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S_{dG/G}(f) = S_{ds/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 hightech 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 singleelectron 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 spinflip 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 in its spin projection, has quantum fluctuations according to Eq. (1). The current change eDv causing bremsstrahlung is here (D)/e, the change in the rate of demagnetization caused by the emission of a energy quantum. This yields
S_{dG/G}(f) = 4a(Dv)^{2}/3pfc^{2} = 4a(D)^{2}/3pfe^{2}c^{2};  (2) 
Let m_{nucl} be the mass of a nucleus, and let N be the number of elementary magnetic dipoles m=ge/m_{nucl}c. Applying a variation Dn=1, we get
Dn/n = D/, or D=/n = Hge/(m_{nucl}c·n).  (3) 
Substituting D into Eq. (2), we get
G^{2}S_{G}(f) = 4a[Hge/(m_{nucl}c·n)]^{2}/3pfe^{2}c^{2} = 4a[Hg/(m_{nucl}c^{2}·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 T_{1} 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 T_{1}. Only those electrons which lose coherence, being subject to a spinflipping 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 quantumelectrodynamic 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 electroelectron 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, semiconductor, sample or device down to nanoscale dimensions. The transition from coherent to conventional Q1/fE is given by the relation
a_{H} = (1/1+s)a_{conv} + (s/1+s)a_{coher} = (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 = 2nQr_{0}.  (6) 
Here r_{0} = e^{2}/mc^{2} is the classical radius of the electron, r_{0}= 2.84 10^{13}cm. 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 a_{H} through the Hooge relation
S_{dj/j}(f)= a_{H}/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 a_{H} changes its value.
References
^{*}Corresponding Address:
Prof. Peter H. Handel
Dept. of Physics & Astronomy, University of Missouri
St. Louis, MO 63121 USA
Tel. 314/5165021; FAX 314/5166152
Email: handel@jinx.umsl.edu
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