ЛЕКЦИИ ФШФС_2007 / НАШИ СТАТЬИ / JCTE477_РЭ_5_05_Англ

INTRODUCTION

Reliability has become a major concern in the design of electronic systems containing hundreds of large-scale integrated circuits (LSICs) and very largescale integrated circuits (VLSICs); this is particularly true of ICs intended for long-term operation. Note also that the reliability of modern ICs is mainly determined by the quality of their passive constituent parts: interconnections, contact pads, resistors, vias, and dielectric films.

In microelectronics high-quality thin films of metals or alloys are widely used to make interconnections, resistors, and ohmic contacts. Interconnections and contacts experience high current densities and thermal stress during operation. These enhance different degradation processes, which gradually change the electrical characteristics of films and contacts until a sudden failure occurs. Investigations have revealed that interconnections and contacts are involved in most failures [1, 2]. It is therefore highly desirable to develop rapid nondestructive methods for the quality control of metal films and film and ohmic contacts. As the term is usually understood in relation to an IC, a nondestructive method is one that detects latent defects by indirect evidence while preserving the capabilities of the IC. An advantage of nondestructive quality control is that it can be readily incorporated into fabrication processes and product tests.

In recent decades, an approach has emerged whereby IC reliability is estimated by measuring the power spectral density (PSD) of flicker noise and the

nonlinearity of the current–voltage characteristic (CVC) for passive circuit parts (metal films, film and ohmic contacts, vias, solder bonds, MOS structures, etc.) [1–4]. Flicker noise is characterized by power spectral density varying as 1/f γ, where γ is a constant. The procedure is most effective if performed in process. Usually, nondestructive control methods are assumed to be the methods using indirect features for detection of latent defects without any changes in quality, parameters, and characteristics of the tested device. Nondestructive control offers some advantages over other control methods. It can be directly introduced into the IC manufacturing processes and test procedures. The most effective method is the IC quality control at the manufacturing stage that uses measurements of the PSD of the 1/fγ noise and nonlinearity parameters of CVCs of test structures for various types of passive components: metal films, film, and ohmic contacts, interconnections, soldered connections, metal–oxide– semiconductor (MOS) structures, etc. There is considerable experimental evidence that latent defects in and damage to passive circuit elements or interconnections can appear as elevated flicker noise. A related strategy is to measure fluctuations in current consumption. It applies to both finished ICs and a system as a whole.

However, the reduction of latent-defect density due to advances in manufacturing technology has made dc measurements of 1/f noise inadequate for this purpose. The following example illustrates this point. Figure 1 shows the effect of annealing measured under a direct current of I0 = 3 mA on two different Ni–Cr film resis-

Fig. 1. Effect of annealing on two different Ni–Cr film resistors, R1 and R2, that originally displayed the same noise power spectral density. Measurements were performed under a direct current of 3 mA [4] for 420 h at an annealing temperature of 520 K. Curves 1 and 2 represent the power spectral densities for R1 and R2, respectively, before the annealing. Curves 1' and 2' refer to R1 and R2, respectively, after the annealing (dR/R = 0.04, dR/R < 0.005).

tors, R1 and R2, that have originally displayed the same power spectral density of 1/f noise [4]. The annealing temperature and duration were 520 K and 420 h, respectively. The heat treatment caused the 1/f noise in R1 to increase by two orders of magnitude and the resistance to deviate by 4%, indications of annealinginduced damage. The noise in R2 remained unchanged and the resistance drift was less than 0.5%. These indicate that two resistors with the same power spectral density of flicker noise observed under dc conditions may differ greatly in temperature stability. Thus, the strategy has to be modified.

