A Comparative Study on the Influence of Elastic ... - Journal Repository

British Journal of Applied Science & Technology 4(15): 2241-2250, 2014 SCIENCEDOMAIN international www.sciencedomain.org

A Comparative Study on the Influence of Elastic Scattering Mechanisms on the Electron Mobility in Wurtzite and Zincblende Gallium Nitride K. Alfaramawi1,2* 1

Science Department, Teachers College, King Saud University, Riyadh, Saudi Arabia. 2 Physics Department, Faculty of Science, Alexandria University, Alexandria, Egypt. Author’s contribution This whole work was carried out by the author KA.

th

Original Research Article

Received 26 December 2013 th Accepted 9 March 2014 th Published 7 April 2014

ABSTRACT The electron mobility of zincblende (ZB) and wurtzite (WZ) structures of n-type GaN is investigated as a function of both temperature and donor concentration. Numerical calculations of the mobility are carried out in a temperature range from 10 K up to 400 K 19 -3 27 -3 and donor doping concentrations from 10 m to 10 m . Elastic types of scattering mechanisms are considered. These types include ionized impurity scattering, neutral impurity scattering, deformation potential acoustic phonons scattering and piezoelectric potential scattering. The electron drift mobility versus temperature shows peak behavior for both zincblende and wurtzite GaN structures. Below approximately 120/140 K (for ZB and WZ GaN respectively), the mobility increases when the temperature is increased. In this temperature range, the ionized impurity scattering is assumed to be the dominant scattering mechanism. Above 120/140 K, the mobility is lowered down by raising the temperature. In this regime, the lattice scattering is considered to be the dominant mechanism. The variation of the electron drift mobility with donor concentration at room temperature shows a continuous decrease with the increase of impurity doping concentration. This is probably due to Coulomb scattering present particularly at low temperatures.

Keywords: Relaxation time; mobility; impurity scattering; lattice scattering. ____________________________________________________________________________________________ *Corresponding author: E-mail: [email protected];

British Journal of Applied Science & Technology, 4(15): 2241-2250, 2014

1. INTRODUCTION Gallium Nitride (GaN) is one of the III-V semiconductors that has a direct band gap and is considered as a promising material for high-power, optoelectronics and high temperature devices. It has a valuable application in blue light emitting diodes (LED) and blue laser diodes [1]. GaN crystallizes in two structure modes. One of them is the zincblende (ZB) structure (also known as cubic or β-GaN) and the other is wurtzite (WZ) structure (sometimes referred to as hexagonal or α-GaN) [2]. The two modern techniques that are commonly used for fabrication of the most GaN advanced devices, including LEDs, are molecular beam epitaxy (MBE) and metal organic chemical vapor deposition (MOCVD) [3,4]. In GaN, charge carriers can suffer scattering by many centers depending on the temperature and type and density of foreign impurities. The scattering mechanisms contain polar optic phonon scattering, ionized impurity scattering, neutral impurity scattering, acoustic phonon deformation potential scattering and acoustic phonon piezoelectric scattering mechanisms [5]. In addition, scattering by dislocations can also take place. Some of the previous scattering mechanisms are elastic in nature and others are inelastic. The elastic processes are being the acoustic phonon deformation potential scattering, acoustic piezoelectric scattering, neutral impurity scattering and ionized impurity scattering. Mobility evaluation of charge carriers in GaN is an important key of other parameters such as resistivity, electron drift velocity and free carrier concentration. The mobility of GaN has been investigated by several groups [6-11]. Rode and Gaskill [6] have used an empirical formula to calculate the Hall mobility as a function of the doping concentration. The model was found to agree with the experimental data at low electron concentrations, but it resulted in large deviations at higher concentrations. Look et al. [7] and Look and Molnar [8] have developed a two-layer model for the electron mobility. Weimann and Eatman [10] and Ng et al. [11] have investigated the role of dislocation in mobility, where the decrease of the electron mobility at lower carrier concentrations is attributed to the scattering of electrons by charge threading dislocations, which act as Coulomb scattering centers. Temperature and impurity dependence of the electron mobility taking into account the scattering by dislocations were studied [12,13]. In this work, we report on a numerical calculation of the electron mobility as a function of temperature and doping concentration for GaN semiconductor of both zincblende and wurtzite crystal structures. Elastic scattering processes including acoustic phonon scattering, neutral impurity scattering and ionized impurity scattering are considered.

2. THEORETICAL MODEL The temperature-dependent electron mobility due to ionized impurities is calculated according to the expression [14] = / ln 1 + − = =

64√ 2/ √

∗

96 ∗ ℎ

1+

1 − 1 − " 1 − %

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where is the semiconductor permittivity, e is the electronic charge, is Boltzmann constant, T is the temperature, is the density of ionized impurities, ∗ is the effective mass of the electron and ℎ is the Planck’s constant. On the other hand, electrons can suffer scattering by non-ionized impurities. The mobility due to neutral impurities is estimated by the temperature independent relation on the form & =

∗ 10& ℎ

2

where & is the concentration of the neutral impurities. Carriers in semiconductor are scattered by lattice thermal vibrations. The elastic types of the lattice vibration is the deformation potential acoustic mode. The mobility due to deformation potential acoustic phonons is expressed as [14] ()*+ =

.

