Detailed Balance

The principle of detailed balance is important in describing the properties of the equilibrium state. In the particular case of electron devices it assures us that the current density is zero when the applied voltage (as measured by the difference in chemical potentials) is zero. The reader may have noticed that the concept of equilibrium has played no part in the development of the present open-system model, and indeed the only place where the chemical potential can appear is in the boundary-condition distribution function. In this context it may not be surprising that the discrete model does not exactly satisfy the detailed-balance condition. This was discovered by Jensen and Buot (1989a), who noticed that if the steady-state curves were computed for a structure lacking inversion symmetry (having unequal barrier widths), a nonnegligible current density was obtained at zero bias. Because it is precisely detailed balance which leads us to expect zero current in equilibrium, the spurious equilibrium current is a measure of the violation of this condition.

Given the observation that the discrete model does not exactly satisfy detailed balance, we should determine whether this is a consequence of the discretization or of the open-system boundary conditions themselves. A simple way to do this is to compute the zero-bias current density for an asymmetric RTD structure using varying mesh spacings and . This was done for a structure which was identical to that described in section 5, except that the widths of the barriers were 3.4 nm and 2.3 nm. It was found that was essentially independent of and , as illustrated in Fig. 23. Thus, the violation of detailed balance is entirely a result of the discretization, and the continuum formulation will apparently satisfy the detailed balance principle.

Figure 23. Violation of the principle of detailed balance in the discrete open-system model. The current density calculated for an asymmetric structure in equilibrium is plotted versus the mesh spacing used in the calculation. The results show that the current density (which measures the departure from detailed balance) is of and is thus a result of the discretization, not of the open-system boundary conditions.

Let us examine this issue in more detail. To begin, let us see what detailed balance implies about the equilibrium density operator or Wigner function. Because the processes occurring in equilibrium must be reversible, the density operator must equal its time-reversed value , or must be purely real. This implies that the equilibrium Wigner distribution must be a symmetric function of p. Thus, an alternative measure of the departure from detailed balance is . Evaluating this measure for computed with various mesh spacings leads to the same conclusion: the irreversible model violates detailed balance to , and the error is independent of .

The procedure of solving for the steady-state Wigner function and then examining the scaling properties of various features of those solutions is, in the absence of a well-developed and thoroughly checked mathematical analysis, the most reliable way to address such questions as the departure from detailed balance. However, if one is to compare alternative discretization schemes for a particular probelem, as is attempted below, it is much more desirable to be able to determine the order of the errors from a knowledge only of the equations (as was done with the moment equations), rather than the solutions. In particular, we want to be able to examine a discretization of the Liouville superoperator and determine the order of error in detailed balance. At present, no simple criterion has been identified which would permit such an analysis. However, we may again examine the factors which bear upon this problem.

Let us again consider the purely classical example of an open system with no internal dissipation. Then the particles will follow their classical trajectories and along those trajectories, the distribution function f will be constant. Detailed balance follows from the presence of a time-reversed trajectory for any given trajectory. Because the energy is constant along a trajectory, the density at an outflowing boundary will be equal to the corresponding inflowing density if, and only if, the distribution functions in the two reservoirs are identical functions of energy ( i.e., in equilibrium). If we focus upon a differential element of the trajectory, the condition that there exists a time-reversed trajectory can be expressed as

which becomes when transformed back to the density-matrix representation, leading to the unsurprising conclusion that time-reversibility is equivalent to the Hermiticity of . In fact, the irreversible model (4.48) satisfies the condition (7.124) if we include the boundary terms (4.49). It would appear appropriate to include these terms in the detailed-balance test, whereas we neglect them in the stability analysis. However, this argument leads to the conclusion that the model ought to exactly satisfy detailed balance.

A further consideration of the classical case suggests that the departure from detailed balance might be traceable to discretization errors in the classical trajectories. That is, when we restrict the distribution function to a discrete mesh of points, a particle cannot exactly follow the proper trajectory, and the time-reversed trajectory might not exactly balance it. The way to correct such a situation is adopt the Lagrangian coordinates discussed in Appendix 11. Then the upwind difference would be applied to the directional derivative along a trajectory and would exactly satisfy time-reversibility. However, this does not help in cases such as quantum-mechanical tunneling, in which trajectories cannot be defined. Discretization errors in the trajectories would presumably lead to the conclusion that both and contribute to the error, contrary to what has been observed. If the error were of the form and the terms had coefficients of different magnitudes, the numerical experiments might easily have overlooked the weaker dependence.

Another way to view the problem of detailed balance in a completely quantum-mechanical context is to note that the equilibrium distribution should satisfy the Bloch equation (6.102). The stationarity of such a distribution under time evolution by the Liouville operator would follow from . We have noted that this is necessarily true in a closed system, but it is not true for an open-system model. In the present case the commutator has nonzero elements adjacent to the boundaries of the system. These might be removed by including the inhomogeneous terms, but the meaning of an inhomogeneous term in a commutator is far from clear.

The connection between detailed balance and reversibility or Hermiticity suggests the following conjecture: that it is impossible to exactly satisfy both detailed balance and the stability condition (irreversibility) in a model with a finite number of degrees of freedom (such as a bounded, discrete model). That this is possible in a model with an infinite number of degrees of freedom, as in unbounded or continuous models, is the thrust of the conventional theories of irreversibility. If this conjecture is correct, this is a significant limit on the accuracy achievable with discrete open-system models.