Monte Carlo study of transport properties in copper halides

Physica B 315 (2002) 201–209

Monte Carlo study of transport properties in copper halides W. Sekkal, A. Zaoui* Condensed Matter Group, International Center for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy Received 22 January 2001; received in revised form 23 August 2001; accepted 2 October 2001

Abstract We present a study of transport properties in copper halides, by means of a Monte Carlo method. A negative differential resistance is observed at elevated field (B50 kV=cm) with a decreasing of mobility. The same behavior appears in CuBr and CuI at lower field. Results also indicate a negligible effect of ionized scattering in such materials. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Monte Carlo; Gunn effect; Copper halides

1. Introduction The electron transport properties of III–V and II–VI materials [1–5] have been studied extensively for many years. GaAs and InP have received a great deal of attention because they have demonstrated great potential for many devices applications. Indeed, these materials have been found to produce Gunn type oscillations. The excellent electrical characteristics of these materials exploit the small G valley effective mass and relatively large G to L separation. In recent years, considerable research have been developed toward the study of I–VII materials. As compared with the III–V compounds, the copper halides have attracted particular interest because they are characterized by large bandgaps, negative *Corresponding author. Address for correspondence: MaxPlanck-Institut fur . Metallforschung, Seestr. 92, 70174 Stuttgart, Germany. E-mail address: [email protected] (A. Zaoui).

spin orbit [6], an unusually large temperature dependence [7] and diamagnetism behavior [8], a large ionicity, and finally new high pressure phases that have not yet been observed for III–V and II– VI semiconductors [9]. However, to date a few theoretical investigations have been made on the electronic transport properties in these compounds. From atomistic simulation methods, the Monte Carlo (MC) is probably the most complete and exact way to predict transport phenomena in semiconductors [10,11]. The applications of this method have brought enlightenment on many aspects more or less known of carriers dynamics, for instance on the diffusion phenomena in strong field conditions, or on the energy and velocity relaxation. The accent has always been put on the fact that the MC method allows more and more numerous and varied complex energy bands and scattering mechanisms to be taken into account. In this paper, we study the electronic transport properties of CuCl, CuBr and CuI. Calculations

0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 0 4 3 - 2

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have been performed using an ensemble Monte Carlo simulation with the inclusion of the first four conduction bands in each material, determined by the empirical tight binding method [12]. Results of steady-state simulation are presented and the difference between the transport properties of the three compounds is discussed. We present in Section 2 the computational model. In Section 3, results are presented. Section 4 contains the conclusion.

2. Model The MC procedure used in this work [10,11] consists of modeling the motion of an electron as a sequence of free flights interrupted by collisions. Following scattering mechanisms have been considered in the analysis: (a) Polar-optical scattering: the scattering rate is given by
n 1=2 15 m l0 ðEÞ ¼ 5:616  10 me
 1 1 1  E0  F0 ðE; E 0 Þ k0 ks E 1=2 ( ) N0 ðabsorptionÞ  ; ð1Þ N0 þ 1ðemissionÞ where _$0 ; E0 ¼ q ( ) E þ E ðabsorptionÞ 0 E0 ¼ ; E  E0 ðemissionÞ  1=2 
E þ E 0 1=2    ; 4 ln 1=2 F0 ðE; E Þ ¼ 4 E  E 0 1=2  0

N0 ¼

1

1 : expð_$0 =KB TÞ  1

ð2Þ

ð3Þ

ð4Þ ð5Þ

E is the electron energy (in eV), mn =me is the relative electron effective mass, and k0 and ks are high and low frequency dielectric constants. T is the lattice temperature, N0 is the optical phonon occupation number, and o0 is the optical phonon frequency.