2√2, - 0

33

/

1

∗ 4/ /

3

where , is the average longitudinal elastic constant of the semiconductor and 3 is the acoustic deformation potential. The mobility due to scattering by piezoelectric potential acoustic mode is given by [15] ()+5 =

.

16√2, - 0

3 ℎ+5

/ ∗ / /

4

where ℎ+5 is piezoelectric constant. Using equations 1 - 4, the total mobility (electron drift mobility) can be calculated using Mathiessen’s rule [14] 1 1 1 1 1 = + + + & ()+5 ()*+

5

3. RESULTS AND DISCUSSION Direct numerical calculations of the mobility due to different elastic scattering mechanisms are carried out for WZ and ZB modes of GaN. The assumed elastic types of scattering centers are the ionized impurities, neutral impurities, deformation potential acoustic vibration and piezoelectric acoustic vibration. The parameters used in the calculations for the two types of GaN structures are listed in Table 1. It is noticed that the calculated results do not change with the weak temperature dependence of the physical parameters presented in Table 1. It is assumed that the semiconductor is non-compensated so that 7 = + & and = 8, where 7 is the doping donor concentration, & is the non-ionized donor concentration (concentration of neutral impurities) and 8 is the free carrier concentration. For GaN, the o effective Bohr radius is 32.9 A [16], by using Mott criterion [17], one can estimate the critical

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concentration of donors which determines the border value between high and light doping 23 -3 levels. The critical donor concentration is estimated as 4.38 x 10 m . In our work, the 23 -3 concentration of the donor impurities was chosen as 1 x 10 m when calculating the temperature dependent mobility. Table 1. Physical parameters of ZB and WZ GaN (extracted from [18] and [19]) Parameter * Effective mass, m Dielectric constant, 2 Longitudinal elastic constant, , (N/m ) Piezoelectric constant, ℎ+5 Acoustic deformation potential, 3 (eV)

Value (ZB) 0.15 mo 9.7 10 2.2x10 0.7 9.2

Value (WZ) 0.22mo 8.9 10 2.65x10 0.7 9.2

Equations 1a – 1c are used to calculate the electron mobility due to scattering by ionized impurities. The temperature dependence of the mobility is shown in Fig. 1 for GaN WZ and ZB structures in the temperature range from 10 K up to 400 K. It is noticed that mobility decreases by lowering down the temperature for both WZ and ZB types. The reduction of the mobility at lower temperatures might be due to strong Coulomb interaction and this may indicate that ionized impurity scattering is becoming the dominant scattering process at low temperatures.

1.4

ZB

1.2

WZ

0.8

2

µion (m / V.s)

1.0

0.6 0.4 0.2 0.0 0

50

100

150

200

250

300

350

400

450

T (K)

Fig. 1. Mobility due to ionized impurity scattering versus temperature for GaN zincblende (solid lines) and wurtzite (dashed lines) crystal structures. Curves represent the numerical calculation according to equations 1a-1c and data presented in Table1 for the two GaN structures

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The mobility due to neutral impurities is estimated using equation 2. The neutral impurity scattering process is a temperature independent and then the mobility associated to it is expected to be a constant. The mobility due to neutral impurity scattering for ZB GaN is 2 estimated as 0.61 m /V.s. On the other hand, for WZ structure the mobility of the electron as 2 a result of neutral impurity scattering is found to equal 0.92 m /V.s. These values are somehow in agreement with the values obtained by many authors [20 and references therein]. = ()*+ + ()+5 , the mobility due to Using equations 3 and 4 and by aid of the relation () acoustic phonon scattering is calculated. Fig. 2 shows the calculated mobility for WZ and ZB structures. The average longitudinal elastic constant of GaN is determined from the relation , = 9:; where 9 is the mass density of GaN and :; is the speed of sound in the material. The mobility due to acoustic phonon scattering decreases by increasing the temperature. From Fig. 2, it seems that this type of scattering affects strongly the electrons and yields an abrupt decrease in the mobility as the temperature is increased. The decrease of the mobility due to acoustic phonon scattering when the temperature is raised indicates that this scattering mechanism is the dominant one particularly at high temperatures.

