A comparative study of changes on electronic properties between Si and Ge nanowire along [110]

direction by application of Uniaxial Strain

Md. Alamgir Kabir, Nahid Akter, Zahid Hasan Mahmood University of Dhaka Dhaka, Bangladesh.

Email: m.alamgir_kabir@yahoo.com, nahid_nstu@yahoo.com, zahid@univdhaka.edu

Abstract- In this review paper, we compare the effect of uniaxial strain on the electronic properties of Si and Ge nanowire along [110] direction based on density-functional theory (DFT). Generally, bulk Si & Ge possess indirect bandgap while both nanowires possess a direct bandgap. The diameters of the nanowires from 1.2-3.7 nm are studied. As bandgap is inversely proportional to wire diameter it can be modulated by uniaxial strain that is applied to the nanowires. Compressive strain increases the bandgap while tensile strain reduces the bandgap for smaller diameter nanowires (~1.2 nm). But when the diameter is increased, bandgap shows parabolic behavior in the compressive strain region. In addition, effective masses of charge carriers can be changed by applying uniaxial strain. Tensile strain increases the effective mass of the hole while compressive strain increases the effective mass of the electron. This paper comparatively shows the bandgap and effective masses of charge carriers are changed due to uniaxial strain for various wire diameters (1.2-3.7 nm) of Si & Ge nanowires. Keywords- Nanowire, Strain, Bandgap, Effetive mass, charge carrier.

I. INTRODUCTION

Si and Ge crystals both have diamond structures and tetrahedral networking. A Si-Si bond is stronger than a Ge-Ge bond. Bulk Ge has an indirect band gap of 0.66 eV while the indirect band gap of Si is at a value of 1.12 eV. Bulk Ge has higher electron/hole mobility ( nμ = 3800

112 −− sVcm and pμ = 1800 112 −− sVcm ) compared to Si

( nμ =1500 112 −− sVcm and pμ = 450 112 −− sVcm ) at room temperature [1]. One-dimensional Si and Ge nanowires (NWs) have attracted extensive research efforts over the past decade. They are expected to play important role for future nanoscale electronic and optical devices, such as light-emitting diodes (LEDs) [2], field-effect transistors (FETs) [3], inverters, logic circuits, memory, quantum computing and nanoscale sensors [4-6]. So it is important to study electronic properties of those nanowires, such as band structure, bandgap, effective masses of

charge carriers and density of state. For nanoscale applications, the quantum confinement effects on Ge nanostructures are more prominent than on Si nanostructures [1].

Strain is a very important factor from the growth and application aspect of nanodevices. During epitaxial growth strain is not avoidable if there is a lattice mismatch between grown nanostructure and substrate. When nanowires are embedded in some materials, then coatings bring strains to the wires. Researcher found that strain can enhance the device’s performance by increasing the effective masses of the electron and hole but in general, less effective masses of charge carriers lead to the less mobility pretending to slow devices. Positive values of strain refer to uniaxial tensile (expansion) while negative corresponds to compressive strain. This paper shows the lateral x- and y-coordinates [110] of the hydrogen (H) passivated wire are optimized at a uniaxial (tensile and compressive) strain. Fig. 1 shows the electronic structure of H-passivated Si & Ge NW. Si & Ge NW along 110 direction expand axially upon relaxation and this axial expansion become negligible when the size of the nanowires is beyond 4 nm and 2 nm respectively [1, 8, 9]. Theoretically, the bandgaps and effective masses of electron and hole of Si and Ge NWs are dependent on their size, crystalline orientation, doping etc. In this review paper, bandgaps and effective masses of charge carriers of nanowires are predicted by DFT. In both cases of Si & Ge, the band gap is increased when the size of the wire is reduced. This effect is primarily due to quantum confinement. This predicted size-dependence of the bandgaps and effective masses in Si & Ge NWs possess good agreement with the literature and references. However, uniaxial strain comprising of compreesive and tensile affects the nanowires behavior making a significant relationship with nanowire size [1].

