Electrónica Nova Scientia Simultaneous hole scattering in a biased simple barrier Dispersión simultánea de huecos en una barrera simple

A theoretical analysis of the (heavy/light)-hole propagating fluxes through a simple barrier het erostructure of III-V semiconductors is developed considering an external perturbation. We include the interaction with an external electric field parallel to the direction of propagation Diago-Cisneros, L. y S. Arias-Laso Revista Electrónica Nova Scientia, No 10 Vol. 5 (2), 2013. ISSN 2007 0705. pp: 01 16 3 normal to the interfacesin the Multicomponent Scattering Approach and study the relevant magnitudes -transmission, conductance and phase timeof hole quantum transport. This formalism allows us to deal with all propagating channels simultaneously. For the simple barrier system we study the dependence of the conductance with the applied bias through the different hole channels. We also analyze how the increasing of the in-plane momentum κT affects the results, which provides an idea of how the band-coupling influences the transmission with applied bias. Additionally, a brief study of the phase transmission time curves is carried out as a function of the applied voltage in the heterostructure, fixing different values of the band mixing. Our results are successfully compared with some trends obtained previously in the hole tunneling through semiconductor heterostructures using different approaches to this applied here.


Introduction
Transport-magnitude measurements related with tunneling of quasi-particles through semiconductor heterostructures have an extensive data in the literature.More specifically, the study of transmission properties in low-dimensional systems under external perturbations has played a fundamental role in the design of tunneling devices (Wessel and Altarelli 1989, 17).
There is a broad range of experiments and theoretical developments devoted to study inter-band tunneling under external bias (Morifuji and Hamaguchi 1991, 19;Kiledjian et al. 1992, 24;Bertran et al. 1994, 23;Pereyra and Anzaldo-Meneses 2005, 419;de Carvalho et al. 2006, 155317;de Carvalho et al. 2006, 041305;Leo et al. 1990, 11;Dragoman et al. 2008, 1;Ertler and Pötz 2011, 165309).Mathematical treatments based on the k.p model and the transfer matrix formalism are usual to solve this kind of problem when analyzing the influence of an external bias in tunneling properties [11].Propagation of holes through heterostructures with applied bias has a young history in comparison with other charge carriers, which is expected in some sense, since the study of holes propagation demands more powerful and complex mathematical treatments than other charge carriers without band mixing.
In the last decades, a renewed interest for treating holes as the main charge carriers in resonant tunneling has arose (de Carvalho et al. 2006, 041305;Ertler and Pötz 2011, 165309;Lee and Hu 2001, 7;Dragoman and Dragoman 2003, 10;Diago-Cisneros et al. 2006, 045308).Recently, through semiconducting-polymers experiments was demonstrated the strong influence of holes mobility on the charge carriers density, unifying the hole transport picture in field-effect transistors and light-emitting diodes (Tanase et al. 2003, 21).In addition, attractive findings in structures of graphene offered new insights for the design of electronic and optoelectronic devices based on hole tunneling through structures of graphene (Mak et al. 2009, 256405).
Group delay calculations under applied bias have brought a new interest for the study of low dimensional systems.The response time of several nano-structured devices is directly related with the tunneling time of the leading charge carriers.Due to this fact it is of high urgency the knowledge of the time curves as a function of some parameters of a particular physical system, which can predicts the conduct of a particular device.Tunneling time, particularly for holes, has received great attention since it was shown in experiments with GaAs-AlAs superlattices (Schneider et al. 1989, 14) that resonant tunneling through hole subbands occurs faster than transport due to non-resonant tunneling of electrons, in spite of their mass.Tunneling times in asymmetric quantum structures with applied bias have been explored using experimental and theoretical methods, which include luminescence measurements (Leo et al. 1990, 11;Oberli et al. 1989, 5;Brockman et al. 1991, 24) and novel formalisms where analytical expressions of the delay time were founded to be the threshold for anomalous events like the Hartman premonition (Hartman 1962, 12;Sepkhanov et al. 2009, 245433).
With this report we attempt to study the transmission and phase time for holes tunneling a simple barrier under an external electric field.In Section 2 we explain in detail the applied formalism to include the external perturbation in the Multicomponent Scattering Approach.Section 3 contains the numerical simulations that were carried out and the main results we have obtained.Finally, in Section 4 we arrive to the principal conclusions.

