:

Eslab(pol)=Pslab(0)εεo
(5a)

and the spontaneous polarization is given as:

Pslab(pol)=Pslab(0)ε
(5b)

Naturally, the spontaneous polarization is smaller than Berry phase polarization as the induced electric field is directed opposite to the polarization that decreases the polarization value. All these above values are listed in Table I. From these it may be noted that polarization of AlN is much higher than its values for both GaN and InN.

TABLE I.

Group III nitrides polarization obtained from Berry phase approach P(o), the spontaneous polarization P(pol) and the related electric field E(pol).19 Static dielectric permittivity was assumed to be: 10.31, 10.28 and 14.61 for AlN, GaN and InN respectively.

Berry phase polarizationSpontaneous polarizationSpontaneous polarization
Compound Ref. 19 P(0) (in C/m2)P(pol) (in C/m2)related electric field E(pol) (in V/Å)
AlN 0.081 0.0079 8.87·10-2 
AlN 0.090 0.0087 9.86·10-2 
GaN 0.029 0.0028 3.19·10-2 
GaN 0.034 0.0033 3.75·10-2 
InN 0.032 0.0022 2.47·10-2 
InN 0.042 0.0029 3.25·10-2 
Berry phase polarizationSpontaneous polarizationSpontaneous polarization
Compound Ref. 19 P(0) (in C/m2)P(pol) (in C/m2)related electric field E(pol) (in V/Å)
AlN 0.081 0.0079 8.87·10-2 
AlN 0.090 0.0087 9.86·10-2 
GaN 0.029 0.0028 3.19·10-2 
GaN 0.034 0.0033 3.75·10-2 
InN 0.032 0.0022 2.47·10-2 
InN 0.042 0.0029 3.25·10-2 

Note that in linear regime the susceptibility formula for polarization is:

P=P(o)+εoχE=P(o)+εo(ε1)E
(6)

where the reference value is Berry polarization P(o). The above relation is useful for derivation of the fields in well-barrier superlattice, and such will be used in Appendix  B.

As indicated, direct determination of polarization in infinite systems, based on periodic boundary conditions, suffers from fundamental difficulties. In this manuscript, we use the direct determination of polarization in slabs as proposed by Ramprasad and Shi for the calculation of the dielectric properties of HfO2 based on nanoscale thick slabs.61,62 Using this approach they were able to extract the bulk component of polarization of HfO2, which was then used for the calculation of the dielectric permittivity of this material. They distinguished electronic and lattice contributions to the dielectric constant for various slab sizes. The large size limit of total polarization of HfO2 was obtained. Subsequently, they obtained the dielectric permittivity of other systems, such as Cu-phtalocyanine,63 and also several Si based single-component (Si, polymer, SiO2) and two-component (Si-SiO2, polymer-SiO2) systems.64 

In the present investigations we will also use a finite-size slab system. The recently developed surface modeling techniques allow us to verify the surface charge by direct simulations of slab representing AlN, GaN and InN.65 By appropriate termination of both sides of the slab, the surface states are shifted to valence or conduction band, so the surface could be set electrically neutral. Naturally, in the case of a polar system, the additional contribution from polarization may introduce additional field which will affect bands, skewing them which finally, in the case of very thick slabs, closes the gap and introduces screening by band charge i.e. by electrons and holes.66 The band skewing may be avoided by application of additional external field. Therefore successful modelling of separate surface charge and polarization effects requires careful design of the electronic and geometric properties of the slab, as it is explained below.

As we are not interested in the investigations of surface structure, the simulations employed 1 x 1 slab of different thickness, i.e. the number of Me-N (Me = Al, Ga or In) double atomic layers (DALs). In the simulations the band-gap correction was used to obtain proper values for the three nitrides.52–54 In order to separate surface and polarization effect, two nitride systems, polar and polarization-free, were used. The structure of both systems has to be very close so that the electronic states associated with the surface states should be identical or close to. The two natural candidates are a cubic slab with MeN{111} faces and a wurtzite slab with MeN{0001} faces. These slabs differ by stacking only, which translates on the rotation of the third DAL, affecting the surface states only slightly.

First, the nitride cubic lattice slab was considered. The slab represent periodic two-dimensional nitride layer parallel to (111) surface. In Figure 3 the band diagrams of three cubic nitride (AlN, GaN and InN) slabs are presented. The electronic temperature for these calculations was 300 K, which is sufficiently low to avoid screening by the band charge arising due to interband excitations. The band diagram shows no presence of surface states in the bandgap. Accordingly, the Fermi level is not pinned at surfaces, both surfaces are not charged (effective surface charge is zero) and the bands are almost flat. Naturally, the electronic charge density associated with the surface states extends into the slab interior. As it was shown by Kempisty et al., the surface states decay exponentially into the slab interior, thus no critical slab thickness could be defined.67 The exponential far distance decay of the wavefunction and the associated charge density difference is universal consequence of linear character of both Schrodinger and Kohn-Sham equations, and cannot be avoided. Thus one may expect additional nonlinear term in averaged electric potential distribution which is presented in Figure 2. In order to remove potential oscillations due to the atomic nuclei and the cores, the potential was averaged in the plane parallel to the slab. Additionally, the adjacent-averaging of the period equal to half of the lattice constant was used in the direction perpendicular to the slab. As already discussed,65 the procedure generates smooth profile, which could be approximated by parabolic fit.

FIG. 3.

Band diagram in momentum (left) and real (middle) space obtained for cubic AlN (top) GaN (middle) and InN (bottom) slabs, parallel to (111) surface. The diagrams on the right present density of states (DOS) of N, Al, and the two termination H pseudoatoms, attached to nitrogen and aluminum surface atoms. The termination hydrogen pseudoatoms have Z = 0.75e and Z = 1.25e charge on nitrogen and metal side, respectively.

FIG. 3.

Band diagram in momentum (left) and real (middle) space obtained for cubic AlN (top) GaN (middle) and InN (bottom) slabs, parallel to (111) surface. The diagrams on the right present density of states (DOS) of N, Al, and the two termination H pseudoatoms, attached to nitrogen and aluminum surface atoms. The termination hydrogen pseudoatoms have Z = 0.75e and Z = 1.25e charge on nitrogen and metal side, respectively.

Close modal

The potential profiles, presented in Figure 4, have several common features. Due to the FFT procedure used to solve Poisson equation, the potential profile is periodic. At both sides of the slab the regions of high electric fields are present. They emerge due to existence of dipole layers, i.e. the electron charge centers are shifted from the location of the positively charged nuclei. As such the electron charge creates a potential barrier of several volts that prevents escape of the electrons from the surface. The presence of hydrogen termination atoms changes the height of these barriers. These barriers depend also on the distance between hydrogen termination atoms and the surface nitrogen and metal atoms for nitrogen and metal surfaces respectively, thus they need not to be identical. They are denoted as ΔV1 and ΔV2 respectively. The net nonzero surface dipole contribution creates the additional potential difference between both sides of the slab that is independent of the polarization and surface charge. In order to compensate this difference, the dipole correction designed by Neugebauer and Scheffler was applied which is visible as sudden potential drop in the vacuum region.68 In the regions corresponding to the slab interior and the vacuum far from the slab, the potential profile should be linear.65 In order to verify this, the potential profiles were approximated by parabolic fit in these two regions, i.e.

Vc(𝑧)=a+b𝑧+c𝑧2
(7)

The approximation procedure based on Eq. 7 generates the data which are listed in Table II. It follows that nonlinear terms are comparable for the vacuum region and for the slab interior. Thus the above-described procedure leads to slabs which are free of the volume charge. Also the slopes of the slab are close to zero for both cases. As shown in Appendix  A, the zero field state in the slab interior and in the vacuum is possible only for polarization-free system for surface charge equal to zero (since polarization is excluded due to the zinc-blende lattice symmetry, the surface charge vanishes).