This review begins with an overview of mechanisms underlying different types of electrical noise in metal films and contacts. An analysis is then given for theoretical and experimental results concerning equilibrium and nonequilibrium 1/f noise and the nonlinear behavior of the current–voltage characteristic (CVC) in the context of metal and alloy films, ohmic contacts, Auwire bonds to Cu contact pads, flip-chip solder bonds, and film resistors. This analysis provides the groundwork for a discussion of nondestructive quality control as applied to passive circuit parts. Measurement techniques are then examined for nonequilibrium flicker noise in one-port passive networks. The review also covers the characterization of CMOS LSICs and electronic systems by measured fluctuations in current consumption.

1.ELECTRICAL NOISE IN SOLIDS AND SOLID-STATE DEVICES

In solids there are several types of electrical noise that differ in nature and mathematical description. In solid-state electronics, major noise sources are resistors, contacts, diodes, and transistors. The most impor-

The thermal motion of free electrons in a conductor produces currents on a microscopic scale. At any point of time, the flow of electrons in one direction may be stronger than in the others, owing to the chaotic character of electron motion. As a result, the voltage across the conductor is subject to faint fluctuations, which depend on the conductor resistance R and temperature T. These fluctuations are known as thermal noise. As electrons travel over the conductor, they collide with ions at lattice sites; with lattice defects; and, more rarely, with one another. The electron velocity is estimated at 107 cm/s, even at room temperature. The corresponding collision rate is so high the current pulse produced by an electron has a duration as short as τ ~ 10–13 s. The fluctuating voltage is thus made up of numerous random pulses, and its power spectral density is flat over

frequencies up to fmax ≈ 0.5πτ ~ 1013 Hz. For this reason, thermal noise is characterized as white noise.

Thermal noise exists whether or not a direct current flows through the conductor. It depends on the atomic structure of the material, and its level is a function of R and T only.

At thermal equilibrium, the Nyquist noise theorem applies [1, 2]. With common operating temperatures, this theorem implies that the mean-square thermal-

noise voltage component U2T , measured in square volts, obeys the relation

U2T = 4kT ReZ∆f ,

where the overbar means averaging over time, k is the Boltzmann constant (k = 1.38 × 10–23 J/K), T is the absolute temperature; Z is the conductor impedance (in ohms); and ∆f is the bandwidth of the circuit or instrument (in hertz).

In the case of a nonlinear CVC, the impedance must be replaced with the differential resistance at the operating point.

Note that the Nyquist noise theorem is true only for systems at thermal equilibrium. Furthermore, it applies only at frequencies f such that 2π f/kT 1, where is the Planck constant ( = 6.62 × 10–32/2π J s). Otherwise,

the above equation must be modified to include quantum phenomena [1].

For a specimen of resistance R, the power spectral density of thermal noise is given by

1B. Hot-Electron (Diffusion) Noise

If a semiconductor is placed in a sufficiently strong electric field, its electron gas becomes a nonequilibrium system because the mean kinetic energy of electrons is raised above their thermal energy, 3/2kT. Electrons in this state are called hot electrons. With such fields, Ohm’s law fails and the current through the semiconductor varies in a nonlinear manner with field strength. The reason for such behavior is the electron mobility being a function of field strength. The Nyquist noise theorem does not apply in this case. Aside from thermal noise, the semiconductor plasma exhibits the noise associated with random electron–lattice energy transfer and with the fluctuating absorption of externally supplied power [1]. This type of noise is known as diffusion noise. Its power spectral density is also flat over a wide frequency range. Diffusion noise is the most common type of electrical noise in solids. With diffusion noise, dissipation occurs by fluctuations, whereas thermal noise is due to the fluctuations associated with the internal energy of the system. Diffusion noise can be used for device quality assessment in some cases.

1C. Shot Noise

Shot noise results from the discrete nature of electric charge. It is common in solid-state and vacuum electronics. Consider, for example, a vacuum diode in which electrons leave the cathode and hit the anode at random and independent points of time. If the diode is at saturation, every electron emitted will reach the anode to produce a current pulse through an external circuit, the pulse width being determined by the transit time τ. The resulting fluctuating component of anode current is given by the Schottky equation:

i2sh = 2eI0∆f ,

where I0 is the mean anode current (in amperes) [1, 2].

