Influence of electron-phonon interaction on piezoresistance of single crystals n-Ge

Piezoresistance of uniaxially deformed along the crystallographic direction [100] single crystals n-Ge for different fixed temperatures has been investigated. Presence of the significant piezoresistance for given conditions of the experiment is explained by increase in the effective mass and decrease in the relaxation time for electrons at the expense of the inversion of (L1-∆1) type of an absolute minimum in n-Ge. Temperature dependencies of resistivity for undeformed (L1 model of conduction band) and deformed under uniaxial pressure P=3 GPа (∆1 model of conduction band) single crystals n-Ge are obtained. Resistivity for undeformed single crystals n-Ge is changed according to the law ρ~T 1.66 . Resistivity for uniaxially deformed single crystals n-Ge is changed as ρ~T 1.53 . The given dependencies show that for L1 model of the conduction band in contrast to ∆1 model one must equally with intravalley scattering of electrons on acoustic phonons take into account scattering of electrons on optical and intervalley phonons. Therefore reduction of the magnitude piezoresistance n-Ge with the increasing temperature is associated with “turning-off” at the expense of the inversion of (L1-Δ1) type of absolute minimum under uniaxial pressure P>2.7 GPа mechanism for scattering of electrons on intervalley and optical phonons. Comparison of results of theoretical calculations with relevant experimental data shows that peculiarities of piezoresistance n-Ge under uniaxial pressures 1.6<P<2.7 GPа for (L1-Δ1) model of conduction band n-Ge can be described only taking into account nonequivalent intervalley scattering of electrons between the minima L1 and Δ1.


INTRODUCTION
Monocrystalline germanium is widely used as a raw material for the manufacture of different kinds of semiconductor devices such as diodes, triodes, power rectifiers, dosimetric devices, optical elements of infrared technique [1]. Hot electrons in n-Ge is a source of terahertz radiation, which is widely used in engineering, medicine [2]. Optical and electrical properties of multivalley semiconductors depend significantly on lattice strain. Sharp growth of the intensity of exciton absorption in crystals of germanium under hydrostatic pressure more than 0.6 GPа has been revealed in work [3]. The authors of the work [4] showed that the life time of excitons in germanium depends on the hydrostatic pressure and intervalley scattering of electrons between the minima of conduction band of different symmetry. Strained germanium is also a promising material for Nan electronics. Elastic fields of deformation on the boundary of heterojunction in Si/Ge heterostructures arise due to the discrepancy of lattice constant of germanium and silicon [5]. The use of nanostructures with the self-induced Ge/Si Nan islands starts the new prospects for the development of opto-and Nan electronics [6]. The spatial arrangement of these Nan islands depends on the distribution of deformation fields in these nanostructures [7]. That's why the study of different kinetic and optical effects in the deformed single crystals n-Ge is actual both in the theoretical and applied sense.
Experimental and theoretical calculations for a wide temperature range show that the effect of piezoresistance n-Ge, which is associated with deforming redistribution of electrons between the minima of L1 type (effect Smith -Herrìng piezoresistance) will be no longer missed under uniaxial pressures P > 1.5 GPa [8]. Giant piezoresistance in n-Ge under uniaxial pressure P>2.1 GPa along the crystallographic direction in [100] was first experimentally achieved in [9]. In this case was obtained a deformationally-induced phase transition metal-nonconductor which is associated with deforming redistribution of electrons between the minima of L1 and Δ1 type with different effective mass and emergence due to the uniaxial pressure of energy gap between the admixture area and conduction band. Resistivity and Hall coefficient n-Ge with double-charging deep level of gold depending on the hydrostatic pressure at room temperature has been investigated in work [10]. Obtained experimental results are explained by the authors with the presence of intervalley scattering between the minima L1 and Δ1.

EXPERIMENTAL RESULTS
We investigated piezoresistance of single crystals n-Ge which had been alloyed by the impurity Sb with concentration  Such uniaxial deformation leads to the simultaneous displacement upward at a scale of energies of four L1 minima and descend of two Δ1 minima [11]. The availability of a plateau for these dependencies indicates implementation of the (L1-Δ1) type of inversion of an absolute minimum for such magnitudes of uniaxial pressures and temperatures. As it can be seen The given temperature dependencies of resistivity n-Ge are explained by the additional mechanism of scattering of electrons on optical and intervalley phonons in the model of L1 and lack of this scattering mechanism in model of Δ1.
Intervalley scattering between the minima L1 and Δ1 should be taking into account along with the examined mechanisms of piezoresistance n-Ge (the same as in the case of hydrostatic pressure). On the basis of theory of the anisotropic scattering calculations for curves of piezoresistance n-Ge with consideration and without consideration of intervalley scattering between L1 and Δ1 minima have been done by us to determine the relative contribution of this scattering mechanism in piezoresistance n-Ge.