FIG. 4.

Averaged electric potential profiles, presented as electron energy (i.e. multiplied by electron charge) obtained for cubic AlN (a) GaN (b) and InN (c) slabs, parallel to <111> direction. The slab were terminated by hydrogen pseudoatoms having Z = 0.75e and Z = 1.25e charge on nitrogen and metal side respectively.

FIG. 4.

Averaged electric potential profiles, presented as electron energy (i.e. multiplied by electron charge) obtained for cubic AlN (a) GaN (b) and InN (c) slabs, parallel to <111> direction. The slab were terminated by hydrogen pseudoatoms having Z = 0.75e and Z = 1.25e charge on nitrogen and metal side respectively.

Close modal
TABLE II.

Eq. 7 approximation parameters derived from the potential profiles in the cubic slab interior and in the vacuum. The double value of the constant value parameter (a) arises due to the difference in the potential jumps at both edges of the slab.

Slab regionabcΔb
AlN (interior) 10.64349 0.00465 1.1386E-5 0.0037 
AlN (vacuum) 1.73743/0.98216 9.619E-4 4.02451E-6 
GaN (interior) 8.39645 0.00481 1.09923E-5 0.0031 
GaN (vacuum) 1.01057/1.18081 0.00171 6.99906E-6 
InN (interior) 10.88952 0.02684 2.72742E-4 0.0259 
InN (vacuum) 0.50573/0.50481 9.21674E-4 3.599E-6 
Slab regionabcΔb
AlN (interior) 10.64349 0.00465 1.1386E-5 0.0037 
AlN (vacuum) 1.73743/0.98216 9.619E-4 4.02451E-6 
GaN (interior) 8.39645 0.00481 1.09923E-5 0.0031 
GaN (vacuum) 1.01057/1.18081 0.00171 6.99906E-6 
InN (interior) 10.88952 0.02684 2.72742E-4 0.0259 
InN (vacuum) 0.50573/0.50481 9.21674E-4 3.599E-6 

The above-described procedure proves that the designed cubic slab, devoid of the surface states in the bandgap, and is characterized by zero net electric charge at both surfaces. According to the arguments presented in Appendix  A, the result is independent of the value of the potential differences ΔV1 and ΔV2. It is therefore natural to use exactly the same slab termination for the wurtzite lattice. The difference between zinc blende and wurtzite structure is related to the shift of the third layer in zinc blende, which induces minor difference in the surface states, not affecting charge balance between the surface and the bulk. The examples of such calculations for wurtzite AlN, GaN and InN slabs are presented in Figure 5.

FIG. 5.

Band diagram in momentum (left) and real (middle) space obtained for wurtzite AlN (a) GaN (b) and InN (c) slabs, parallel to <0001> direction. The diagrams on the right present density of states (DOS) of N, Al, and the two termination H pseudoatoms, attached to nitrogen and aluminum surface atoms. The slab were terminated by hydrogen pseudoatoms having Z = 0.75e and Z = 1.25e charge on nitrogen and metal side respectively.

FIG. 5.

Band diagram in momentum (left) and real (middle) space obtained for wurtzite AlN (a) GaN (b) and InN (c) slabs, parallel to <0001> direction. The diagrams on the right present density of states (DOS) of N, Al, and the two termination H pseudoatoms, attached to nitrogen and aluminum surface atoms. The slab were terminated by hydrogen pseudoatoms having Z = 0.75e and Z = 1.25e charge on nitrogen and metal side respectively.

Close modal

Similarly to the cubic slabs presented in Figure 3, the wurtzite diagrams show no presence of surface states in the bandgap, thus no Fermi level pinning is observed and the band charge contribution is small and can be neglected. To avoid side effects in the determination of polarization, the slab length has to be relatively large. Unfortunately, the strong polarization field will eventually close the gap for longer slabs. In order to avoid generation of the band charge due to Fermi level penetration into the skewed bands of long slabs the field inside the slab is set to zero by adding the external electric field. In that case very long slabs could be employed.

The diagrams in Figure 6 prove that different surface dipole layer does not affect the slope of the potential in the vacuum obtained when the electric field inside GaN slab is force to zero. The criterion for zero charge may be obtained from fitting formula given by Eq. 7 for the cases (c) and (d) presented in Figure 6 in the interior and vacuum region. From this procedure the following values were obtained for the slab interior: b = 8.55 10-4 V/Å and b = 1.46 10-4 V/Å, respectively. From this procedure applied to the vacuum region the following values were obtained: b = −0.247 V/Å and b = −0.247 V/Å, respectively. The first pair of results proves that the electric field was equal to zero in the interior (Fig. 6c and d) with quite good precision. From the second pair we have obtained the same electric field outside the semiconductor with surfaces of different dipole layer. Thus these results confirm that the surface charge, capable to affect fields in the system, do not arise due to the changes of the hydrogen passivation layer at the slab edges.

FIG. 6.

Averaged electric potential profiles along [0001] direction in the wurtzite GaN slab, obtained for: (a) periodic boundary conditions, (b) periodic boundary condition and nonequilibrium distance for hydrogen termination pseudoatoms, (c) additional external field so that the field in the slab vanishes, i.e. Eslab= 0, (d) as for case (c) employing nonequilibrium distance for hydrogen termination pseudoatoms.

FIG. 6.

Averaged electric potential profiles along [0001] direction in the wurtzite GaN slab, obtained for: (a) periodic boundary conditions, (b) periodic boundary condition and nonequilibrium distance for hydrogen termination pseudoatoms, (c) additional external field so that the field in the slab vanishes, i.e. Eslab= 0, (d) as for case (c) employing nonequilibrium distance for hydrogen termination pseudoatoms.

Close modal

The averaged potential profiles for the three nitrides are compared in Figure 7. Since the z coordinates values are limited to 60 Å, the parabolic terms contribute to less than 0.1 V for vacuum. Thus this term contribution is limited to a few percents. Therefore the electric field in the vacuum, given by linear term, is well established being evidently nonzero in all presented cases. According to the derivation in Appendix  A, the electric field is related to spontaneous polarization field by the following relation (Eq. A6a):

Evac(β)=εEslab(pol)
(8)

From the parabolic approximation given by Eq. 7, the spontaneous polarization, the spontaneous polarization related electric fields, and the Berry phase polarization were determined with the results presented in Table III.

FIG. 7.

Averaged electric potential profiles along [0001] direction in the wurtzite AlN; GaN and InN slabs with external electric field applied so that the field in the slab is force to zero. The electronic temperature was set equal to 300 K for AlN and GaN and 30K for InN.

FIG. 7.

Averaged electric potential profiles along [0001] direction in the wurtzite AlN; GaN and InN slabs with external electric field applied so that the field in the slab is force to zero. The electronic temperature was set equal to 300 K for AlN and GaN and 30K for InN.

Close modal
TABLE III.

Electric field and polarization related properties of group III nitrides. Static dielectric permittivities for AlN, GaN, InN were assumed to be: 10.31; 10.28; 14.61, respectively.

