Superlattices and M icrostructures, Vol. 8, No. 4, 1990 413 INFLUENCE OF QUANTUM WIRE CROSS SECTION FLUCTUATIONS ON TRANSPORT PHENOMENA V.V. Mitin Department of Electrical & Computer Engineering, Wayne State University, Detroit, MI 48202 On leave from the Institute of Semiconductors, Academy of Sciences of Ukrainian SSR, Kiev (Received 30 July 19!Xl) For all currentlv available methods of auantum wire (OWI) fabrication the cross section of the QWI varies along the length, producing fluctua~ons in the energy position of the subband enerav alone the OWL This fluctuation leads to a smearine of the oeak-like structure of the dependence Gf the average density of states on carrier energy. Thi’s, in turn, results in a smearing of the peak-like structure of the dependence of the resistivity of a QWI on the Fermi energy and/or total electron concentration, and in an increase of resistivity. In previous attempts to obtain agreement between theory and experiment the subband energy fluctuation was not taken into account, that is why the importance of other factors, such as impurity scattering, have been overestimated. Interest in quantum wires (QWI) (systems containing a 1-D electron gas) is now increasing very rapidly [l-7] because they could serve in a variety of high-speed electronic devices. A 1-D electron gas implies that electrons are free to move only in one direction. Let us define this as the x-direction. The energy in the y- and r-directions is fixed because electrons are confined to a QWI of small cross sectional area A = d,dz , where the linear dimensions d, and d, along y and z are of the order of the de Broglie wavelength, &,,oftheelectrons.(Wecat~alsohaved,,d~<&, [1,2].)The experimental results, which are presented in Refs. 1 and 2, clear1 aQ WI demonstrate the well pronounced variation of A along . We propose that this cross sectional variation has a serious effect on transport phenomena so our emphasis will be on transport properties related to the dependence of A on x in long 1-D samples (we are not interested in ballistic transport, which can occur in a short 1-D sample). Proposed Model In any theoretical calculation, the electron energy, E,(k)=(ti)*/(2m)+,!$; &=E,+E,; i,j=1,2,3., (1) is a function of the longitudinal electron wave vector k = k, , and two quantum numbers, i and j. Here m is the effective mass of the electron, h is Planck’s constant, and Ei and E, are quantized levels corresponding to the confinement of the electron motion in the y- and z-dhections, respectively. Different theoretical models emolov different assumu- tions about Ei and E,. Practically all of them [3-61 assume that the QWI of thickness d, is a part of a quantum well structure. The models for E, are different. In this presentation we will emphasize one of them despite the fact that several models were exploited in our calculations. For the z-direc- tion, a quantum well of thickness d, is assumed [4], and for simplicity we consider a QWI with square cross section [7], d, = d, = d and introduce [4] E, = E,(i* + j’)/Z; E, = (fix)*/(md*); i, j = 1,2,3., (2) where& is theenergypositionofthefirst subbandat i = j = 1 (note, that for i + j a subband has a two-fold degeneracy). In this approach we have only one parameter, d (x), which defines E,(x) and hence E,(x). (It is necessary to stress that the approach d, = d, = d does not change the major results we want to demonstrate. We considered, for example, the case when d, a d,, so that either d, or d, was x-dependent. Theresults were qualitatively the same, because a fluctuation in d, or d, produces a fluctuation of E,(x).) For each subband given in Eqs. 1 and 2 the spectrum is parabolic, so that the density of states (DOS) per unit length, g(E), can be easily calculated [4-71: g(E) = -&z2q C(E - Ep, B where the difference E -EO should always be positive. The energy Ed depends on d(x); so does g (E ). If E4 varies smoothly with x (so that the characteristic size of the fluc- tuations of d(x) along x is larger than &, ), Eq. 3 defines the local DOS. The average over the sample DOS < g(E) > can be determined by averaging Eq. 3 for a known distribution of d on x. We analyzed different functions of d(x), starting fromthesimplestd(x) ==z d > (1 +asit@)), witha a 1 and ye: h, as a parameters. We found that the results deoend on the normal&ed-to-unity probability of having each value of d in the OWL That is whv for demonstrative ~umoses we I 1 A use a Ga&ian distribution of d(x) around its average value < d >, with the characteristic half width Ad: ’ (4) where the probability 0(d) is normalized to unity. 0749%6036/90/080413+03$02.00/0 0 1990 Academic Press Limited