AlxGa1-xN nanowire structure

Accepted Manuscript Title: Effect of biasing voltage on Quantum Confinement in GaN/Alx Ga1−x N Nanowire Structure Authors: Ulhas S. Sonawane, E.P. Samuel, C.K. Kasar, D.S. Patil PII: DOI: Reference:

S0030-4026(17)30267-X http://dx.doi.org/doi:10.1016/j.ijleo.2017.03.008 IJLEO 58931

To appear in: Received date: Revised date: Accepted date:

3-8-2016 23-2-2017 3-3-2017

Please cite this article as: Ulhas S.Sonawane, E.P.Samuel, C.K.Kasar, D.S.Patil, Effect of biasing voltage on Quantum Confinement in GaN/AlxGa1-xN Nanowire Structure, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2017.03.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Effect of biasing voltage on Quantum Confinement in GaN/AlxGa1-xN Nanowire Structure Ulhas S. Sonawane1, E. P. Samuel2, C. K. Kasar1, D. S. Patil1* 1Nano-Simulation Laboratory, Department of Electronics, North Maharashtra University, Jalgaon [Maharashtra], India 2Department

of Mechanical Engineering, Korea University, Seoul, South Korea

*Corresponding author *[email protected]

Abstract: Effect of biasing voltage on the quantum confinement has been studied to analyse performance of GaN/AlGaN nanowire structure. Here, rectangular model of quantum nanowire with GaN active region surrounded by AlxGa1-xN barrier region has been considered. Schrodinger equation in two dimensions has been solved for this model to study electron confinement. The solving approach has been considerately utilized for sufficient accuracy and efficiency. Moreover, wave function intensity and FWHM for different biasing voltages has been obtained along with the potential profile in quantum nanowire structure. Biasing voltage was increased from 1V to 2.5V. The confinement factor has been increased from 0.81 to 0.838 with an increase in bias of 1.0 V to 2.5 V at Al mole fraction of 0.25. Our analysis shows that increasing biasing voltage increases the confinement of carriers in the wire region which will be advantageous for lasing action. Significant decrease in escape time of the carriers with biasing voltage has been obtained.

Keywords: Quantum confinement; GaN/AlGaN; Nanowire

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1. Introduction: III-V nitride semiconductor quantum wire (QWR) laser diode has potenetial to overcome the bottleneck of optoelectronics devices for high speed and large data density. Optical and electrical properties of Gallium Nitride (GaN) and Aluminum Gallium Nitride (AlGaN) have been subject of applied research for optical devices particularly light emitting diode (LED) and laser diodes [1-2]. These nitride semiconductors has immense and necessary capability for better performance of laser diodes from visible to ultraviolet wavelegnth [3-4]. Nitride semicondutors with their wide bandgap heterostructure have siginficant advantages such as two photon absorption (TPA) do not interefere with saturable absorption. The recovery time of absorption is substantially low (<1ps) and the shorter dephasing time brodens the homogeneous line width, low surface state densities and robust spin coherence along with excitonic effect at room temperature for polariton lasers [5]. Moreover, in modern optical communications systems and digitalized data read/write devices needs low power operations. Subsequently, for obtaining such properties, the choice of optical material and low dimensional (quantum structures) are strongly required [6-7]. Quantum wire structures provide better optical properties [8] than quantum well structures with controlled growth [9]. On the other hand, quantum dot structure leads to smaller optical confinment. In quantum wire lasers small volume of active region and large density of states is responsible for lower threshold voltage. This lower threshold voltage value is required in communication system for lower power consumption and high modulation bandwidth. These nanowire structures have vast applications in different fields such as photonics, nanoelectronics, bio-medical, chemical sensing, gas sensing etc [10-16]. The semiconductor QWRs attains spectral braodening due to thinning of wire dimensions concerns in practical laser diode and optical applications [17]. The variation in QWR dimension causes the singficantly energy fluctuations in confinement region. As far as optoelectronics and photonics applications are concerned, it is needed to control polarization in many applications like compact laser based 2