(b) Non-polar optical scattering (intervalley scattering):
n 3=2 5 m lon ðEÞ ¼ 1:129  10 me ( ) Non ðabsorptionÞ D20 0 1=2  E  ; ð6Þ rEon Non þ 1ðemissionÞ Non ¼

1 ; expðqEon =kB TÞ  1

ð7Þ

where D0 is the intervalley deformation potential, r is the crystal density, and Non is the intervalley phonon occupation number. All energies are measured in (eV) and are counted from the bottom of the conduction band minimum. E 0 ¼ E8Eon :

ð8Þ

(c) Acoustic phonon scattering: la ðEÞ ¼

0:449  1018 ðmn =me Þ3=2 TE12 1=2 1 E ðs Þ; ru2

ð9Þ

here r is the crystal density in g=cm3 ; E1 is the acoustic deformation potential (in eV), T is the lattice temperature (K), and u is the sound velocity (cm/s). (d) Intervalley scattering between non-equivalent valleys:
n 3=2 2 Dij 1=2 m 5 lij ðEÞ ¼ 1:129  10 Zj E me rEij ij ( ) Nij ðabsorptionÞ  ; ð10Þ Nij þ 1ðemissionÞ Nij ¼

1 ; expð_$ij =kB TÞ  1

E 0 ¼ E  Eij  Wij ;

ð11Þ ð12Þ

where Wij is the energy separation between the valleys and oij is the intervalley phonon frequency. Eij ¼ _$ij =q is the energy of the intervalley scattering phonon in (eV), Zj is the number of the equivalent valleys, Dij is the intervalley deformation potential and Nij is the intervalley phonon occupation number. This scattering is caused by the non-polar optical phonons. The energies are accounted from the bottoms of the minima of the conduction band.

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(e) Intervalley scattering between equivalent valleys:
n 3=2 2 m De 5 lij ðEÞ ¼ 1:129  10 ðZe  1Þ ðE 0 Þ1=2 me rEe ( ) Ne ðabsorptionÞ ðs1 Þ;  ð13Þ Ne þ 1ðemissionÞ Ne ¼

1 ; expð_$e =kB TÞ  1

ð14Þ

Ee ¼

_$e ; q

ð15Þ

203

NB Þ for AC–BC alloy. jDuj is the scattering potential. The model includes ionized impurity scattering [13] also. This effect is felt mainly at low fields where electron energies and velocities are small. The reason for this lies in the form of the scattering rate for an electron with wave vector k; which can be written as follows: pffiffiffi RðkÞ ¼ ð2 2pne4 m * 1=2 =e2 _2 b2 Þ  ½E 1=2 ð17Þ with

where Ze is the number of equivalent valleys, De is the deformation potential, _$e is the intervalley phonon energy (oe is the intervalley phonon frequency), and Ne is the intervalley phonon occupation number. (f) Alloy scattering: The alloy scattering rate is given by pffiffiffi 1 3p 2ðmn Þ3=2 ¼ CA ð1  CA ÞjDuj2 ðKB TÞ1=2 ; tAl 8NA _2 e1=2 ð16Þ where e ¼ E=KB T; NA is the numbers of A atoms, NB ; the number of B atoms and CA ¼ NA =ðNA þ

b2 ¼

4pne2 : eKB T

ð18Þ

Here n ¼ ND is the free-electron density for uncompensated systems, mn is the effective mass, E is the electron energy, T is the lattice temperature, KB is the Boltzmann constant, and e the dielectric constant. The simulation parameters for CuCl, CuBr, and CuI at ambient temperature are presented in Table 1. These parameters can be classified as a bulk material parameters and band-dependent one. The last parameters are calculated from a tight binding method [12] for CuCl, CuBr, and CuI. The velocity components vi are selected from

Table 1 Parameters for CuCl, CuBr, CuI and CuCl1x Brx CuCl Bulk material parameters Density (g=cm3 )a Velocity of sound ðcm=sÞa High-frequency dielectric constanta Low-frequency dielectric constanta

4.136 3.63 3.61 7.90

Valley-dependent material parameters Optical phonon frequency ð103 rad=sÞa Equivalent intervalley phonon frequency ð103 rad=sÞ Intervalley phonon frequency ð103 rad=sÞ Acoustic deformation potential (eV)a Equivalent intervalley deformation potential (109 eV=cm) Intervalley deformation potential (109 eV=cm) Valley separation (eV)b Central valley effective mass ðm0 Þb Satellite valley effective mass ðm0 Þb

1.19 1.0 1.0 0.425 1.0 1.0 0.97 0.43 2.2

a b

From Ref. [14]. From Ref. [12].