978-1-4799-0400-6/13/$31.00 ©2013 IEEE

(b) Figure 1. (a) Electronic structure of H-passivated 1.7 nm diameter [110] Si NW. (b) Electronic structure of H-passivated 1. 8nm and 3 nm diameter [110] Ge NW [9].

II. DETAILS STUDY OF RESULTS

A. Strain effect on bandgap:

Band structures are modulated with strain. Most electronic properties are related to the bottom of the conduction band and the top of the valence band. Generally, strain has dominant effects on the band structure near Γ while it has negligible effects on wave vectors far away from Γ [1].

For 1.7 nm [110] Si NW, the bandgap decreases linearly with increasing tensile strain ( 0>α ). The linear relationship 7.11.0)( +−= αeVEg fits the bandgap in the tensile strain region, where the slope is approximately 100 meV/%. Fig. 2(a) shows from 0 to -1% compressive strain, bandgap is increased but after −1% strain, it is decreased with further increase in strain. Moreover, for a strain larger than −4%, the bandgap becomes indirect [7]. The band structure of the Ge nanowire with a diameter of 1.8 nm is shown in fig. 3(b) and (c). The strain modifies the energies of conduction band edge (CBE) and valance band edge (VBE) dramatically near Γ , and has negligible energy shifts on wave vectors far away from Γ . Table I shows the DFT predicted diameter vs bandgap for Si & Ge NW [1, 9].

TABLE I. DIAMETER VS BANDGAP FOR SILICON & GERMENIUM NW

Si Ge Diameter (nm)

Bandgap (eV)

Diameter (nm)

Bandgap (eV)

1.7 1.74 1.8 1.02 2.3 1.58 2.5 0.73 3.1 1.477 3.0 0.61

Figure 2. (a) Bandgap changes for different strain for 1.7 nm [110] Si NW [7]. (b) Ge NW of 1.8 nm with uniaxial strain. The energy variations of the bottom of the conduction band (c) the top of the valence band [1].

B. Strain effect on bandgap with different diameter nanowire:

For Ge NW, the variation of band gaps as a function of uniaxial strain for several different sized wires is plotted in Fig. 3(a). For the wire with a diameter of 1.2 nm, the band gap variation with strain is almost linear (green line). The gap decreases with tensile strain and increases with compression strain. For wire diameter of 1.8, 2.5 and 3.7 nm are also presented in the same figure [1, 10]. For Si NW, the variation of band gaps as a function of uniaxial strain for several different sized wires is plotted in Fig. 3(b). For Si NW 1.7 nm wire diameter, the band gap linearly decreases with increasing tensile strain (blue line). It also increases for -1% compression strain again further decreases linearly with increasing compression strain (-1% to -5%). When the wire diameter increases (2.3 and 3.1 nm), it shows parabolic behavior in the compressive strain region

Ban

dgap

(eV

)

[1, 11]. That means both tensile and compression strain the bandgap drops with different manner.

Strain (%)

Figure 3. (a) Bandgap vs strain for three different [110] Ge NW diameter (1.2, 1.8, 2.5, 3.7 nm) [1]. (b) Bandgap vs strain for three different [110] SiNW diameter (1.7, 2.3, 3.1 nm) [7]. Finally, we can say both tensile and compression strain the bandgap drops with different manner. So, strain affects on the bandgaps of Si and Ge NW with strongly dependent on diameter.

C. Variation of effective masses with different diameter:

The smaller effective masses of the charge carriers in a material implies larger mobility of charge carriers, thus increasing the operating speed of devices made from the material. The effective masses of the charge carriers ( *m ) can be calculated according to the formula from the band structure of the nanowires [1].

1222 )/(* −= dkEdm (1)

We reviewed the effective mass of the electron ( *em )

and the effective mass of the hole ( *hm ) for both Si and

Ge NW in the units of electron mass ( em ) without strain shown in table II [1, 8].