Berry phase polarizationSpontaneous polarizationSpontaneous polarization
CompoundP(0) (in C/m2)P(pol) (in C/m2)related electric field E(pol) (in V/Å)
AlN 0.08959 0.008689 0.09814 
GaN 0.01924 0.001872 0.02114 
InN 0.02860 0.001957 0.02211 
Berry phase polarizationSpontaneous polarizationSpontaneous polarization
CompoundP(0) (in C/m2)P(pol) (in C/m2)related electric field E(pol) (in V/Å)
AlN 0.08959 0.008689 0.09814 
GaN 0.01924 0.001872 0.02114 
InN 0.02860 0.001957 0.02211 

In the case of AlN, the obtained values are in good agreement with Ref. 19, whereas the value in the second paper in Ref. 19 is different. In case of GaN, the present result is lower than both values obtained by Bernardini et al.19 Similarly, the InN polarization is lower than those obtained previously. Note that the difference between InN values in Ref. Ref 19 is of the same order as the difference with presently obtained value. The difference between our results and those obtained previously is not large which confirms validity of both approaches. One of the possible sources of error may be related to the use of high electron temperature in Fermi-Dirac distribution and associated with interband transitions and screening by the band charge. That is consistent with the AlN results that are in good agreement. In order to verify this effect, the two calculations for electronic temperature equal to 300 K and 30 K were made. The difference between the obtained electric fields was below 1% of the obtained value. Thus the error related to this effect has to be excluded as possible source of the discrepancy.

Another source of error may be related to penetration of the surface state wavefunction into the slab which led to the presence of the negative charge in the interior and the excess positive charge at the surface. The magnitude of this contribution may be directly obtained from the potential distribution. More specifically, the magnitude of the parabolic term, as compared to the linear change is the measure of unbalanced charge in the interior.65 Such effect should be present in both cubic and hexagonal lattices. The cubic value of the parabolic term is comparable to hexagonal, that confirms this identification. The magnitude of this term indicates that the “charge” error is not higher than 5% of the value of a field. Thus it also should be excluded. The electric fields obtained within above slabs are related to spontaneous polarization which will be proven by theoretical and experimental verification using AlN/GaN MQWs systems. Careful verification of the assumptions used in the derivation of these values needs theoretical and experimental approach which are presented in the paragraphs IV and V, respectively.

The ab initio calculations of AlN/GaN MQW systems offer the opportunity to compare directly the determined polarization and polarization-related electric fields, and those obtained from DFT data.69,70 Systematic studies of both zinc-blende and wurtzite AlN/GaN MQW structures are shown in Figures 8 and 9, respectively. We assume that the MQW system is embedded in a uniformly doped semiconductor environment, which fixes the location of the Fermi level. The nature of the dopant is not relevant for the calculations.

FIG. 8.

Spatial distribution of the bands obtained by projection of quantum states on atomic wavefunctions for the system consisting of 12 AlN- 12 GaN atomic layers (i.e. 3 nm thick GaN well & 3 nm thick AlN barrier). The metal and nitrogen atoms are located in perfect AlN zinc blende lattice without any relaxation (a) and fully relaxed (b).

FIG. 8.

Spatial distribution of the bands obtained by projection of quantum states on atomic wavefunctions for the system consisting of 12 AlN- 12 GaN atomic layers (i.e. 3 nm thick GaN well & 3 nm thick AlN barrier). The metal and nitrogen atoms are located in perfect AlN zinc blende lattice without any relaxation (a) and fully relaxed (b).

Close modal

We consider the simplest case of the MQWs far from p-n junction or other sources of internal electric field. In such a case, the potential drop across the MQWs is zero. Neglecting the edge effects that appear due to the finite number of the wells and barriers, it could be assumed that potential difference across a single well-barrier period is zero, i.e. the electric potential is periodic. Thus, such single well-barrier structure is ideally suited for DFT calculation, assuming periodicity of an electric potential, necessary to solve Poisson equation by Fast Fourier Transform (FFT) method. The two such examples are presented below.

The example of the MQWs GaN/AlN system having zinc blende lattice is presented in Figure 8. The system consists of 12 AlN and 12 GaN atomic layers. In order to remove the influence of the deformation on the system, the atomic configuration of the system was kept in original GaN lattice without geometry optimization. Thus zinc blende cubic symmetry was preserved and no polarization is present. In the second diagram the ion positions and c lattice constant are relaxed and piezoelectric effects induce some polarization in the system. The ab intio obtained bands are plotted in Figure 8. As it is shown, for the nonrelaxed case the bands are flat, showing no indication of the presence of electric fields in the system.

As it is shown, the rigid zinc blende AlN/GaN superlattice is devoid of any fields both in the quantum well (QW) and in the quantum barrier (QB). The presence of tetragonal strain breaks the system symmetry, allowing the field to arise, as it is visible from the spatial band diagram. The exact value of the fields may be obtained from the averaged electric potential shown in Fig. 9.

FIG. 9.

Average electric potential of the structure presented in Figure 8: (a) – all atoms are in GaN lattice, (b) – with geometry optimization applied. Solid red line represents average potential in AlN/GaN zinc blende superlattice, blue broken line represents potential of superposition of atomic orbitals charge.

FIG. 9.

Average electric potential of the structure presented in Figure 8: (a) – all atoms are in GaN lattice, (b) – with geometry optimization applied. Solid red line represents average potential in AlN/GaN zinc blende superlattice, blue broken line represents potential of superposition of atomic orbitals charge.

Close modal

The potential profile in Figure 9 confirms the absence of electric field for the perfect zinc blende symmetry, and the emergence of the field in the system with broken symmetry. In the case of a relaxed geometry system the fields were equal to -1.36 10-2 V/Å (1.36 MV/cm) and 1.30 10-2 V/Å (1.30 MV/cm) for AlN and GaN respectively. That indicates the relatively large piezoelectric fields in the nitride structures.

The second considered case is the polar wurtzite AlN/GaN MQWs system consisting of 16 AlN and 16 GaN atomic layers. As before the lattice was either not relaxed, preserving symmetry of GaN lattice or relaxed, attaining the average a lattice constant. The band diagrams of the system are presented in Figure 10. A low precision estimate of the electric fields could be obtained from the band diagram directly. More precise determination of the fields is based on average potential profiles,70 depicted in Figure 11.

FIG. 10.

Electron band diagram (left) and the spatial distribution of the bands obtained by projection of quantum states on atomic wavefunctions for the system consisting of 16 AlN/16 GaN atomic layers. The metal and nitrogen atoms are located in perfect wurtzite lattice of bulk material of GaN (a) and AlN (b) without geometry optimization procedure. For comparison, the structure adjusted to the experimentally measured relaxed a lattice constant of AlN/GaN superlattice with c-stress relaxed by geometry optimization procedure (c).

FIG. 10.

Electron band diagram (left) and the spatial distribution of the bands obtained by projection of quantum states on atomic wavefunctions for the system consisting of 16 AlN/16 GaN atomic layers. The metal and nitrogen atoms are located in perfect wurtzite lattice of bulk material of GaN (a) and AlN (b) without geometry optimization procedure. For comparison, the structure adjusted to the experimentally measured relaxed a lattice constant of AlN/GaN superlattice with c-stress relaxed by geometry optimization procedure (c).

Close modal
FIG. 11.

Average electric potential of the structure presented in Figure 8 adjusted to (a) – GaN lattice, (b) – AlN lattice, and with geometry relaxed (c). The solid red line represents the averaged potential in a AlN/GaN wurtzite superlattice, whereas the blue dashed line represents the potential of superposition of atomic orbitals charge.

FIG. 11.

Average electric potential of the structure presented in Figure 8 adjusted to (a) – GaN lattice, (b) – AlN lattice, and with geometry relaxed (c). The solid red line represents the averaged potential in a AlN/GaN wurtzite superlattice, whereas the blue dashed line represents the potential of superposition of atomic orbitals charge.

