On the relationship between the quantum Langevin model and the Landauer formula

We show that, in the weak coupling limit, the main formalism of the phenomenological quantum I.angevin model for single electron tunneling in small tunnel junctions is derivable within the framework of the rigorous generalized Landauer formula. The study of Coulomb blockade, a suppression of single electron tunneling (SET) (for a review see Ref. [ 1 ] ) in small capacitance junctions, is attracting much experimental and theoretical interest. Likharev et al. [ 1 ] proposed the semiclassical orthodox theory, according to which the tunneling current is totally suppressed at zero temperature and voltages V< Vo = e/2C, where C is the capacitance. It turns out that Coulomb blockade is most pronounced if the influence of the environment is weak. More recent studies show that for the Coulomb blockade of single electron tunneling [2,3], there are two sources, the leads connecting the junction to the external circuit (the environment) and the discrete charge transfer across the junction, which reduce the effective Coulomb barrier i.e., partially smear off the Coulomb blockade. It has been demonstrated that the quantum Langevin (QLE) model [ 2,3 ] captures the basic physics of quantum smearing of Coulomb blockade (finite I at V< e /2C, where C is the junction capacitance) by taking into account the zero-point fluctuations of the instantaneous charge on the junction [2,3]. Nevertheless, the QLE model is phenomenological in nature. Thus, our first goal will be to show that, at least in the weak coupling limit, it leads to results which are identical to those obtained by use of the rigorous generalized Landauer formula [4-6 ]. We start with a brief review of the main aspects of the quantum Langevin equation (QLE) model for the environmental effects on single electron tunneling. First, one solves the quantum Langevin equation [ 2, 3 ] for the Fourier transform q (co) of the charge fluctuation q(t) . Then, using the fluctuation dissipation theorem, one obtains the mean-square charge fluctuation [2,3 ] oo ( q 2 ( t ) ) = f dco hcoC2 7~ 0 coth ( 1⁄2/~co) X Re(icoC+ Z , (o9)) ' (1) where fl = kBT, and Z(co) is the impedance of the environment (including the contribution from the tunnel junction). An interesting observation is that if the Ohmic model (Z(co) =constant) is applied to ( 1 ), at T = 0 one gets a divergent (q2) , and at T-,~ (classical limit) one gets (q2)=kBTC. The above 0375-9601/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0375-9601 ( 94 ) 00242-H G. Y. Hu, R.F. O'Connell / Physics Letters A 188 (1994) 384-386 analysis indicates that the Ohmic model is not applicable to ( 1 ) in the low temperature limit since there is an f-sum rule divergence problem. To calculate the effect of quantum smearing of Coulomb blockade in small tunnel junctions, it is assumed that the charge fluctuations q obey a Gaussian distribution. After accommodating the spread in values of q, and in terms of the effective tunneling rates ( F -+ >, the tunneling current is [2,3] I=e[ (F-(Q) ) ( F +(Q) ) ] e i dq [F-(a+q)-F+(a+q)]P(q),