Основы наноелектроники / Основы наноэлектроники / ИДЗ / Книги и монографии / Наноматериалы, методы, идеи. Сборник научных статей, 2007, c.206

The function f (px ); satisfies the same normalization condition as the equilibrium function f0 (px)

and, therefore, it makes the integral of right-hand side of formula (3) vanish. Besides, the integral of left-hand side of the Boltzmann equation

(3) vanishes too, because of the periodicity condition mentioned. The condition (8) can be proved by direct calculation using formulae from [7]. The distribution function f(px) at several values of Ex and T is shown in figure 1.

The current density j in the direction of 1SL axis can be found (in dimensional units) by a conventional way

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The value of A(E x ,T) can be estimated numerically with high accuracy. Expanding the exponent in a power series we get

where functions Gn(Ex) are defined by recurrent formula

As Gn Ex [0,1/2n 3)), series (13) converges quickly. As

numerical experiments show, first four terms of series (13) give good approximation at T > 0.5.

At |EX| 0 we have A(EX,T) 0, so in low fields (|EX| « 1) in linear approximation on Ex we have

where angle brackets mean averaging over the equilibrium distribution. Note that the conductivity temperature dependence in low fields (the expression within round brackets in (15)) is close to the analogous dependence for the miniband cosine model (I1(1/2T)/I0(1/2T), In(z) being the modified Bessel function).

In high fields (|EX| > 1) we have

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(17)

As numerical experiments show, formula (17) gives good approximation at T < 0.07. In limiting case T 0 we get the expression that was found in [6]

Equation (10) determines the current-voltage characteristic for the parabolic miniband 1SL with the current density temperature dependence taking into account.

From (15,16) it follows that j ~ Ex at |EX| « 1 and j ~ 1/Ex at |Ex| » 1. Therefore at fixed temperature T = fix the function j(Ex,T) reaches its maximum at some value Ex = ExC(T) > 0 (see figure 5).

Thus the parametric representation of dependence Exc(T) is defined by

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and it is sufficient to solve this equation at Ex > 0. 1

The numerical solution of equation (20) at T versus Ex is presented in figure 2. Note that dependence Ex = ExC(T) is monotone so the inverse function Tc F= TC(EX) exists.

Figure 2. The dependence T — Tc(Ex) (solid line). Regarding the SL2 model this curve represents local separatix on controlling parameters plane. The dashed lines 1 an 2 represent the approximations (21) and (24) respectively. The icons show the behaviour of synergetic potential Ф(ЕУ) at marked points in controlling parameters space.

To investigate behaviour of function Tc(Ex), consider first the case of high temperatures T»1. Expanding all functions in a power series on 1/T and neglecting all terms o(l/T2), we get

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Consider now the case of low temperatures T«1. Using (17), we ge,

Note, that Exc{T) decreases with increasing temperature. Essentially, that Exc value does not depends on the temperature at all in the cosine model. Meanwhile, in experiment [8] a shift of Exc to lower fields has been found with increasing temperature. (In [8] the miniband transport in 1SL GaAs/AlAs was investigated in temperature range 10-300 К under fields ut to 1.5 kV/cm, and negative differential conductivity was found surely enough.)

3. Nonequilibrium phase transitions in 2SL

The dispersion law of the square 2SL mentioned in Section 1 takes the form

Where o — const is electron energy in the lowest size-quantized level, here is miniband width.

If the coordinate axes are directed along the SL principal ones, then the current density along X axis is given by (11), while that along Y axis is given by the same formula with x у change.

Now a situation is considered, where the coordinate axes are rotated by angle 45° with respect to principal axes of the square 2SL. In this case we have

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where the field is expressed in units of 2 /ed ,, while the current in units ofту d / 2 , function j(Ex,T) is defined by (11). Note that

Let the sample be open in Y direction, so that the equation of state of nonequilibrium electron gas is

Consider the transverse field Ey as an order parameter and EXl T as controlling parameters. Then for given Ex and T condition (31) defines the stationary states Ey = Ey(Ex,T). As it follows from (30), the states Ey = 0 are stationary Ip addition to this trivial solution, which may be stable or unstable against small fluctuations, equation (31) has also nontrivial ones (Ey 0), which are solutions of equation j y /Ey = 0.

The stability of transverse field Ey is determined with the following inequality [9]:

where Ф is a synergetic potential (the entropy production [10]). Thus the stable solutions of equation (31) by fixed EX,T correspond to minima of the synergetic potential Ф.

Using the synergetic.potential, we shall analyse the behaviour of nonequilibrium electron gas using the concepts and methods of catastrophe theory [11]. The degenerate pririca] points of potential Ф are defined by conditions

Equations (33) determine implicitly the points in the controlling parameters space (EX,T > 0), which parametrize the bifurcation set (local separatrix). At Ey = 0 the solutions of system (33) determine the set of bifurcation points, consisting of two curves symmetrical with respect to the line Eх = 0:

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= 0, so each bifurcation point defines also the extremum of 1SL current density. Therefore the set of bifurcation points coinsides with that presented in figure 2.

Numerical experiments show that local separatrix consists only of bifurcation points. As a result the controlling parameters space is decomposed to two open subregions I and II, parametrizing the structurally stable potential functions of qualitatively different type. So it is sufficient to choose any point from each region I, II and to study behaviour of appropriate function Ф. In figure 2 the form of potentials, corresponding to come singly points on controlling parameters set is presented. Thus if the point (T, Ex) is located in region I, then synergetic potential has only one critical point Ey = 0, at which it reaches its minimum. So for such values of controlling parameters the only stable state Ey = 0 exists. If the point (T, Ex) is located in region II, then synergetic potential has three critical points and two local minima, corresponding to two nontrivial solutions of equation (31), located on the upper and lower sheets of critical manifold and corresponding to stable states Ey 0. In this case the state Ey = 0 is unstable. Along the local separatrix the states Ey = 0 become degenerate. Therefore when any path in the controling parameters space intersects the local separatrix, the slable states Ey 0 appear and NPT2 occurs. Thus the function TC(EX) defines the "Curie temperature" for Ex G (E1,E2).

The numerical solutions of (31) are presented in figures 3 and 4, in which different situations are illustrated.

Consider now the behaviour of nontrivial transverse field in tne vicinity of bifurcation point. Let |E y | « 1, then

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Figure 3. The dependence of spontaneous transverse field Ey on the temperature T (normalized). At fixed applied field the path Ex = fix on the controlling parameters space intersects the local separatrix at point T = Tc(Ex) and NPT2 occurs. The curves correspond to the following values of Ex: 1) Ex = 0.31; 2) Ex = 0.33; 3) Ex = 0.36; 4) Ex = 0.374 > E2 . The dotted lines represent approximation (37).

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Figure 4. The dependence of spontaneous transverse field Ey on the applied field Ex {normalized). At fixed temperature the path T'= fix on the controlling parameters space intersects the local separatrix at point Ex = Exc(T) and NPT2 occurs. The curves correspond to the following values of T: 1) T = 0.05; 2) T = 0.1; S) T= 0.5. The dotted lines represent approximation (38).

From (35) and (36) we get the approximate solution of equation jy/Ey = 0

figure 3). Let T = fix and |EX — ExC(T)| «1, then in a similar way

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