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Quantum 1/f Effect in Semiconductor and Magnetic Structures
with Nanoscale Dimensions

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
Sixth Foresight Conference on Molecular Nanotechnology.
There will be a link from here to the full article when it is available on the web.


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. (1). 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 depending 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 conside-red 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.

1. Quantum 1/f Effect in Quantum Dots

Quantum dots are also known as artificial atoms or single-electron transistors (SET). A small portion of the the length of the 2-degrees-of-freedom (2D) channel of a HFET is pinched off with two barriers, and for a certain value of the gate potential, the discrete energy level of a free electron in the SET corresponds with the fermi-energy of the metal in the leads, allowing the electron to perform resonant tunneling through the SET. For other values there is no current tranport, at least if the temperature is sufficiently low and if the Kondo-effect is negligible. The Kondo effect causes the channel resistivity to increase at low temperature, because of the coupling with localized impurity spins in the channel. Moreover, the Kondo interaction also causes energy contributions which broaden the line-shape of the resonance mentioned above, thereby broadening the transmission line in the current dependence on gate voltage, introducing a characteristic double peak structure due to spin interactions. The quantum 1/f effect is present in the current transmitted through the SET off-resonance, and should be studied in order to better quantify the thermal and Kondo energy and momentum differences.

Indeed, the quantum 1/f effect depends on the momentum change Dp of the electrons in the tunneling process, while the off-resonance current depends on the energy change of the electrons. This allows the effective spectrum of elementary excitations to be studied by comparing these two dependences. Furthermore, using the conventional quantum 1/f formula, one can then optimize the device for practical applications which require stability of the current flowing through the open SET. Since N is 1 in the SET, the Q1/fE is expected to be relatively large in this devices.

2. Quantum 1/f Effect in Quantum Wells

A two-dimensional all optical processor which multiplexes a rectangular array of parallel incoming signals into a series output transmitted through a single optical fiber was suggested for the first time by the author along with a corresponding 2-dimensional series to parallel transformer. The system uses four-wave interaction in a hologram projected on a multiple quantum well (MQW) device, both for time-division multiplexing and for demultiplexing. It can be used for the direct fiber-optical transmission of many time-dependent images, without going through the usual video-electronic serialization, as well as for multiple analog or digital, video and audio-transmissions, or multiplexing and demultiplexing of any nature.

The system is very sensitive to quantum 1/f amplitude and phase noise introduced by the quantum wells in the holographic medium and by the semi-transparent mirrors. The MQW device is a semi-insulating MQW, or SI-MQW. It consists for instance of a succession (e.g., 150) of (e.g. 100Å wide) GaAs quantum wells sandwiched between AlGaAs barrier planes (e.g., 35Å thick). Large diffraction efficiency of the red-out beam can be obtained when a real-time hologram is recorded in the material. The diffraction efficiency is the ratio of the diffracted beam power to the incident power.

The recording of the hologram is performed by the generation of fringes of high conductivity (e.g., p-type) sandwiched by semi-insulating (intrinsic) fringes at places of destructive interference in the otherwise semi-insulating MQW device. The high-conductivity fringes appear where the two coherent beams have interference maxima: the parallel optical information carrying beam and the reference beam.

The Q1/fE will be influenced by the discrete energy level sheme present in each quantum well and by any coupling between wave functions allowing tunneling between wells. Due to the small average concentration of carriers n in the MQW structure, the quantum 1/f conductivity fluctuations ds are large, the spectral source term Ss(f,r,r') =<dsrdsr'>f being proportional to 1/n. Due to the small concentration n, the magnetic contribution to the energy of the electronic drift motion is negligible, and therefore the conventional quantum 1/f formula is applicable. The conventional quantum 1/f formula, gives a spectral source term of Ss(f,r,r') = (4aDv2/3pfnc2)d(r-r'), where a =1/137 is the fine structure constant, Dv the velocity change in the scattering process dominating the resistivity, and n the local concentration of carriers.

The effect of this large quantum 1/f noise on the diffraction efficiency is amplified, because the diffraction efficiency is proportional to the difference in s between the diffraction maxima and minima. The relative fluctuation spectrum of the diffraction efficiency is given by [Ss + S s']/(s-s')2. When a constant image is transmitted, i.e., if the n¥m incoming channels are each locked in the endless repetition of a certain bit of information (e.g., 0 or 1), the received image emerging from the demultiplexer will flicker due to the quantum 1/f noise in both the multiplexer and demultiplexer. This will determine how large the logic swings are and how large the minimal power levels in the hologram are, compared with the ones acceptable. Certain holographic media (e.g., semi-insulating CdZnTe/ZnTe multiple-quantum-well photorefractive devices), may be eliminated on this basis. The quantum 1 /f refraction index modulation is related to the quantum 1/f conductivity fluctuation through the Cramers-Cronig dispersion relations.

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 ms 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, semiconductor, 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), (2)

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. (3)

Here r0 = e2/mc2 is the classical radius of the electron, r0= 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 (see Fig. 1). 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. (4)

This resulting total dependence on N is shown qualitatively in Fig. 2 below for the transition from macroscopic dimensions to nanoscale samples. It 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.

thickness slice

Fig. 1: To define the parameter s, a slice as thick as the classical electron radius is considered. The number of carriers in it is s.


thickness slice

Fig. 2: The quantum 1/f parameter aH and the resulting spectral density Sj = aH/Nf as a function of the number of carriers in the sample, N or of the cross section size L.



  • Handel (1975) Phys. Rev. Letters 34, p.1492-1494. 1/f Noise -an 'Infrared' Phenomenon
  • Handel (1980) Phys. Rev. A22, 745-757. Quantum Approach to 1/f Noise
  • Handel (1994) IEEE Trans. on Electr. Devices 41, 2023-2033. Fundamental Quantum 1/f Noise in Semiconductor Devices

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

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