CuBr 4.72 3.38 4.062 7.90

0.92 1.0 1.0 0.65 1.0 1.0 1.72 0.28 1.74

CuI 5.667 3.146 4.58 6.5

0.96 1.0 1.0 1.4 1.0 1.0 1.32 0.33 1.58

x ¼ 0:1

x ¼ 0:2

x ¼ 0:3

4.1944 3.605 3.655 7.90

4.2528 3.58 3.70 7.90

4.3112 3.555 3.7456 7.90

1.19 1.0 1.0 0.425 1.0 1.0 1.045 0.415 2.154

1.19 1.0 1.0 0.425 1.0 1.0 1.12 0.40 2.108

1.19 1.0 1.0 0.425 1.0 1.0 1.19 0.385 2.062

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the following equation: Z vi f ðv0i Þ dv0i ; R¼ N

ð19Þ

where R is a uniformly distributed number, and f ðvi Þ is the probability distribution. A different random number is used for each velocity compo-

Fig. 1. Band structures of CuCl (a), CuBr (b), and CuI (c).

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nent. Effective masses have been determined by fitting the tight binding calculated band structure. The values of the sound velocities were calculated from the elastic constants [14] of the zinc blende structure using the approach of Wiley [15].

3. Results and discussion Tight binding band structures along the main symmetry points of the Brillouin zone are shown in Figs. 1(a), (b) and (c) for CuCl, CuBr, and CuI, respectively. Although the three compounds are very similar (all have a direct fundamental gaps at the G point) their conduction bands appear to be sufficiently different to cause significant differences in their transport properties. For example, the nearest satellite valley in CuCl (L point) is about 0:97 eV above the conduction band minimum, while the nearest satellite valley in CuBr and CuI (L point) is more than 1:72 and 1:32 eV above the minimum, respectively. Fig. 2 shows the calculated electron drift velocity for CuCl in the zinc-blende structure as a function of electric field strength at 77 and 300 K: A negative differential resistance is observed with a sudden decrease in the mobility at elevated field (around 50 kV=cm). Probably the most striking feature of the temperature-threshold field dependence and temperature-peak velocity (Fig. 3) is the strong dependence of temperature with the threshold field for the onset of negative differential mobility. It is shown that this changing is from 50 kV=cm at 300 K to 30 kV=cm at 4 K: This dependence is less important for GaAs (only from 2:3 kV=cm at 77 K to 3:6 kV=cm at 500 K). In order to investigate the effect of ionized impurity scattering on the Gunn effect, we display in Fig. 4 the velocity versus the electric field for CuCl for ND ¼ 0 and 106 =cm3 : Even if these effects are important in III–V materials (in GaInAs [16] for instance), a small change in the curve is noticed. The dependence of the fraction of electron in the central valley versus electric field is shown in Fig. 5. In connection with the velocity-field curves, these characteristics provide some more physical insight into the transport processes. Occupancies

Fig. 2. Temperature dependence of the electron velocity-field characteristic in CuCl.

versus electric field dependences show a typical behavior for compound semiconductors [17–20], for which scattering due to polar optical phonons is most effective below the threshold field. If field strenghts is greater than the threshold field, the electrons significantly populate the satellite valley, where new intervalley deformation scattering mechanisms become effective. As a result of the increasing scattering rate, the increase of energy with electric fields slows down in both cases. Consequently, for CuCl compared to GaAs, the combined effect of high effective mass, a high satellite valley separation energy and a slightly higher phonons scattering rate within the gamma valley induce a great threshold electric field and the decrease of the velocity. In the second part of this work, we study the drift velocity of CuBr and CuI under electric field and we will treat the effect of ionicity on the peak velocity. Fig. 6 shows the same behavior for CuBr and CuI at low field. Besides, a slight increasing in the peak velocity is noticed for CuBr as compared to the case of CuCl and CuI. This increasing could

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Fig. 3. (a) The peak velocity dependence of the temperature; (b) the threshold field dependence of the temperature.