Multicomponent Scattering Approach with applied electric field
We consider the tunneling of holes through a semiconductor barrier of nanometric dimensions (d = 10-20 Å), which is embedded in two volumetric layers of III-V materials.Both layers are properly doped to assure that one of the electrodes provides holes that pass through the barrier.In Figure 1, the heterostructure with applied bias under consideration is presented.As we will be dealing with holes, the energies are considered as positive in the valence band.According to (1) and (3), it is evident that obtaining the transfer matrix is essential in order to study the physical magnitudes we are interested in.In the case of a simple barrier with applied electric field, we need to integrate numerically the KL differential equation system.On the other hand, the form of the matrices and is already known, although it is important to mention that with applied electric field, the parameters that characterize both regions L and R do not coincide, because of the electric potential applied to the heterostructure (as one can see in Figure 1).
We are interested in finding the transfer matrix corresponding to a region of constant parameters with uniform electric field applied along the direction of propagation.We consider the sub-system up (2x2) of the KL Hamiltonian to find the matrix , and then we will be able to calculate the transfer matrix of the low sub-system and the matrix corresponding to the (4x4) space of KL through some well known methods.
Afterwards, with expression (3), we can obtain the matrix of interest to calculate the transmission magnitudes.
From the Schrödinger equation corresponding to the sub-system up , we have where is the null matrix of second order.The potential corresponding to the physical situation showed in Figure 1 has the form (5) where F represents the applied electric field and e is the modulus of the electron charge.When we write in the matrix expression (4), using its matrix form (García-Moliner and Velasco 1992), we obtain Here we have assumed the matrices y are antihermithic, and with constant parameters.
It is usual, when solving numerically a second or higher order differential equation system, to intend to simplify it obtaining an equivalent first order system with a number of equations that agrees with the highest order derivative of the original system.This can be useful since the integration, as a problem of initial values, is straightforward when one consider the first order system and the implemented methods to solve this system are better optimized.We now need to turn the system (6) in the form (7) At this point, it would be useful to define the vector , which converts the N-equation system (6) into a 2N-differential equation system of first order.From Equation ( 6), the first derivative of the proposed vector can be expressed as ), although it is important to take into account there is a difference of potential between regions L and R, so the transfer matrix will differ for given fixed energies and .On the other hand, we apply the definition of canonical basis for the barrier region, as one in which the TM of function and derivative is the identity matrix, then we solve Equation ( 9) after apply repeatedly, with a set of values for the energy and , the initial vectors (Press et al. 1997) . (11) The expression obtained for must be conveniently multiplied by (1) and then, by Equation ( 3), be used in order to study the transmission properties of the heterostructure in resemblance with the usual mechanism without applied external bias (Diago-Cisneros 2006).We are concerned in calculating the transmission probabilities and the conductance of the system, in addition with the phase transmission time for hole bands.If we consider the incoming particles from the left, the transmission probability from the j channel to the i channel is defined by where represent the transmission amplitudes for the incoming particles given by the definition of the transfer matrix and the direct relation that connects this matrix with the scattering matrix .
The two-probe Landauer conductance will be obtained as follows

Numerical Results
The sketched scheme in Figure 1 shows the hole transport process we intend to study.For the    In Figure 4 the two-probe Landauer conductance for the simple-barrier system is exhibited in dependence with the applied voltage for different values of .It is clear that the transmission of the system falls rapidly with the increasing of the band mixing.This behavior evidences the significant effect of the band mixing over the volt-amperic (V-I) curve, as was reported in theoretical studies for the double barrier resonant tunneling (DBRT) (Rosseau et al. 1989).Even though we do not show here V-I curves strictly speaking, it is still valid to analyze the conductance, since this global magnitude is related straightforwardly with the tunneling-density current (Landauer and Martin 1994, 1). Figure 4 shows clear evidences that for a fixed incoming energy, the rising of the applied voltage above 25 mV practically closes the whole transmission.In this case, the incident energy was fixed at 0.02 eV and as the electric field energy increases, this magnitude can reach a value for which the propagating states in the L region (see Figure 1) remain in the forbidden energy range which provokes the transmission to vanish (Diago-Cisneros 2006).