Close modal

For the rigid structure the obtained fields were equal to -1.26·10-2 V/Å and 1.29·10-2 V/Å for GaN and AlN respectively. These fields are much different from the fully relaxed case: -5.49·10-2 V/Å and 6.26·10-2 V/Å, which confirms the dominant role of piezoelectric effects in nitride structures.

The ab intio calculations offer an opportunity of direct comparison of the electric field obtained using the present model and that of Fiorentini et al., with the fields obtained from calculations.71,72 Therefore, Table IV summarizes the polarization-related properties of AlN/GaN MQWs with a width that matches to the samples investigated experimentally in this work.

TABLE IV.

Electric fields in GaN/AlN MQWs.a

GaN well widthElectric field difference -Electric field difference -Electric field difference -
(in nm)GaN adjusted ΔE(pol) (in V/Å) -AlN adjusted ΔE(pol) (in V/Å) -relaxed ΔE(pol) (in V/Å) -
0.0711 0.1040 0.10763 
1.5 0.0688 0.0982 0.08662 
0.0712 0.0981 0.08822 
0.0664 0.0996 0.08525 
0.0633 0.0994 0.08051 
0.0621 0.0928 0.06951 
GaN well widthElectric field difference -Electric field difference -Electric field difference -
(in nm)GaN adjusted ΔE(pol) (in V/Å) -AlN adjusted ΔE(pol) (in V/Å) -relaxed ΔE(pol) (in V/Å) -
0.0711 0.1040 0.10763 
1.5 0.0688 0.0982 0.08662 
0.0712 0.0981 0.08822 
0.0664 0.0996 0.08525 
0.0633 0.0994 0.08051 
0.0621 0.0928 0.06951 
a

From the above data it follows that these fields are strongly affected either by the strain induced by substrate or by relaxation to Vegard’s law value.

A series of 40-period AlN/GaN MQW structures were grown by plasma-assisted molecular beam epitaxy (PA-MBE) on GaN/AlN/sapphire templates.56 These structures have 4-nm-thick AlN barriers, whereas the GaN well width varied from 1 to 6 nm. Time resolved PL was measured using third harmonic of Ti:Saphire laser (300 nm) with pulses frequency from 100 kHz to 80 MHz and a streak camera. The results show that MQW emission in function of the QW thickness, covering broad range from 2.4 to 3.7 eV and the recombination rates from 105 to 109 s-1. The recombination rates are higher for higher energies what was observed already by other experimentalists.23,73,74 The time resolved PL spectra are shown in Fig. 12.

FIG. 12.

Time resolved PL spectra of GaN/AlN MQWs structures of different GaN well thickness, as indicated in the graphs. The AlN barrier thickness is 4 nm for all samples.

FIG. 12.

Time resolved PL spectra of GaN/AlN MQWs structures of different GaN well thickness, as indicated in the graphs. The AlN barrier thickness is 4 nm for all samples.

Close modal

The PL spectra revealed that these MQWs emit in a few modes. These modes can arise from the step-like changes of the QW thickness by one atomic layer or due to local strain relaxation caused by dislocation as proposed in Ref. 75. However, these additional states are either short-living or very weak, so we can distinguish principal emission modes in the investigated structures. Since the high-energy states have much shorter lifetimes, it was necessary to measure every sample in few time ranges. The time scales presented in Fig. 12 cover span from 1 ns to 3 μm and are chosen to show both high- and low-energy states. It is visible that high-energy states decay faster what can be explained by nonradiative relaxation to lower states and by better overlap of electron and hole functions leading to faster radiative recombination. The better overlap in the high-energy states can be due to smaller local width of QWs or smaller piezoelectric effect that could be decreased by strain relaxation due to dislocations. The later effect is more pronounced for wider wells. Such determined transition energies are listed in Table V.

TABLE V.

Optical transition energies in GaN/AlN MQWs.

GaN well width (in nm)Transition energy ω (in eV) -
3.72 
1.5 3.12 
3.18 
2.89 
2.63 
2.45 
GaN well width (in nm)Transition energy ω (in eV) -
3.72 
1.5 3.12 
3.18 
2.89 
2.63 
2.45 

Using these data the emission energies from PL measurements and the ab initio calculations for MQWs system adjusted to GaN, AlN and averaged composition relaxed are compared in Figure 13. This diagram shows basic agreement between the PL data and ab initio results.

FIG. 13.

GaN bandgap corrected optical transition energy (Fig. 11) in variable thickness GaN QWs with 4 nm thick AlN QBs, obtained from PL measurements (black stars) and ab initio data for the structure adjusted to GaN (solid blue), AlN (dash red) and averaged composition relaxed (dot green).

FIG. 13.

GaN bandgap corrected optical transition energy (Fig. 11) in variable thickness GaN QWs with 4 nm thick AlN QBs, obtained from PL measurements (black stars) and ab initio data for the structure adjusted to GaN (solid blue), AlN (dash red) and averaged composition relaxed (dot green).

Close modal

The QW width dependence of the PL energies, presented in Figure 13, suggests existence of the two main different factors that are influencing the optical transition energies. They have opposite influence: the quantum confinement in the well, which increases PL energy, and the electric field, which red shifts the PL. Thus the deviation of the transition energy from the bandgap value may be expressed as the sum of two contributions:

Δ(hν)=hνEg(GaN)=ΔEpol+ΔEloc
(9)

where the GaN bandgap energy is Eg(GaN)=3.47eV and ΔEpol, ΔEloc are the polarization and confinement energy shifts, respectively. The main factor is the electric field in the QWs which shuffles the electrons and holes towards opposite ends of the wells and lowers their energy difference. Thus it is negative and may be approximated by the electric field induced electron energy at both ends of the wells:

ΔEpol=eEww=e(εwEw(pol)εbEb(pol))bwεbw+εwb=AzGaNzGaN+c
(10)

where c constant is close to 4 nm, polarization fields in the uniform materials of the barrier and the well are Eb(pol) and Ew(pol), their thickness b (b = zAlN) and s (s = zGaN), respectively. In order to recover the polarization contribution, the PL energy shift Δ(hν) has to be multiplied by inverse expression to derive the magnitude of the polarization fields in the well as

A(zw)=Δ(hν)zGaN+czGaN
(11)

The result is plotted in Figure 14. It is expected that for large QW width the quantum confinement contribution is negligible and the A(zGaN) approximates the polarization factor.

FIG. 14.

The polarization coefficient A (Eq. 11) in transition energy in variable thickness GaN QWs with 4 nm thick AlN QBs obtained from PL measurements: blue squares - PL measurements, red circles - DFT data, solid green line - approximation according to Eq. 12.

FIG. 14.

The polarization coefficient A (Eq. 11) in transition energy in variable thickness GaN QWs with 4 nm thick AlN QBs obtained from PL measurements: blue squares - PL measurements, red circles - DFT data, solid green line - approximation according to Eq. 12.

Close modal

The diagram shows saturation at wide QWs confirming the predicted polarization picture. The perfect adjustment of the experimental and DFT relaxed system is obtained for c = 2.8 nm which indicates that the efficient width of the QWs is much lower than nominal 4 nm. From the diagram far field value of A is obtained A = 1.8 eV. Taking into account the barrier width b = 40 Å, and neglecting the difference in permittivities, the direct estimate of the polarization field difference is: ΔEPL(pol)=EAlN(pol)EGaN(pol)=0.045 V/Å. This has to be compared with the difference obtained in the present simulations giving ΔE(pol)=EAlN(pol)EGaN(pol)=0.077 V/Å. For comparison, the Berry phase data give ΔE(pol)=EAlN(pol)EGaN(pol)=0.067 V/Å (Ref. 59) and ΔE(pol)=EAlN(pol)EGaN(pol)=0.050 V/Å (Ref. 60). Generally the best agreement is obtained for Ref 60. Note that if the effective width c = 2.8 nm is used, the polarization field difference gives ΔEPL(pol)=EAlN(pol)EGaN(pol)=0.064 V/Å which is in reasonable agreement with the obtained data.