displays, non-linear optics and optical communications. Although use of gold substrate induces the high propogation attributing the higher threshold values; there are some reports about linear polarization in GaN nanowire laser on Gold substrate [18]. It is possible to obtain inherant and linearly polarized lasing in rectangular shaped nanowire. The intrinsic polarization can be controlled using rectangular cross section. It has been noted that polarization was along short dimensions due to higher transverse confinement factors resulting from the rectangular cross section. The main advantage of using the rectangular shaped wire is that it does not require the lossy substrate so the threshold values can be decreased. Furthermore, the barrier region of AlGaN can be grown along polar c-plane or non polar a-plane. Although devices fabricated with polar c-plane, AlGaN having the reduced optical transition probability and device efficiency due to the large Quantum Confinement Stark Effect (QCSE). So it is advantageous to consider non polar a-plane AlGaN for higher device efficiency to reduce or cancel the polarization and piezoelectric effects [19-21]. Different modern fabrication techniques like chemical vapor deposition (CVD), Molecular beam epitaxy (MBE), metal organic chemical vapor deposition (MOCVD), can be used to grow complex hexagonal, triangular, or rectangular shaped GaN nanowire. The process of fabrication of nanowire structure is time consuming and costly. To make it cost and time effective, it is obligatory to carryout simulation prior to actual fabrication. This simulation process can predict the actual behavior and one can get the optimized parameters for materials and designing the nitiride based QWRs. The solution of Schrodinger equation for quantum wire is more complex due to existance of two dimensional electron confinement. Therefore, efforts have been carried out to compute eigen energy by using the effective mass approximations. In this paper, we have applied the boundary conditions and continuty at the interfaces between confinement and barrier regions [22-24] to solve the physical equations. By using wave functions appropriately

3

for the QWR, solution of Schrodinger effective mass equation with biasing has been worked out.

Solution of Schrodinger equation and electron distribution properties have been studied under biasing condition for nitride nanowire as shown in Fig. 1. Theory is presented in section 2. The results are discussed in section 3 and section 4 concludes the paper.

2. Mathematical modelling and simulation: To analyze the electron density and related properties in the quantum wires, the effective eigen value due to two dimensional confinement has been deduced. There is a significant need to realize the effective eigen energy as electrical properties are affected by low dimensional QWR.

4

Numerical approach has been employed to obtained the Eigen energy, which is explained through the flow chart as shown above. To calculate the Eigen energy, wave functions and related parameters such as Confinement factor and Escape time of electron, we have used the effective mass equation. Hence, we have started with Schrodinger equation in slowly varying enevelop approximation. In a quantum wire, Schrodinger equation for an electron with a lateral potential 𝑣𝑝 (𝑥, 𝑦), has the following form ∂2 ψ(x, y) ∂2 ψ(x, y) 2m∗ [ + ] − (𝑣 (x, y) − eF(x, y) − Ex,y )ψ(x, y) = 0 ∂x 2 ∂y 2 ℏ2 𝑝

(1)

where, m* is the effective mass with respect to the material of each layer, e is the electric charge and F is the applied electric field strength. The electric field potential can be calculated using the Eq. (2). 𝐹 = −𝑒𝑉𝑎 /𝐿

(2)

Here, Va is the applied bias voltage and L is the distance where the electric field is applied. The general solutions of the Schrodinger equation are given in Eq. (3), ψ(𝑥, 𝑦) = Ae(q(𝑥,𝑦)) + Be(−q(𝑥,𝑦))

for (𝑥, 𝑦) ≤ 0

(3a)

ψ(𝑥, 𝑦) = Csin(k(𝑥, 𝑦)) + Dcos(k(𝑥, 𝑦))

for(𝑥, 𝑦) ≤ a

(3b)

ψ(𝑥, 𝑦) = Fe(q(𝑥,𝑦)) + Ge(−q(𝑥,𝑦))

for (𝑥, 𝑦) ≥ a

(3c)

In Eq. (3), A, B, C, D, F and G are arbitrary constants, we had applied boundary conditionsto the general solutions as indicated in Eq. (3) to determine arbitary constants. These arbitary constants in succession required to evaluate Eigen energy using transcidental equation.‘a’ is wire width (we assumed the cross-section of wire to be square), q and k are the wave vectors in barrier and wire region respectively. The wave vectors in the wire region and the barrier region are related to effective mass by the following relation,

5

𝑞=

∗ 𝐸 √2𝑚𝑤

𝑘=

and

ℏ

∗ (𝑣𝑝 −𝐸) √2𝑚𝑏

(4)

ℏ

∗ At this juncture, ħ is reduced Planck's constant, 𝑚𝑤 is the effective mass of the GaN, while 𝑚𝑏∗

is the effective mass of the AlGaN which varies with the compostion of the aluminium. So, at interface first i.e. at z =0 Ae(qz) + Be(−qz) = Csin(kz) + Dcos(kz)

(3)

