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Quantum 1/f Effect in the Nuclear Spin Decoherence Rate and in Nanodevices

Peter H. Handel*

Department of Physics and Astronomy, University of Missouri
St. Louis, St. Louis, MO 63121

This is an abstract for a presentation given at the
Seventh Foresight Conference on Molecular Nanotechnology.
There will be a link from here to the full article when it is available on the web.


1. Introduction. The quantum 1/f effect is a fundamental fluctuation of all currents j of charged particles, of all scattering cross sections ss, recombination cross sections sr, tunneling rates or transition rates of any kind G, in short of all physical cross sections s and process rates G, given by the universal formula S(f) = 2aA/fN (conventional quantum 1/f equation; Handel 1975, 1980) for process rates in small devices, and = 2a/pfN (coherent quantum 1/f equation; Handel 1994) for electric currents in large devices. These two forms can be combined into a single general formula, as shown below in Eq. (5). Here S(f) is the spectral density of fractional fluctuations in current, dj/j, in scattering or recombination cross section ds/s, or in any other process rate dG/G. a=e2/hbarc=1/137 is Sommerfeld's fine structure constant, a magic number of our world de-pending only on Planck's constant hbar, the charge of the electron e and the speed of light in vacuum c. A = 2(Dv/c)2/3p is essentially the square of the vector velocity change Dv of the scattered particles in the scattering process whose fluctuations we are considering, in units of c. Finally, N is the number of particles used to define the notion of current j, of cross section s or of process rate G.

The way in which quantum-mechanical behavior at the microscopic level generates classical behavior at the macroscopic level is still open to investigation. Macroscopic quantum effects are considered to be the exception, but the study of such effects can shed light on this problem.

The conventional quantum 1/f effect (Q1/fE) in any rate G or cross section s is a macroscopic quantum phenomenon described by the quantum engineering formula explained above,

SdG/G(f) = Sds/s(f) = (4a/3pfN)(Dv/c)2. (1)

Planck's constant is in the denominator here, and this causes unusual behavior. Instead of being relevant at high frequencies, this quantum effect is important at very low frequencies, where it diverges, thereby becoming macroscopically observable in any laboratory or electronic device, although it is a genuine quantum fluctuation. Its quantum expectation is zero. Experiments have verified the 1/f spectrum of fundamental 1/f noise to below the frequency of 10-7 Hz. However, its presence in most dissipative parameters which enter VHF and UHF generators, mixers, resonators, amplifiers, attenuators, etc, causes it to limit the stability of high-tech devices and systems at any frequency in the form of phase noise or flicker of frequency.

The main purpose of this paper is to show how the quantum 1/f effect affects the operation of quantum engineering devices, i.e., quantum dots (or single-electron transistors), quantum wells, quantum wires, spin transistors or arrays of all these devices.

2. Quantum 1/f Fluctuations of the Nuclear Spin Decoherence Rate. Decoherence is caused by the elementary spin-flip of a nucleus due to its interaction with the rest of the world. The rate of this process, which reduces the total magnetic moment M of the sample by the magnetic moment m of a single nucleus undergoing a change of one hbar in its spin projection, has quantum fluctuations according to Eq. (1). The current change eDv causing bremsstrahlung is here (DMdot)/e, the change in the rate of demagnetization caused by the emission of a energy quantum. This yields

SdG/G(f) = 4a(Dv)2/3pfc2 = 4a(DMdot)2/3pfe2c2; (2)

Let mnucl be the mass of a nucleus, and let N be the number of elementary magnetic dipoles m=gehbar/mnuclc. Applying a variation Dn=1, we get

Dn/n = |DMdot|/|Mdot|, or DMdot=Mdot/n = Hgehbar/(mnuclc·n). (3)

Substituting DMdot into Eq. (2), we get

G-2SG(f) = 4a[Hgehbar/(mnuclc·n)]2/3pfe2c2 = 4a[Hghbar/(mnuclc2·n)]2/3pf (4)

This is the spectral density of fractional quantum 1/f fluctuations in the rate G of decoherence (electrodynamical Q1/fE only).

3. Quantum 1/f Effect in the Decoherence Time of Spin Transistors. The decoherence time of spin transistors is given by the spin relaxation time T1 which is of the order of 45 µs in the metal of the base, and which is strongly affected by the Q1/fE. A bottleneck is created in the spin transistor due to the scarcity of electrons with the right sign of the spin which are accepted by the collector. Therefore, current flow is proportional with the value of T1. Only those electrons which lose coherence, being subject to a spin-flipping decoherence interaction, can pass into the collector.

Decoherence is a very important process, because it causes what had earlier been simply called collapse of the wave function in the quantum mechanical measurement process, and because it limits the accessibility of quantum computing. Therefore it is interesting to note that the decoherence rate fluctuates with a 1/f spectral density. The quantum-electrodynamic part of these quantum 1/f fluctuations is caused by bremsstrahlung in the elementary interaction processes causing the decoherence. This, in turn, can be for instance electro-electron scattering, with a quantum 1/f effect given by Eq. (1) above.

4. Quantum 1/f Effect in Semiconductor Structures with Nanoscale Dimensions
At this point we ask how the Q1/fE changes when we scale a macroscopic conductor, semi-conductor, sample or device down to nanoscale dimensions. The transition from coherent to conventional Q1/fE is given by the relation

aH = (1/1+s)aconv + (s/1+s)acoher = (1/1+s)(4a/3p)(Dv/c)2 + (s/1+s)(2a/p), (5)

where s is a parameter which governs the transition and depends on the concentration n of carriers and on the transversal cross section area Q of the conductor, semiconductor, sample or device, perpendicular to the direction of the current. Specifically,

s = 2nQr0. (6)

Here r0 = e2/mc2 is the classical radius of the electron, r0= 2.84 10-13cm. Therefore, s is the number of carriers in a salami slice of thickness equal with the classical diameter of the electron, normal to the direction of current flow. The resulting spectral density of fractional quantum 1/f fluctuations is then given by the quantum 1/f coefficient aH through the Hooge relation

Sdj/j(f)= aH/fN. (7)

This shows that although the spectral density varies monotonously when the size of the cross section is lowered down to nanoscale dimensions, there is a plateau on which the spectral density remains constant, while aH changes its value.


  1. P.H. Handel, "1/f Noise -an 'Infrared' Phenomenon," Phys. Rev. Letters, vol. 34, p. 1492, 1975.
  2. P.H. Handel, "Quantum Approach to 1/f Noise," Phys. Rev. vol. A22, p. 745, 1980.
  3. P.H. Handel, "Fundamental Quantum 1/f Noise in Semiconductor Devices," IEEE Trans., vol. ED-41, 2023, 1994.
  4. M. Tacano, "Quantum 1/f Noise and other Low Frequency Fluctuations" AIP Conf. Proceedings #282, P.H. Handel and A.L. Chung Editors, A.I.P. Press, 1992, p. 21.

*Corresponding Address:
Prof. Peter H. Handel
Dept. of Physics & Astronomy, University of Missouri
St. Louis, MO 63121 USA
Tel. 314/516-5021; FAX 314/516-6152


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