This A(zw) dependence may be used in the analysis of the quantum confinement energy, which is positive and may be approximately expressed as:

Δ(ω)expApol=Γ2k22m=Γ22m(2πzGaN)2=GzGaN2
(12)

where Γ describes approximations used, such as incomplete localization, different band offset, etc. From this dependence, the value of the constant G was determined to be G = 2.70 eV.

Polarization in the solid is analyzed considering spontaneously polarized crystals. It is shown that Berry phase case assumption of zero field in the solid is not compatible with continuity of dielectric displacement vector. The physically realizable case is attained by addition of the electric field inside the polarized media and the correction of polarization using susceptibility of the material. Such corrected polarization attains the spontaneous polarization value. The value of the field induced by the spontaneous polarization is derived. As the polarization induced field is directed opposite to the Berry phase polarization, spontaneous polarization is always smaller than the Berry phase polarization.

Slab model for direct determination of the polarization of the bulk nitrides: AlN, GaN and InN is introduced and critically assessed. By application of this slab model to the zinc blende nitride crystals it is shown that by appropriate determination of the boundary conditions it is possible to enforce zero charge at the surfaces. Thus the model allows obtaining the value of spontaneous polarization related electric fields in these three nitrides. The obtained fields are: 0.098 V/Å, 0.021 V/Å and 0.022 V/Å for AlN, GaN and InN, respectively. Using formalism described above, the spontaneous polarization of the nitrides were 8.69·10-3 C/m2, 1.88·10-3 C/m2, and 1.96·10-3 C/m2 for AlN, GaN and InN. Also the Berry phase polarization values were determined to be: 8.69·10-2 C/m2, 1.92·10-2 C/m2, and 2.86·10-2 C/m2, for these three compounds. These values are in reasonably good agreement with the values obtained by Bernardini et al.59,60

AlN/GaN superlattices was analyzed, using the above derived polarization and polarization related fields in nitrides. The values of the fields in the lattice were derived using continuity of dielectric displacement vector and the minimal electric field energy. It was shown that both approaches give consistent value provided that the above derived polarization relations are obeyed. Ab intio simulations were conducted for the GaN/AlN MQWs of different well and barrier thicknesses. It was shown that the fields derived from the bulk fields and those obtained directly for simulations of MQWs are in good agreement.

MBE grown GaN/AlN MQWs of good crystalline quality were used in time dependent PL measurements. It was shown that the obtained emission energies are in good agreement with the derived change of the energy of quantum states thus confirming the nature and the magnitude of QCSE and related polarization fields.

This work was supported by the Polish National Science Centre on the basis of the decisions No. DEC-2012/05/B/ST3/02516, DEC-2011/03/B/ST5/02698, and DEC-2012/05/B/ST3/03113. This research was supported in part by PL-Grid Infrastructure. The calculations reported in this paper were performed using computing facilities of the Interdisciplinary Centre for Modelling of Warsaw University (ICM UW) and on cluster Nostromo, i.e. a computing facility of Center for Preclinical Research and Technology (CePT). We would like to thank Prof. Piotr Boguslawski for critical reading of the manuscript.

Consider the two finite slabs of the uniform material of zinc blende or wurtzite lattice oriented perpendicularly to [111] in zinc blende and in wurtzite [0001] directions respectively. In principle these slabs may be surrounded by the infinite extent vacuum that in the ab initio calculations are replaced by the finite thickness vacuum supplemented by the periodic boundary conditions.

The electron energy (i.e. the potential profile multiplied by negative electron charge (-e)) in the zinc blende slab is schematically presented in Fig. 15. It is assumed that due to wide gap of the nitrides, the absence of doping and without the conduction and valence bands contributions, the bulk charge in the system is absent. Therefore the potential profiles may be represented by straight lines associated with the constant uniform electric fields. The dominant features of the potential profiles are the potential jumps present at both sides of the slab denoted as ΔV1, ΔV2. They are related to the extension of the electronic charge outside the positively charged nuclei. That indicates on higher electron energy outside the slab, thus they represent the electron energy necessary to escape from the slab to the vacuum. These potential jumps may be modified by the termination of the slabs, both by the type of the hydrogen pseudoatoms and also by the distance between them and the surface metal and nitrogen atoms. Thus these potential drops may be changed independently and need not to be identical. Assuming that no surface charge is present the potential profile in absence of the field outside is represented by line (a) in Fig. 15. Naturally this is due to the polarization field absence in zinc blende material and that the external field is assumed to be zero (Eext = 0). Therefore the total potential difference across the presented region is:

ΔVtot(a)=ΔV1+ΔV1
(A1)

where superscript ‘a’ denotes line (a). The left and right potential jumps ΔV1 and ΔV2 are independent of the fields in the bulk, thus they are identical for all cases (a) - (d). Therefore their superscripts were dropped.

FIG. 15.

Electron energy within finite thickness zinc blende slab and the vacuum: (a) without any charge at the surfaces (solid black line); (b) with interface charge at the surfaces (blue dash-dotted line); (c) case (b) compensated by external field Eext so that the field inside the slab vanish (green dotted line); (d) case (b) compensated by external field so the total potential difference is zero (red dashed line).

FIG. 15.

Electron energy within finite thickness zinc blende slab and the vacuum: (a) without any charge at the surfaces (solid black line); (b) with interface charge at the surfaces (blue dash-dotted line); (c) case (b) compensated by external field Eext so that the field inside the slab vanish (green dotted line); (d) case (b) compensated by external field so the total potential difference is zero (red dashed line).

Close modal

Both surfaces may be charged due to the charge transfer between these surfaces changing the occupation of surface states. This leads to emergence of the electric field inside the slab, as represented by line (b) in Fig. 15. Naturally the field outside is absent as above. Thus the potential difference changes to:

ΔVtot(b)=ΔV1+ΔV1Eslab(b)zslab
(A2)

where subscript ‘‘slab’’ denotes slab thickness and field.

Further changes may be obtained when the external field is applied, adding to the field in and outside the slab. By appropriate choice the field may compensate the field in the slab (represented by line (c) in Fig. 15). Accordingly the potential difference is represented by:

ΔVtot(c)=ΔV1+ΔV1Evac(c)zvac
(A3)

Finally, the other external field choice may be enforced by ab initio simulations, so that the potential is periodic (line (d) in Fig. 15). As both the potential jumps and the fields contribute to the total potential, difference is zero according to the equation:

ΔVtot(d)=ΔV1+ΔV1Evac(d)zvacEslab(d)zslab=0
(A4)

The latter case (d) is realized in ab intio simulations where FFT solver of Poisson equation enforces periodic boundary conditions. By appropriate addition of external field, the transition from the case (d) to only one of the three conditions (a) - (c) may be realized. The other two are thus excluded. For zero charge at the surfaces the case (a) is realized, both (b) and (c) are not possible. For any two other cases the surface charge is present. As shown in Figure 1, the zinc blende slabs are characterized by absence of surface charge for AlN, Gan and InN.

Quite different representation of the fields and potentials is obtained for polar wurtzite slab as shown in Fig. 16.

FIG. 16.