As z = 0 and neglecting the term with B as it is representing the reflected wave, A=D

(4)

Taking derivative of equation (3) and putting z=0 leads to, C=

qA

(5)

k

Similarly, at second interface i.e. interface z = a Csin(kz) + Dcos(kz) = Fe(qz) + Ge(−qz)

(6)

The term with arbitrary constant F can be neglected as it is again representing the reflected wave. Putting z = a along with value of C and D in terms of A, gives (7)

G = AL Where, q

1

L = ((k) sin(ka) + cos(ka)) e−qa

(8)

The complete wave function with the boundary conditions is defined as follows, 0

a

∗

b

∫ ψψ dz + ∫ ψψ dz + ∫ ψψ∗ dz = 1 −c 0

∗

0 2

(9)

a a

b

2

∫ (Ae(qz) ) dz + ∫ (Csin(kz) + Dcos(kz) )2 dz + ∫ (Ge(−qz) ) dz = 1 −c

0

(10)

a

This integration is solved in following way, A2 M + A2 N + A2 R = 1

(11)

6

where, M=(

1−e(−2qc) 2q

)

(12)

q 2 u v q N = (( ) ( ) + ( ) − ( ) w) k 2 2 k

(13)

L2 (e(−2qb) − e(−2qa) ) R= −2q

(14)

In equation (13), sin2ka u = (a − ( )) 2k

(15)

sin2ka )) 2k

(16)

v = (a + (

cos2ka − 1 w = (( )) 2k

(17)

from equation (10), 1 A=√ M+N+R

(18)

From equations (4), (5) and (7), D=A C=

qA k

G = AL Here the arbitary constants are deduced followed by dtermination of Eigen energy to form a complete wavefunction in nanowire.

3. Results and Discussion: Here, we present the analysis of the biasing effect on the electron properties of GaN/AlxGa1-xN quantum wire nanostructure. The material parameters used for this modeling and simulation are summarized in Table 1. The electric field 7

siginificantly changes the concentration and tunelling of electron in nanostructure due to lower dimensions. Thus, study of various electrical properties under the influence of applied bias becomes essential for optimization of physical, compositional and electrical parameters.

Figure 2 shows the potential profile of GaN/AlxGa1-xN quantum wire nanostructure for different Aluminum mole fraction in barrier region for different applied bias voltage. The Aluminum mole fraction has been resrticted to 0.1 to 0.25 for synchronizing work with the exeprimental data.To reduce the polarization and piezoelectric effects, grwoth of AlGaN has been conisdered along non polar a-plane axis. The analysis of these results depicts that increment of applied bias increases the potential energy since wave vector constants 'k' and 'q' significantly changes. The bandgap energy of the barrier region increases with the increase in Aluminum composition resulting in higher band offset between barrier and wire region. Thus higher composition of Aluminum in AlxGa1-xN increases eigen energy due to higher band offset between barrier and wire region as clearly observed in Fig. 2. When the applied biasing is of 1V for aluminum mole fraction of 0.17, the minimum potential energy in first barrier is computed to be 0.18 eV and for aluminum mole fraction of 0.25 it is 0.4 eV. The potential energy along with the distance across the barrier regions increases from lower edge to higher edge of the nanostructure. Finally, at the edge of the second barrier it reaches upto 0.38eV and 0.61eV for aluminum mole fraction of 0.17 and 0.25 respectively. When applied bias is 2.5 V the potential energy increases significantly in the barrier regions from edge to edge and slenderly in the wire regions. The potential energy increases from 0.2 to 0.64 eV and from 0.42 to 0.72 eV for Aluminum mole fraction of 0.17 and 0.25 respectively. The increase in potential energy for different Aluminium concetration across the nanostructure reveals that the probability of electron tunneling from wire region will be minor and better confinement will apparently emerge.