Figure 1 .
Figure 1.Schematic representation of the heterostructure considered.Physical situation particularly for holes, the energies are considered as positive in this case.
notice, Equation (10) contains the definition of the function and derivative transfer matrix .The problem to be solved has three regions: L, barrier and R. In regions L(R) the full transfer matrix can be obtained in a similar way as in the case without electric field (Diago-Cisneros et al. 2006, 045308; Arias-Laso and Diago-Cisneros 2011, 1730; Diago-Cisneros 2006 the phase transmission time is calculated based on the expression (14) where stands for the phase-transmission amplitudes, which are directly related with the complex transmission amplitudes (Diago-Cisneros et al. 2006, 045308; Arias-Laso and Diago-Cisneros 2011, 1730; Diago-Cisneros 2006).
Figure 2 exhibits the conductance dependences with the applied voltage as the band mixing is increased from to .As one can see, when mixing between the propagating modes is too low (panels 2(a) and 2(d)) the scattering channels are uncoupled.For this reason, it is expected the predominance of transmission probabilities corresponding to direct transitions over the ones from crossed transitions.Therefore, in every curve of conductance in Figure 2 the principal component is the one corresponding to the direct transition.Nevertheless, even for this small value of , there are some crossed paths that remain open, which leads to obtain some non-zero transmission probabilities for crossed transitions.This behavior is a direct consequence from the breaking of the time reversal invariance symmetry in the (4x4) dimension due to the electric bias effects.Another remarkable feature in this figure is the more powerful influence of the light holes in comparison with the heavy holes, independently of the voltage values.This last trend was already found in a study of the inter-band tunneling of holes through a similar heterostructure(Morifuji and Hamaguchi 1991, 19).

Figure 2
Figure 2 Conductance through the four accessible channels of the system as function of the applied voltage, increasing the band-mixing level with the incoming energy fixed at Ei = 0.02 eV .Subfigures 2(a) and 2(d) correspond to = , while for 2(b) and 2(e) was fixed = , and finally sub-figures 2(c) and 2(f) hold a strong mixing between channels with = .

Figure 3
Figure 3 Transmission probabilities corresponding to three crossed transitions for = and = , as a function of the applied voltage.

Figure 4
Figure 4 Conductance of the simple barrier system as a function of the applied voltage for different levels of band mixing ( ).

Figure 5
Figure 5 presents the phase transmission time for hole-propagating channels through the simple barrier with applied electric field.Sub-figures 5(a), 5(b) and 5(c) shows the phase time dependences with the applied voltage for different values of the band mixing.When = the channels mixing is practically null and the phase time for the direct transitions is dominant.One can notice from panel 5(a) there is a degeneracy among the curves depending on the effective mass.The phase times for direct transitions (dotted red line) and (dotted purple line) are degenerated in spin-up and spin-down respectively.This behavior is caused for the low mixing between hole bands, in this case hole channels are uncoupled and the phase time of a transition from one state to another with the same effective mass and certain spin polarization matches with the transition corresponding to the opposite spin polarization.It is easy to notice the breaking of this degeneracy in panel 5(b), which means that for a greater mixing the hole channels are not uncoupled anymore and previously forbidden crossed paths start to be opened for tunneling.For a strong mixing, = , the phase time shows trends noticeably different than the previous ones, even appearing negatives values of this magnitude.As one can see in panel 5(c), phase time curves corresponding to heavy holes, as to light holes, show a symmetric behavior with reference to = 0.This trend demands a more deep study to figure out whether is caused for numerical reasons or have an explanation from the physical point of view.

Figure 5
Figure 5 Phase transmission time for the simple barrier structure as a function of the applied voltage for increasing values of .