Electron energy within finite thickness wurtzite slab and the vacuum with no surface charge present: (α) only polarization field present (solid black line); (α) compensated by external field Eext so that the field inside the slab vanish (blue dash-dotted line); (γ) compensated by external field Eext so that total potential difference is zero (green dotted line); (δ) additional external field so both fields are present (red dashed line).

FIG. 16.

Electron energy within finite thickness wurtzite slab and the vacuum with no surface charge present: (α) only polarization field present (solid black line); (α) compensated by external field Eext so that the field inside the slab vanish (blue dash-dotted line); (γ) compensated by external field Eext so that total potential difference is zero (green dotted line); (δ) additional external field so both fields are present (red dashed line).

Close modal

Accounting the above arguments it is assumed that the surface charge is zero. Since the polarization field is present, the simultaneous field disappearance in and outside the slab is not possible, the only case realized is the of the zero field in the vacuum (case α). The total potential difference is:

ΔVtot(α)=ΔV1+ΔV1Eslab(α)zslab=ΔV1+ΔV1Eslab(pol)zslab
(A5)

where we have taken into account that the field in the slab is equal to spontaneous polarization induced electric field. Application of external field Eext(β) may cancel the field in the slab (case β). From the continuity of electric displacement vector at the slab surface it follows that for any additional field the following relation is obeyed ΔEvac=εΔEslab which, for case β, may be translated into:

Evac(β)=εEslab(pol)
(A6a)

and

ΔVtot(β)=ΔV1+ΔV1Evac(β)zvac=ΔV1+ΔV1+εEslab(pol)zvac
(A6b)

Therefore, for zero slab field inside, the field outside the slab is sufficient for direct determination of the polarization related field in the bulk Eslab(pol). Note that the result is not dependent on the potential jumps at the interfaces which could be potentially the main source of error. That allows precise determination of the polarization related field form ab initio calculations.

The other two possible cases include the zero total potential (case γ) and the additional field which may increase or decrease field inside and outside the slab (case δ). The zero total potential is obtained for the electric field Eext(γ)=Evac(γ) added to field

ΔVtot(γ)=ΔV1+ΔV1Eslab(γ)zslabEvac(γ)zvac=ΔV1+ΔV1(Eslab(pol)+Evac(γ)ε)zslabEvac(γ)zvac=0
(A7)

The external field is the field that is automatically added by ab initio procedure to solve Poisson equation by FFT employing periodic boundary conditions. The field may be readily obtained measuring the slope of potential function outside the slab. Actually additional external field may be applied in addition to the field obtained from FFT solver to get any field as represented by case (δ). Actually, this procedure was used to transform case (γ) into the cases (α) and (β) that were used in determination of polarization related Eslab(pol).

The relation between polarization in the bulk polar semiconductors and the field in periodic well-barrier system may be derived using different formulations. The basic assumption used employs connection between dominant impurity level and the Fermi energy. In most simple setup, the MQW structure is embedded in the bulk system in which the potential difference is zero, i.e. the structure is located outside the electric field induced by p-n junction, by the surface states or by heterostructures. Naturally, introduction of the potential difference induced by these factors is straightforward and will not be discussed here. Using the absence of such electric fields and assuming that the Fermi level is controlled by the dominant impurity level having well defined energy with respect to the band states, leads to the conclusion that the overall potential difference across MQWs structure is zero. Naturally, the edge effects are critical for the emergence of the potential distribution in MQW superstructure by screening them from the surrounding bulk. The screening will be accounted for in the field intensity and neglected in geometric factors. Invoking periodicity of the well-barrier superlattice leads to the basic results stating that potential difference across well-barrier period is zero. Expressing these difference by electric fields in the well Eb and the barrier Ew we obtain:

Ebb+Eww=0
(B1)

where b and w denote the barrier and well widths, respectively. It is also assumed that the structure is perpendicular to the polarization direction so the scalar notation may be used in which the fields have the components perpendicular to the structure only. It has to be noted that the component parallel to the well/barrier is screened by the free carriers in the wells, so the perpendicular component is of interest only.

Assumption that displacement field is continuous in the QW and the QB leads to:

Dw=Pw+ε0Ew=Pb+ε0Eb=Db
(B2)

which can be expressed in terms of zero field polarizations of the well and the barrier using Eq. 3, as

Pw(o)+ε0εwEw=Pb(o)+ε0εbEb
(B3)

Originally, Bernardini and Fiorentini derived electric fields in equal width well-barrier structure. i.e. b = w,65 simplifying Eq. B1 to:

Eb+Ew=0
(B4)

Using Eqs. B3 and B4, the electric field in the equal width well/barrier, obtained by Bernardini and Fiorentini was

Eb=Ew=Pw(0)Pb(0)ε0(εb+εw)
(B5)

In subsequent publication,66 Fiorentini et al. introduced more general rule for various width of the well and the barrier using Eq. A1, that gives

Ew=(Pb(0)Pw(0))bε0(εbw+εwb)=Pb(0)Pw(0)ε0(εby+εw)
(B6a)
Eb=(Pw(0)Pb(0))wε0(εbw+εwb)=(Pw(0)Pb(0))yε0(εby+εw)
(B6b)

where y denotes well to barrier width ratio, i.e. y = w/b. These values may then be used along with Eq. 4b to express these fields in terms of spontaneous polarization of the well and the barrier, as

Ew=(εbPb(pol)εwPw(pol))bεo(εbw+εwb)=εbPb(pol)εwPw(pol)ε0(εby+εw)
(B7a)
Eb=(εwPw(pol)εbPb(pol))wε0(εbw+εwb)=(εwPw(pol)εbPb(pol))yε0(εby+εw)
(B7b)

Finally these formulae may be converted into dependence on the polarization induced field as:

Ew=(εwEw(pol)εbEb(pol))bεbw+εwb=εwEw(pol)εbEb(pol)εby+εw
(B8a)
Eb=(εbEb(pol)εwEw(pol))wεbw+εwb=(εbEb(pol)εwEw(pol))yεby+εw
(B8b)

The above formulae create a complete description of the fields in periodic well-barrier superlattice in terms of Berry phase polarization, spontaneous polarization and polarization related electric fields.

Alternative derivation of the relation between the electric field and polarization in periodic superlattice may be based on energy functional as proposed by us in Ref. 69. Spontaneously polarized system arise due to forces that are not purely electrostatic. Thus the electromagnetic energy functional refers to the excess energy:

ΔW=ε02d3rεi(EiEi(pol))2
(B9)

In case of periodic well-barrier structure perpendicular to the polarization direction, the field is uniaxial and disappears outside the structure. Additionally it is uniform in the QW and in the QB which reduces energy functional to:

ΔW=ε02d2s[εw(EwEw(pol))2w+εb(EbEb(pol))2b]
(B10)

with the integration of the surface of the structure equal to S. For single layer, e.g. for disappearing barrier b = 0, the functional is zero for Ew=Ew(pol)thus it fulfills the compatibility requirement with single polarized layer. Periodicity condition (Eq. B1) may be used to eliminate field in the barrier and taking into account uniformity of the field in the layers, the functional is transformed to

ΔW=ε0S2[εw(EwEw(pol))2w+εb(Ewwb+Eb(pol))2b]
(B11)

The field in the well is given by minimization of the excess energy i.e. with respect to the field in the well Ew

ΔWEw=0
(B12)

which gives

εw(EwEw(pol))+εb(Ewwb+Eb(pol))=0
(B13)

Solution of this equation determines the field in the QW and via Eq. A1, in the QB. The obtained fields are identical to those given by Eqs B8. Thus by two different arguments we derived identical fields that are different from those given by Fiorentini et al.71 These fields are used in the discussion of the well-barrier structure.