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(a) 1V, (b) 1.5V, (c) 2V and (d) 2.5V Wavefunction analysis is very useful and vital in quantum structures as it articulates about the particle’s complex behaviour and electron confinement. Hence, analysis has been extended to realize the electron confinement within the quantum wire nanostructure. The electron confinement ina quantum nanowire is essential for achieving the b population inversion to obtained desired stimulated emission in quantum nanowire lasing. Waterfall model helps us to relatively analyse electron distribution in GaN/AlxGa1-xN quantum nanowire structure for different biasing. Fig. 3 depicts electron confinement and distribution with varying applied biasing from 1 to 2.5 V. From this result, it is clear that the increase in biasing increases the wavefunction intensity. The wave function intensity was found to be incresed from 2.5x10-3 (a.u.) to 4.72x10-3 (a.u.) for biasing of 1 to 2.5 V respectively. Increase in electron density in active region of GaN gratifies lasing action. Therefore, the investigation of the electron confinement in active region of GaN has been carried out. More the confinement factor better is the laser diode efficiency due to probability of higher photon generation. Fig. 4 shows the confinment factor for various biasing voltages. It has been observed that increase in applied biasing voltage and Aluminium concentration in AlxGa1-xN increases the confinment factor. For Al mole fraction of 0.1,0.17 and 0.25 at 2.5 V applied bias voltage , the confinment factor deduced to be 0.8,0.82 and 0.838 respectively. It has been attributed to the increase in the band offset and availability of more electron in GaN active region for the lasing action. It is important to be considered that the applied electric field is not increased much and hence no asymmetric nature observed in our study. The escape time in the quantum wire structure is very crucial to be estimated for better electron confinement. It is necessary to keep electron density higher enough such that the threshold current density providing better efficiency at lower applied bias voltage. There is an analogy 9

in tunnelling of electrons in quantum structures and α decay in nuclear physics. The Confined electrons ocillates within quantum wire region and attempts to escape as they hits the barrier. The escape rate is proportional to frequency of photons (υ) generating in quantum wire region and the transmission probability of barrier. The escape rate of electron in the nanowire quantum structure fundamentally without carrier concentration consideration given by 1⁄τ = νexp(−2qb), where τ is escape time, q is the wave propagation constant of the barrier region and b is the barrier width. This study is basically to realize significance of applied bias voltage, barrier region thickness, Aluminium mole concentration and wavelength impact on quantum nanowire strucutre for optimizing physical and structural parameters.It has been observed that carriers that escapes from barrier region are captured by quantum confinement region with same rate [25]. Similarly, the carriers that escapes from the quantum region are captured in barrier region with same rate as that of escape rate.Evidently from Fig. 5 increase in wavelength and applied bias voltage,results in the escape time increment and decrement alnogwith the quantum structure distance respectively. As the escape time is inversely proportional to propagation constant, consequently with increase in applied biasing escape time reduces.Thus, electrons scarcity at boundary region long-drawn outs escape time. The influence of bias voltage is much more apparent in comparison to wavelength variation from 300 to 375 nm with interval of 25 nm. Figure 6 shows full width half maximum (FWHM) of the electron distribution wavefunction of the quantum wire nanostruture with respect to the applied biasing voltage.It is important to be consider that these FWHM are along x-axis as shown in Fig. 6. However, FWHM calculated along y-axis, are same due to dimensions along y-axis were kept simliar for symmetricty. FWHM in case of GaN/AlGaN quantum wire nanostructure shows nonlinear decrease with increase in applied bias voltage. For applied biasing of 1 V spread was calculated

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to be ~ 10.8 nm, while the FWHM furthermore decreases to 10.45 nm when applied biasing voltage reaches to 2.5 V. Figure 7 shows the electron confinement and wavefunction distribution pattern within quantum wire structure under variation of applied bias voltage. The image pattern has been obtained for applied bias voltage from 1 to 2.5 V with interval of 0.5 volts. The composition (x) of Aluminium in AlxGa1-xN was 0.25 to achieve better electron confinement and higher intensity of the wavefunction. The images in Fig. 7 clearly reveals the structure of the quantum wire to be square in shape. The shading variation appears to be circular in shape which shows the electron wavefunction to be sinusoidal (wave nature). The hot darker shade in the center region indicates peak values of wavefunction intesity obtained in the central region of the nanowire and approves the better electron confinement. Thus, it has been concluded that the photon generation under high electron density in the central region of the nanowire will improve the near field intensity and lower dispersion can be achieved in far field region.

4. Conclusions: Influence of nanowire geometry and curvature effects is very significant and investigated vigilantly to optimize the physical properties of nanowire structures. The

b which has been FWHM of the wave function decreases awith the increase in applied biasing attributed to higher electron confinement in the nanowire region with the increasing bias voltage. FWHM of 10.45 nm and better electron confinement has been demonstrated for 2.5 volt of applied biasing and for 25% Aluminum concentration in the barrier regions. The

c d biasing has been potential energy profile for various Aluminum concentration and applied obtained for entire quantum wire structure. Our analysis is very useful to demonstrate potential applicability of GaN/AlGaN nanowire for efficient lasers.