Fig. 4. Electron velocity as a function of the electric field for ND ¼ 0 and 106 =cm3 :

Fig. 5. Fraction of electron in the ð0 0 0Þ valley as a function of field strength.

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be due to the different energy band structures and in particular to the energy separation between the central valley and the higher lying minima at L point which has the highest value for CuBr (see Table 1). The augmentation of the threshold field for CuCl compared to CuBr and CuI is may be due to the increase of the strength of the polar scattering rate, which is due to the strong ionic character of CuCl (0.74 [14]). In order to understand better the variation of the peak velocity and the threshold field from CuCl to CuBr, we have plotted (Fig. 7) the threshold field and the peak velocity for CuCl1x Brx as a function of the mole fraction ‘‘x’’ using the parameters listed in Table 1. We observe that both variations are linear according to the following equations:

Fig. 6. Drift velocity versus electric field for CuCl (dashed line), CuBr (dotted line), and CuI (solid line).

E ¼ 57:20  35:76  x;

ð20Þ

V ¼ 1:92  106 þ 0:53  106  x;

ð21Þ

where V is the peak velocity, and E is the threshold field. The high peak velocity corresponds to CuBr.

Fig. 7. The variation of the threshold field with concentration (a); the variation of the threshold field with concentration (b).

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A comparison is made between the transport properties of copper halides and those found for III–V and II–VI materials. We have also reported in Table 2 semiconductors which exhibit a Gunn effect at room or low temperature and/or by applying pressure. As a first conclusion, it seems that the high peak velocity should be achieved in materials with (a) a large low field mobility and (b) a large energy separation between the central valley and the higher minima. For instance, GaAs has relatively small valley separation in the order of 0:31 eV: The separation is somewhat larger at 0:6 eV for InP that is predicted to have a larger peak velocity than GaAs. The large low field mobility needed for high velocity, requires low bandgap materials with small effective masses. For this reason, copper halides materials have a low peak velocities than those found for GaAs or for InP. However, a narrow bandgap, as in InAs, leads to low breakdown voltages. Thus, a compromise must be found between these two requirements. In the following, we try to link the effect of ionicity of materials on peak velocity. As it is known, ionicity increases from III–V to I–VII materials. In Fig. 8, we plot the peak velocity versus ionicity. The variation is non-linear and is fitted to the following equation: V ¼ 2:0770 þ 21:2654 * fi  24:8723 * fi2 ;

Table 2 Materials in which effects due to negative differential mobility have been observed

GaAsa InPb CdTec ZnSed Gee InSbf InAsg

Threshold field (kV/cm)

Peak velocity (107 cm=s)

3.3 7–10 13 38 2.3 (77 K) 0.6 (77 K) 1.4

2. 2.7 1.3 1.50 1.4 (77 K) F >2

a

From Ref. [10]. From Ref. [4]. c From Ref. [5]. d From Ref. [21]. e From Ref. [22]. f From Ref. [23]. b

ð22Þ

where V is the peak velocity and fi is the ionicity of materials taken from Ref. [24] or [25]. Therefore, this analysis may allow us to use such empirical relations to predict the peak velocity of other semiconductors only on the basis of their ionicity values.

4. Conclusion We have presented a Monte Carlo simulation study based on different scattering mechanisms (acoustic, polar optical, impurity and intervalley phonon scattering), for the transport properties in copper halides. Calculations show the presence of a Gunn effect with a negative resistance at B50 kV=cm for CuCl. The same behavior appears in CuBr and CuI at lower field. The increasing of

Fig. 8. The peak velocity versus ionicity.

the threshold field from CuBr, CuI to CuCl can be related to the increase of the strength of the polar scattering rate which is important in CuCl. Results also indicate a negligible effect of ionized scattering in these materials.

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