1.
L. D.
Lanadau
and
E. M.
Lifsic
,
Electrodynamics of Continuous Media
(
Pergamon
,
Oxford
,
1984
).
2.
P.
Bogusławski
and
J.
Bernholc
, in
Polarization Effects in Semiconductors. From Ab initio Theory to Device Applications
ed. by
C.
Wood
and
D.
Jena
,
Springer
N. York
2008
, p.
1
.
3.
C.
Noguera
and
J.
Goniakowski
,
Chem. Rev.
113
,
4073
(
2013
).
4.
J. D.
Jackson
,
Classical Electrodynamics
,
John Wiley & Sons
,
New York
1967
.
5.
7.
R. M.
Martin
,
Phys. Rev. B
9
,
1998
(
1974
).
8.
R. D.
King-Smith
and
D.
Vanderbilt
,
Phys. Rev. B
47
,
1651
(
1993
).
9.
D.
Vanderbilt
and
R. D.
King-Smith
,
Phys. Rev. B
48
,
4442
(
1993
).
10.
R.
Resta
,
Berry Phase in Electronic Wavefunctions, Troisième Cycle de la Physique en Suisse Romande
(
Année Academique
,
1996
).
11.
M.
Posternak
,
A.
Baldereschi
,
A.
Catellani
, and
R.
Resta
,
Phys. Rev. Lett.
64
,
1777
(
1990
).
12.
A. K.
Tagantsev
,
Phys. Rev. Lett.
69
,
389
(
1992
).
13.
M.
Springborg
and
B.
Kirtman
,
Phys. Rev B
77
,
045102
(
2008
).
14.
M.
Springborg
,
V.
Tevekeliyska
, and
B.
Kirtman
,
Phys. Rev B
82
,
165442
(
2010
).
15.
M.
Springborg
,
B.
Kirtman
,
Theor. Chem. Accounts
130
,
687
(
2011
) .
16.
M. E.
Lines
and
A. M.
Glass
,
Principles and Applications of Ferroelectrics and Related Materials
(
Clarendon
,
Oxford
,
1974
).
17.
M. P.
Halsall
,
J. E.
Nicholls
,
J. J.
Davies
,
B.
Cockayne
, and
P. J.
Wright
,
J. Appl. Phys.
71
,
907
(
1992
).
18.
F.
Bernardini
,
V.
Fiorentini
, and
D.
Vanderbilt
,
Phys. Rev. Lett.
79
,
3958
(
1997
).
19.
F.
Bernardini
,
V.
Fiorentini
, and
D.
Vanderbilt
,
Phys. Rev. B
56
,
R10024
(
1997
);
F.
Bernardini
,
V.
Fiorentini
, and
D.
Vanderbilt
,
Phys. Rev. B
63
,
193201
(
2001
).
20.
F.
Bernardini
,
V.
Fiorentini
,
Phys. Stat. Solidi B
216
,
391
(
1999
).
21.
T.
Takeuchi
,
C.
Wetzel
,
S.
Yamaguchi
,
H.
Sakai
,
H.
Amano
,
I.
Akasaki
,
Y.
Kaneko
,
S.
Nakagawa
,
Y.
Yamaoka
, and
N.
Yamada
,
Appl. Phys. Lett.
73
,
1691
(
1998
).
22.
R.
Langer
,
J.
Simon
,
V.
Ortiz
,
N. T.
Pelekanos
,
A.
Barski
,
R.
Ander
, and
M.
Godlewski
,
Appl. Phys. Lett.
74
,
3827
(
1999
).
23.
J. S.
Im
,
H.
Kollmer
,
J.
Off
,
A.
Sohmer
,
F.
Scholz
, and
A.
Hangleiter
,
Phys. Rev. B
57
,
R9435
(
1998
).
24.
M.
Buongiorno Nardelli
,
K.
Rapcewicz
, and
J.
Bernholc
,
Appl. Phys. Lett.
71
,
3135
(
1997
).
25.
S. P.
Łepkowski
,
H.
Teissyere
,
T.
Suski
,
P.
Perlin
,
N.
Grandjean
, and
J.
Massies
,
Appl. Phys. Lett.
79
,
1483
(
2001
).
26.
P.
Lefebvre
,
J.
Allegre
,
B.
Gil
,
H.
Mathieu
,
N.
Grandjean
,
M.
Leroux
,
J.
Massies
, and
P.
Bigenwald
,
Phys. Rev. B
59
,
15363
(
1999
).
27.
Y. C.
Shen
,
G. O.
Muller
,
S.
Watanabe
,
N. F.
Gardner
,
A.
Munkholm
, and
M. R.
Kramers
,
Appl. Phys. Lett.
91
,
141101
(
2007
).
28.
I. V.
Rozhansky
and
D. A.
Zakeheim
,
Phys. Stat. Solidi C
3
,
2160
(
2006
).
29.
M. F.
Schubert
,
S.
Chajed
,
J. K.
Kim
,
E. F.
Schubert
,
D. D.
Koleske
,
M. F.
Crawford
,
S. R.
Lee
,
A. J.
Fisher
,
G.
Thaler
, and
M. A.
Banas
,
Appl. Phys. Lett.
91
,
231114
(
2007
).
30.
J.
Xie
,
X.
Ni
,
Q.
Fan
,
R.
Schimada
,
U.
Ozgor
, and
H.
Morkoc
,
Appl. Phys. Lett.
93
,
121107
(
2008
).
31.
H.
Teisseyre
,
A.
Kaminska
,
G.
Franssen
,
A.
Dussaigne
,
N.
Grandjean
,
I.
Grzegory
,
B.
Lucznik
, and
T.
Suski
,
J. Appl. Phys.
105
,
63104
(
2009
).
32.
H.
Morkoc
,
Handbook of Nitride Semiconductors and Devices
(
Wiley-VCH
,
Berlin
,
2008
), vol. 3.
33.
G.
Kresse
and
J.
Hafner
,
Phys. Rev. B
47
,
558
(
1993
);
G.
Kresse
and
J.
Hafner
,
Phys. Rev. B
49
,
14251
(
1994
).
34.
G.
Kresse
and
J.
Furthmüller
,
Comput. Mat. Sci.
6
,
15
(
1996
).
35.
G.
Kresse
and
J.
Furthmüller
,
Phys. Rev. B
54
,
11169
(
1996
).
36.
P.
Ordejón
,
D. A.
Drabold
,
M. P.
Grumbach
, and
R. M.
Martin
,
Phys. Rev. B
48
,
14646
(
1993
).
37.
P.
Ordejón
,
D. A.
Drabold
,
R. M.
Martin
, and
M. P.
Grumbach
,
Phys. Rev. B
51
,
1456
(
1995
).
38.
J. M.
Soler
,
E.
Artacho
,
J. D.
Gale
,
A.
García
,
J.
Junquera
,
P.
Ordejón
, and
D.
Sánchez-Portal
,
J. Phys.: Condens. Matter
14
,
2745
(
2002
).
39.
S. P.
Łepkowski
and
J. A.
Majewski
,
Phys. Rev. B
74
,
35336
(
2006
).
40.
H. J.
Monkhorst
and
J. D.
Pack
,
Phys. Rev. B
13
,
5188
(
1976
).
41.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
,
Phys. Rev. Lett.
77
,
3865
(
1996
);
[PubMed]
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
, Erratum,
78
,
1396
(
1997
).
42.
P. E.
Blöchl
,
Phys. Rev. B
