shot noise in mesoscopic conductors - Low Temperature Laboratory

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20 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 operators describe electrons in the outgoing states. They obey anticommutation relations a( (E)a( (E)#a( (E)a( (E)" (E!E) , a( (E)a( (E)#a( (E)a( (E)"0 , a( (E)a( (E)#a( (E)a( (E)"0 . Similarly, we introduce creation and annihilation operators a( (E) and a( (E) in incoming states and bK (E) and bK (E) in outgoing states in the right lead (Fig. 2). The operators a( and bK are related via the scattering matrix s, bK 2 2 bK a( . (29) bK a( 2 2 bK a( "sa( The creation operators a( and bK obey the same relation with the Hermitian conjugated matrix s. The matrix s has dimensions (N #N )(N #N ). Its size, as well as the matrix elements, depends on the total energy E. It has the block structure r t s" . (30) t r Here the square diagonal blocks r (size N N ) and r (size N N ) describe electron re#ection back to the left and right reservoirs, respectively. The o!-diagonal, rectangular blocks t (size N N ) and t (size N N ) are responsible for the electron transmission through the sample. The #ux conservation in the scattering process implies that the matrix s is quite generally unitary. In the presence of time-reversal symmetry the scattering matrix is also symmetric. The current operator in the left lead (far from the sample) is expressed in a standard way, IK (z, t)" e 2im dr K (r, t) R Rz K (r, t)! R Rz K (r, t) K (r, t) , where the "eld operators K and K are de"ned as and K (r, t)" dE e K (r, t)" dE e (r ) (2v (E)) [a( e#bK e] H (r ) (2v (E)) [a( e#bK e] . Here r is the transverse coordinate(s) and z is the coordinate along the leads (measured from left to right); are the transverse wave functions, and we have introduced the wave vector,
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 21 k "[2m(E!E )] (the summation only includes channels with real k ), and the velocity of carriers v (E)"k /m in the nth transverse channel. After some algebra, the expression for the current can be cast into the form IK (z, t)" e 4 1 dE dE e v (E)v (E) [v (E)#v (E)][exp[i(k (E)!k (E))z]a( (E)a( (E) ! exp[i(k (E)!k (E))z]bK (E)bK (E)] # [v (E)!v (E)][exp[!i(k (E)#k (E))z]a( (E)bK (E) ! exp[i(k (E)#k (E))z]bK (E)a( (E)] . (31) This expression is cumbersome, and, in addition, depends explicitly on the coordinate z. However, it can be considerably simpli"ed. The key point is that for all observable quantities (average current, noise, or higher moments of the current distribution) the energies E and E in Eq. (31) either coincide, or are close to each other. On the other hand, the velocities v (E) vary with energy quite slowly, typically on the scale of the Fermi energy. Therefore, one can neglect their energy dependence, and reduce the expression (31) to a much simpler form, IK (t)" e 2 dE dE e [a( (E)a( (E)!bK (E)bK (E)] . (32) Note that n( (E)"a( (E)a( (E) is the operator of the occupation number of the incident carriers in lead L in channel n. Similarly, n( (E)"bK (E)bK (E) is the operator of the occupation number of the out-going carriers in lead L in channel n. Setting E"E# and carrying out the integral over gives IK (t)" e 2 dE [n( (E, t)!n( (E, t)] . (33) Here n( (E, t) are the time-dependent occupation numbers for the left and right moving carriers at energy E. Thus, Eq. (33) states that the current at time t is simply determined by the di!erence in occupation number between the left and right movers in each channel. We made use of this intuitively appealing result already in the introduction. Using Eq. (29) we can express the current in terms of the a( and a( operators alone, IK (t)" e 2 dE dE ea( (E)A (¸; E, E)a( (E) . (34) Here the indices and label the reservoirs and may assume values L or R. The matrix A is de"ned as A (¸; E, E)" ! s (E)s (E) , (35) _ _ A discussion of the limitations of Eq. (32) is given in Ref. [10].
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Ya.M. Blanter, M. Bu( ttiker / Phys
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Our starting point is Eq. (51), whi
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introducing the transmission coe$ci
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investigated experimentally. A self
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and the non-equilibrium resistance
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the sample. These electrons are des
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with D ,¹ (2!¹ ). Performing t
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For completeness, we mention that i
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We note here, however, that there i
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The noise power (201) is strongly v
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Fig. 32. Experimental results of Ku
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where Ya.M. Blanter, M. Bu( ttiker
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where the quantity is expressed th
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6.3. Interaction ewects Ya.M. Blant
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one "rst goes from the non-interact
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non-thermalized electrons, the high
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Fig. 34. Geometry of the chaotic ca
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which describes strictly ballistic
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approximately e/C. Between the peak
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Boltzmann}Langevin approach (see Se
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thermal to shot noise. It is, howev
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Note added in proof Ya.M. Blanter,
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and Lee et al. [340] obtain in this
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