THE RESULTS OF THEORETICAL CALCULATIONS
Resistivity of uniaxially deformed along the crystallographic direction [100] single crystals n-Ge can be presented as: where 11 , L nn  , 11 , L   -concentration and mobility of electrons for L1 and Δ1 minima correspondingly, q -electron charge.
For not degenerated electron gas 11   For undeformed single crystals of n-Ge energy gap between L1 and Δ1 is equal 0.18 еV and decreases linearly from uniaxial pressure [12]. Then for the case of the deformed single crystals of n-Ge can be written: Then, taking into account expressions (3) and (5), resistivity for uniaxially deformed single crystals n-Ge is Isoenergetic surfaces for both L1 and ∆1 minima are ellipsoids of rotation. Then mobility of charge carriers in an arbitrary direction can be determined from the ratio [13]: where  -angle between the examined direction and major axis of the ellipsoid;   і  -mobility of charge carriers across and along the axis of the ellipsoid.
Then, according to (1), for the L1 minimum Components for the tensor of mobility   and  can be expressed through the tensor components of relaxation times and effective masses for the corresponding minimums:   (11) (Necessary notations in expressions (11) are listed in the appendix).
Also necessary to take into account conduction bands for scattering of electrons on optical phonons, which frequencies correspond to temperature TC1 = 430 K (intravalley scattering) and intervalley scattering on acoustic phonons with a typical temperature TC2 = 320 K [15] for L1 model of conduction band along with the scattering of electrons on the acoustic phonons and ions of admixture in n-Ge. Equivalent intervalley scattering on acoustic and optical phonons with characteristic temperatures TC1 = 100 K and TC1 = 430 K between the valleys, which are located on the same axis (gscattering) [16] holds for two-valley Δ1 model of conduction band single crystals n-Ge which has been formed by uniaxial pressure P// [100]. Role of the nonequivalent intervalley scattering will increase owing to reduction of the energy gap between L1 and Δ1 minima under uniaxial pressure. Nonequivalent L1↔Δ1 intervalley scattering of electrons is stipulated by their interaction with acoustic phonons with the characteristic temperature TC2=320 K [16]. Scattering of electrons on optical and intervalley phonons can be described by the scalar time of relaxation ij  [16]: where F e b r u a r y 20, 2 0 1 5 ( ; ) For the equivalent intervalley scattering [16]     and for the nonequivalent scattering where j m , Then for the most general case for scattering of electrons on the acoustic phonons, ions of admixture, optical and intervalley phonons expressions for the tensor components of relaxation times for L1 and Δ1 minima will be as following:   For a nondegenerate electron gas  (18) The mobility of electrons for L1 and Δ1 minima can be found taking into consideration the expressions (8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18). This will allow (on the basis of the expressions (3)(4)(5)(6)) to find the dependence of resistivity n-Ge from uniaxial pressure along the crystallographic direction [100]. Constants of optical and intervalley deformation potential, components for tensor of the acoustic deformation potential for intravalley scattering, effective mass for the density of states and tensor components of the effective mass for L1 and Δ1 minima are necessary for this process. A significant number of the given parameters had been found by us in works [12,15,17]. The necessary parameters of L1 and Δ1 minima for calculation are presented in the table 1. F e b r u a r y 20, 2 0 1 5  [17] m  0 0.082m [18] 0 0.32m [17] Effective mass for the density of states 11 , L m  0 0.55m [18] 0 0.88m [17] Components Results of theoretical calculations of the piezoresistance n-Ge with taking into account (solid curves) and without taking into account (dashed curves) of the nonequivalent intervalley scattering between L1 and Δ1 minima are presented in fig. 1.

CONCLUSIONS
Characteristic features of piezoresistance n-Ge (when (L1-Δ1) model of conduction band is implemented) can be described only taking into account the given mechanism of scattering. Such conclusions have been done by us after comparison of these curves with the relevant experimental results. Presence of the significant piezoresistance is associated with "turning-off" at the expense of the uniaxial deformation of mechanism for scattering of electrons on intervalley and optical phonons during the transition from L1 to Δ1 model of the conduction band n-Ge.