50
,
17953
(
1994
).
43.
G.
Kresse
and
D.
Joubert
,
Phys. Rev. B
59
,
1758
(
1999
).
44.
M.
Leszczynski
,
H.
Teisseyre
,
T.
Suski
,
I.
Grzegory
,
M.
Bockowski
,
J.
Jun
,
S.
Porowski
,
K.
Pakula
,
J. M.
Baranowski
,
C. T.
Foxon
, and
T. S.
Cheng
,
Appl. Phys. Lett.
69
,
73
(
1996
).
45.
H.
Angerer
,
D.
Brunner
,
F.
Freudenberg
,
O.
Ambacher
,
M.
Stutzmann
,
R.
Höpler
,
T.
Metzger
,
E.
Born
,
G.
Dollinger
,
A.
Bergmaier
,
S.
Karsch
, and
H.-J.
Körner
,
Appl. Phys. Lett.
71
,
1504
(
1997
).
46.
Yu.
Goldberg
in
Properties of Advanced SemiconductorMaterials GaN, AlN, InN, BN, SiC, SiGe.
Eds.
M. E.
Levinshtein
,
S. L.
Rumyantsev
,
M. S.
Shur
,
John Wiley & Sons, Inc.
,
New York
,
2001
,
31
47
.
47.
A.
Zubrilov
in
Properties of Advanced SemiconductorMaterials GaN, AlN, InN, BN, SiC, SiGe.
Eds.
M. E.
Levinshtein
,
S. L.
Rumyantsev
,
M. S.
Shur
,
John Wiley & Sons, Inc.
,
New York
,
2001
,
49
66
.
48.
N.
Troullier
and
J. L.
Martins
,
Phys. Rev. B
43
,
1993
(
1991
);
N.
Troullier
and
J. L.
Martins
,
Phys. Rev. B
43
,
8861
(
1991
).
49.
P.
Kempisty
,
P.
Strak
, and
S.
Krukowski
,
Surf. Sci.
606
,
571
572
(
2012
).
50.
L. G.
Ferreira
,
M.
Marques
, and
L. K.
Teles
,
Phys. Rev. B
78
,
125116
(
2008
).
51.
M.
Ribeiro
,
L. R. C.
Fonseca
, and
L. G.
Ferreira
,
Phys. Rev. B
79
,
241312
(
2009
).
52.
E.
Silveira
,
J. A.
Freitas
,
S. B.
Schujman
, and
L. J.
Schowalter
,
J. Cryst. Growth
310
,
4007
(
2008
).
53.
B.
Monemar
,
J. P.
Bergman
,
I. A.
Buyanova
,
H.
Amano
,
I.
Akasaki
,
T.
Detchprohm
,
K.
Hiramatsu
, and
N.
Sawaki
,
Solid State Electron.
41
,
239
(
1997
).
54.
Y. C.
Yeo
,
T. C.
Chong
, and
M. F.
Li
,
J. Appl. Phys.
83
,
1429
(
1997
).
55.
K. M.
Yu
,
Z.
Liliental-Weber
,
W.
Walukiewicz
,
W.
Shan
,
J. W.
Ager
,
S. X.
Li
,
R. E.
Jones
,
E. E.
Haller
,
H.
Lu
,
W. J.
Schaff
,
Appl. Phys. Lett.
86
,
071910
1-3
(
2005
).
56.
A.
Kaminska
,
P.
Strak
,
J.
Borysiuk
,
K.
Sobczak
,
J. Z.
Domagala
,
M.
Beeler
,
E.
Grzanka
,
K.
Sakowski
,
S.
Krukowski
, and
E.
Monroy
,
J. Appl. Phys.
119
,
015703
1-10
(
2016
).
57.
P. K.
Kandaswamy
,
F.
Guillot
,
E.
Bellet-Amalric
,
E.
Monroy
,
L.
Nevou
,
M.
Tchernycheva
,
A.
Michon
,
F. H.
Julien
,
E.
Baumann
,
F. R.
Giorgetta
 et al.
GaN/AlN short-period superlattices for intersubband optoelectronics: A systematic study of their epitaxial growth, design, and performance
J. Appl. Phys.
104
,
093501
1-16
(
2008
).
58.
M. A.
Sanchez-Garcia
,
E.
Calleja
,
E.
Monroy
,
F. J.
Sanchez
,
F.
Calle
,
E.
Munoz
,
R.
Beresford
,
J. Cryst. Growth
183
,
23
(
1998
).
59.
Y.
Kotsar
,
B.
Doisneau
,
E.
Bellet-Amalric
,
A.
Das
,
E.
Sarigiannidou
,
E.
Monroy
;
Strain relaxation in GaN/AlxGa1-xN superlattices grown by plasma-assisted molecular-beam epitaxy
J. Appl. Phys.
110
,
033501
1-7
(
2011
).
60.
N. A.
Spalding
and
J.
Solid
,
State Chem.
195
,
2
(
2012
).
61.
R.
Ramprasad
and
N.
Shi
,
Phys. Rev. B
72
,
052107
(
2005
).
62.
N.
Shi
and
R.
Ramprasad
,
Appl. Phys. Lett.
91
,
242906
(
2007
).
63.
N.
Shi
and
R.
Ramprasad
,
Appl. Phys. Lett.
89
,
102904
(
2006
).
64.
N.
Shi
and
R.
Ramprasad
,
Phys. Rev. B
74
,
045318
(
2006
).
65.
S.
Krukowski
,
P.
Kempisty
,
P.
Strak
,
J. Appl. Phys.
114
,
143705-1-12
(
2013
).
66.
B.
Meyer
and
D.
Marx
,
Phys. Rev. B
67
,
35403
(
2003
).
67.
P.
Kempisty
,
S.
Krukowski
,
P.
Strak
,
K.
Sakowski
,
J. Appl. Phys.
106
,
054901
1-10
(
2009
).
68.
J.
Neugebauer
and
M.
Scheffler
,
Phys. Rev. B
46
,
16067
(
1992
).
69.
Z.
Romanowski
,
P.
Kempisty
,
K.
Sakowski
,
P.
Strak
, and
S.
Krukowski
,
J. Phys. Chem. C
114
,
14410
(
2010
).
70.
P.
Strak
,
P.
Kempisty
,
M.
Ptasinska
, and
S.
Krukowski
,
J. Appl. Phys.
113
,
193706
(
2013
).
71.
V.
Fiorentini
and
F.
Bernardini
,
Phys. Rev. B
58
,
15292
(
1998
).
72.
V.
Fiorentini
,
F.
Bernardini
,
F.
Della Sala
,
A.
Di Carlo
, and
P.
Lugli
,
Phys. Rev. B
60
,
8849
(
1999
);
V.
Fiorentini
,
F.
Bernardini
,
F.
Della Sala
,
A.
Di Carlo
, and
P.
Lugli
,
Phys. Rev. B
114
,
14410
(
2010
).
73.
P.
Lefebvre
,
A.
Morel
,
M.
Gallart
,
T.
Taliercio
,
J.
Allegre
,
B.
Gil
,
H.
Mathieu
,
B.
Damilano
,
N.
Grandjean
, and
J.
Massies
,
Appl. Phys. Lett.
78
,
1252
(
2001
).
74.
P. A.
Dróżdż
,
K. P.
Korona
,
M.
Sarzyński
,
T.
Suski
,
R.
Czernecki
and
D.
Wasik
,
Physica Status Solidi (b)
253
,
284
291
(
2016
).
75.
K. P.
Korona
,
J.
Borysiuk
,
C.
Skierbiszewski
, and
S.
Porowski
,
Acta Phys. Pol. A
112
,
243
(
2007
).