TABLE II. DIAMETER VS EFFECTIVE MASS OF CHARGE CARRIER FOR SILICON & GERMENIUM NW

Si Ge Diameter (nm)

me* mh* Diameter (nm)

me* mh*

1.7 0.14 0.18 1.8 0.12 0.09 2.3 0.14 0.26 2.5 0.11 0.15 3.0 0.16 0.32 3.0 0.11 0.19

D. Strain effect on effective masses: For 2.7 nm Si NW, 1.5% compressive strain increases the effective mass of the electron to 0.299 em while the

effective mass of the hole is reduced to 0.167 em . In contrast, under 1.5% tensile strains, the effective mass of the electron is decreased to 0.138 em while the effective mass

of the hole is increased dramatically to 5.6 em [8].

For 3.7 nm Ge NW, 2% compressive strain increases the effective mass of the electron to 0.166 em while the

effective mass of the hole is reduced to 0.133 em . In contrast, under 2% expansive strains, the effective mass of the electron is deceased to 0.102 em while the effective

mass of the hole is increased dramatically to 1.139 em [1]. The change of effective masses of the electron and hole with strain is also dependent on the size of Nanowires. Since, effective mass of charge carrier is inversely proportional to carrier mobility. Fig. 4(a-c) shows, the effective mass of the electron increases rapidly with compressive uniaxial strain, while decreasing mildly with tensile strain. However, the effective mass of the hole reduces under compression, while enhanced dramatically with tensile strain [1, 8]. When the diameter is increased, the mobility of electron is increased with increasing compressive strain. Again the mobility of hole is increased with increasing tensile strain with increasing diameter. This is the significant relationship between mobility and strain in nanostructures which leads to the desired device performance.

(b) (c) Figure 4. (a) Strain causes an abrupt change in the conduction band electron effective mass for Si NW [7]. (b) The change of effective masses of the electron and ( c ) hole are plotted as a function of uniaxial strain for Ge NW at different size [1].

III. COMPARATIVE STUDY In this review, there are two electronic properties named bandgap and effective masses of charge carries in NW. So, comparison of these properties for varying behavior would be discussed under strain effect for both Si and Ge NW.

A. Strain effect on bandgap For both Si and Ge NW, the wires possess direct bandgap at Γ along [110] direction where the bandgap variation with uniaxial strain is wire diameter dependent. For Si NW of approximately 1-2 nm, the bandgap changes linearly with maximum of 5% expansive strain while it shows nearly parabolic behavior for approximately 2-4 nm diameter wire with maximum 5% of compressive strain. Bandgap shows linear behavior under maximum 3% of tensile strain for Ge NW of 1.2-2 nm wire diameter while for 2-4 nm wire, bandgap shows parabolic behavior under maximum 3% of compressive strain. But in exception, 1.2 nm Ge NW shows nearly linear behavior for compressive strain.

B. Strain effect on effective masses Strain affects effective masses of the charge carriers in different manner. Tensile strain increases the effective mass of hole while compressive strain increases the effective mass of electron. The electron effective mass of Si NW increases 0.01 of me and for Ge NW, decreases 0.01 of me as the nanowire diameter increases. The effective mass of hole is increased in undetermined manner with increase in

diameter for both Si and Ge NW. Under compressive strain, the electron effective mass increased greater for Si than Ge for the same diameter of nanowire while under tensile strain, the hole effective mass increased dramatically greater for Si than Ge for the same diameter of nanowire. So, it can be said that the effective masses of the electron and hole can be reduced by tuning the wire diameter and applying appropriate strain.

IV. CONCLUSION

In this review, the electronic properties of hydrogen passivated Si and Ge nanowires under uniaxial strain based on DFT values are studied extensively. It would be concluded that studied results of bandgaps and effective masses of Si & Ge nanowires are predominantly dependent on the diameter and strain. It is observed that the less strain is needed for Ge than Si NW for significant change in bandgap. In contrast, the effective masses for electron and hole of Si NW are increased superiorly than Ge NW. These results may be considered as remarkable and promising factors for applications in highly sensitive optical and electronic devices in nanoscale range. There is a great scope for improving these electronic behaviors with mechanical effect by using the Si/Ge or Ge/Si core-shell nanowires [10, 11].

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