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Acknowledgment: Authors Ulhas S. Sonawane and C. K. Kasar are thankful to University Grants Commission, New Delhi to provide financial support under UGC-BSR (UGC Basic Scientific Research) Scheme for carrying this work.

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[12] Zhao, Y., Han, C., Yang, J., Su, J., Xu, X., Li, S., Xu, L., Fang, R., Jiang, H., Zou, X. and Song, B., Nano lett., 15(3), (2015) 2180 [13] Zhou W., Dai X., Fu., T., Xie C., Liu J., Lieber C.M, Nano Lett., 14 (3), (2014) 1614 [14] Barrelet C., Bao J., Loncar M., Park H., Capasso F., Lieber C.,: Hybrid Single-Nanowire Photonic Crystal and Microresonator Structures, Nano Lett.,6(1), (2006) 11 [15] Chetan K. Kasar, Ulhas S. Sonawane, Jaspal P. Bange, D. S. Patil, Journal of Materials Science:

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[16]Zhang X., Zhang X., Wang L., Wu Y., Wang Y., Gao P., Han Y., Jie J., Nanotechnology,24, (2013) 1 [17] Chetan K. Kasar, Ulhas S. Sonawane, Jaspal P. Bange, D. S. Patil, Journal of Materials Science: Materials in Electronics, 27 (11) (2016) 11885 [18] H. Xu, A. Hurtado, J. Wright, C. Li, S. Liu, J. Figiel, T. Luk, S. Brueck, I. Brener, G. Balakrishnan, Q. Li, and G. Wang, Opt. Express 22 (2014) 19198 [19] F Scholz, Semicond. Sci. Technol. 27 (2012) 024002 [20] Chitnis, Ashay, Changqing Chen, Vinod Adivarahan, Maxim Shatalov, Edmundas Kuokstis, Vasavi Mandavilli, Jinwei Yang, and M. Asif Khan, Applied Physics Letters 84(18) (2004) 3663 [21] Chakraborty, Arpan, B. A. Haskell, S. Keller, J. S. Speck, S. P. DenBaars, S. Nakamura, and U. K. Mishra, Applied Physics Letters 85(22) (2004) 5143 [22] Sonawane U.S., Samuel E. P., Kasar C.K., Patil D.S, Optik - International Journal for Light and Electron Optics, 127(12) (2016) 4937 [23] Samuel E.P., Patil D.S., Optoelectronics and advanced materials-Rapid Communication, 1(12) (2007) 698 [24] Sonawane U.S., Samuel E.P., Zope U., Patil D.S., Optik - International Journal for Light and Electron Optics, 124(9) (2013) 802 [25] Ramey S.M., Khoie R., IEEE Transactions on Electron devices, 50(5) (2003)1179 13

Fig. 1 Schematic representation of Quantum wire III-V Nitride Laser Diode

c

d

Fig. 2 Potential energy for different Aluminium mole fraction and applied biasing

14

Fig. 3 Wavefunction of electron density for different applied biasing alongwith quantum wire dimension.

15

Fig. 4 Confinement factor as a function of applied biasing for different Aluminium Concentration.

Fig. 5 Escape time dependence on Barrier and Wire Width for various wavelength and applied biasing (a) 300 nm, (b) 325 nm, (c) 350 nm and (d) 375 nm.

16

10.8 10.75

FWHM (nm)

10.7 10.65 10.6 10.55 10.5 10.45 1

1.5 2 Applied Bias (V)

2.5

Fig. 6 FWHM of wavefunction intensity as a function of applied bias -10

0

10

10

0

-10

Distance Along Y-axis (nm)

-10

-10

0

0

10

10

10

10

0

0

-10

-10 -10

0

10

10

0

-10

Distance Along X-axis (nm)

Fig. 7 Electron Distribution Pattern for different applied biasing (a) 1V, (b) 1.5V, (c) 2V and (d) 2.5V.

17

S Material Parameter s, Set

Comput Solve the Schrodi En er gy

Assign N o

Y e Compute Wave Function

Flow Chart for Computing QWR Characterstics

18

Table 1 Materials parameters used for simulation Parameter Layer Material Biasing (in Volts) Aluminum mole fraction Electron effective mass Band offset

Barrier Region AlxGa1-xN Va=1-2.5 x= 0.1,0.17,0.25 𝑚𝑏∗ =(3.53x +1.76(1-x))m0 vp=1.45x+ 0.53x2

19

Wire Region GaN x= 0 ∗ 𝑚𝑤 =0.2m0