0.8 0.7

WZ

2

µac (m /V.s)

0.6 0.5 0.4 0.3

ZB

0.2 0.1 0.0 0

50

100

150

200

250

300

350

400

450

T (K)

Fig. 2. Mobility due to acoustic phonons scattering versus temperature for GaN zincblende (solid lines) and wurtzite (dashed lines). Curves are best approximation according to equation 3 for acoustic deformation potential scattering and equation 4 for piezoelectric potential scattering and data presented in Table 1. The two types of mobilities are combined together by using Mathiessen’s rule

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Using Mathiessen’s rule (5) and equations 1 - 4, one can calculate the electron drift mobility 2 for GaN. The room temperature electron mobility is estimated as 0.066 m / V.s for ZB mode 2 23 -3 while it is 0.094 m / V.s for WZ mode for donor concentration at 1 x 10 m . Fig. 3 demonstrates the calculated mobility plotted against temperature in the range from 10 K up to 400 K for WZ and ZB structures compared with experimental data. The solid symbols represent an experimental mobility extracted from Ref. [12] for sample with 23 -3 concentration 1.03 x 10 m while open circles represent experimental data taken from Ref. 23 -3 [21] for sample of doping concentration 1.5 x 10 m . As the mobility possesses peak behavior, this indicates two distinct regions. At low temperature regime till approximately 120/140 K (depending on the type of GaN), the electron mobility increases by raising of T and the ionized impurity scattering may be the dominant process controlling the mobility at this regime. At temperatures higher than 120/140 K the electron mobility gradually decreases as the temperature is increased and the acoustic phonon scattering is probably being the dominant mechanism.

0.10

ZB

0.08 0.07

2

Electron drift mobility (m / V.s)

0.09

0.06 0.05 0.04 0.03

WZ 0.02 0.01 0.00 0

50

100

150

200

250

300

350

400

450

T (K) Fig. 3. Calculated electron drift mobility as a function of temperature for GaN ZB (solid line) and WZ (dashed line) structures. Curves are plotted according to calculations using equations 1 – 5 with the data listed in Table 1. The solid symbols represent experimental data for GaN sample extracted from Ref. [12] for samples with donor 23 -3 concentration 1.03 x 10 m and the open circles represent experimental data taken 23 -3 from Ref. [21] for sample of doping concentration 1.5 x 10 m

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British Journal of Applied Science & Technology, 4(15): 2241-2250, 2014 2

The peak mobility for ZB structure is about 0.081 m /V.s and for WZ structure is 0.0972 2 m /V.s. The difference between the calculated mobility and the experimental results may come from that the calculations consider only the elastic types of scattering mechanisms and ignore the other types (i.e. optical phonon scattering and dislocation scattering). Equations 1-5 are numerically simulated to calculate the electron drift mobility as a function of donor concentration at fixed temperature. Fig. 4 represents the electron drift mobility as a 19 -3 27 -3 function of doping donor concentration in the range from 10 m up to 10 m for ZB phase of GaN at room temperature. The curves are compared with experimental data taken from Ref. [22] for sample of ZB structure at room temperature. The electron mobility is lowered by the increase of doping concentration. This is attributed to the increase of Coulomb scattering either at room or low temperatures. There is a good agreement between our calculated results and the available experimental data.

0.30

ZB

2


0.27 0.24 0.21 0.18 0.15 0.12 0.09 0.06 0.03 0.00 -0.03 10

19

10

20

10

21

10

22

23

10

10

24

10

25

26

10

27

10

28

10

-3

Doping concentration (m ) Fig. 4. Electron drift mobility as a function of donor concentration for ZB phase of GaN at room temperature. Equations 1 – 5 are used to calculate the mobility with the data presented in Table 1 for ZB- GaN. Solid points show experimental data taken from Ref. [22] for sample of ZB-GaN at room temperature Fig. 5 shows the electron mobility versus donor concentration for WZ phase of GaN compared with experimental data taken from Ref. [21]. The same behavior as in ZB structure is noticed.

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0.22 0.20

WZ

2


0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 -0.02 19

10

20

10

21

10

22

10

23

10

24

10

25

26

10

10

10

27

28

10

-3

Doping concentration (m ) Fig. 5. Electron drift mobility as a function of donor concentration for WZ GaN phase at room temperature. The curve is due to simulations of equations 1 – 5 to calculate the mobility with the data presented in Table 1 for ZB- GaN. Solid points show experimental data taken from Ref. [21] for sample of WZ-GaN at room temperature

4. CONCLUSIONS Calculation of the electron mobility due to different types of elastic scattering mechanisms were proceeded at temperature range between 10 K and 400 K and donor concentrations 19 -3 27 -3 from 10 m to 10 m . The electron drift mobility shows peak at approximately 120/140 K, depending on the type of GaN, distinguishes two different regions. From 10 K up to about 120/140 K, the mobility increases by elevation of the temperature. Further increase of the temperature causes a decrease in the electron mobility. The mobility as a function of the donor concentration is also studied at room temperature. The electron drift mobility decreases as the donor concentration is raised due to probable Coulomb scattering.

ACKNOWLEDGEMENTS The author extends his appreciation to the Deanship of Scientific Research at King Saud University, Saudi Arabia, for funding this work.

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COMPETING INTERESTS Authors have declared that no competing interests exist.

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© 2014 Alfaramawi; This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Peer-review history: The peer review history for this paper can be accessed here: http://www.sciencedomain.org/review-history.php?iid=478&id=5&aid=4257

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