The spectrum of shot noise is bounded above by

1/2πτ.

In semiconductor devices, shot noise is generated by biased p–n junctions, and its level depends on the bias polarity and magnitude. Note that both majority and minority carriers contribute to shot noise, the contribution of the latter varying with operating conditions.

Current I through a semiconductor diode can be treated as consisting of two independent components:

the diffusion minority-carrier current Isat exp(U/ϕT) = I + Isat and the drift majority-carrier current Isat , where Isat is the saturation current, U is the applied voltage, and ϕT = kT/e. The two current components fluctuate independently to produce shot noise. At low frequencies, its power spectral density is as follows [5]:

(in A2/Hz).

In homogeneous conductors, shot noise is related to the drift velocity of carriers, whereas thermal noise is determined by their thermal velocity. In metals, shot noise is insignificant because the drift velocity is small compared with the thermal velocity.

From the preceding discussion, it should be clear that thermal and shot noise are inherent in electronic devices. They represent the minimum possible noise level.

1D. Generation–Recombination Noise

Generation–recombination noise occurs in semiconductors owing to fluctuations in free-carrier density as a result of fluctuating carrier recombination and generation via trapping levels, which are produced by impurities and lattice disturbances [1, 2, 5].

For a semiconductor that carries current I0 , the power spectral density of generation–recombination noise is given by

where N is the fluctuating total number of carriers in the semiconductor, N0 is the total number of carriers at

equilibrium, ∆N = N – N0, ∆ N2 is the averaged value of ∆N squared, τ is the carrier lifetime, f is the frequency, and ω = 2πf is the cyclic frequency [1, 5].

Assume that individual carriers are independent of

one another, and N is a Poisson variable. Then ∆ N2 = N0 [5]. Furthermore, we notice that the power spectral

density remains at Sgr0 = (4 I20 τ)/N0 for ωτ 1; if ω2τ2 1, the spectral density varies as 1/ω2, constituting the Lorentz profile. Lifetime τ can be found from frequency f0 such that Sgr( f0) = Sgr0/2, because τ = 1/2πf0 .

Metal films also exhibit a Lorentzian–Debye spectrum if vacancy sinks are distributed uniformly over the film and the probability of annihilation for any vacancy has a uniform distribution over any period within its lifetime. Such a fluctuation spectrum was observed by Celasco in Al films with a mean grain size of ~0.5 µm;

Yet another example is submicrometer field-effect transistors, in which RTS noise results from fluctuations in channel resistance due to migration of single carriers from the channel to oxide traps and back. The resulting current pulses, though low, can be used to explore the dynamics of carrier capture and emission by single traps, as well as to determine trap parameters. The noise becomes increasingly apparent as the feature sizes decrease.

Advances in submicrometer technology furthered research into RTS noise in semiconductor devices. A common mechanism was thus found to underlie three types of excess noise associated with carrier-density fluctuations. These are generation–recombination noise, flicker noise (the ∆n model) [1], and RTS noise. This last type predominates in nanoscale structures. As the size of a device increases, so does the number of traps, and RTS noise changes into noise with a 1/f spectral density. In metal films, the number of vacancy sources and sinks and, hence, quasi-equilibrium vacancy density increase with the minimum feature size. The vacancies have a wide range of activation energies, and fluctuations in their density produce noise with a 1/f spectral density (see Section 2B).

RTS noise is also found in the current consumption of digital CMOS circuits sustaining open faults in the metallization pattern or contact windows (see Section 6).

Like generation–recombination noise, RTS noise has a Lorentzian spectrum:

where A is a constant and τ0 is such that

in which τ1 and τ2 are the mean values of t1 and t2 , respectively, as shown in Fig. 2 [2]. For semiconductors, τ1 and τ2 refer to carrier capture and emission, respectively, by traps. For metal films and contacts, they refer to the creation and annihilation of vacancy defects.

2. PHYSICAL MECHANISMS OF 1/f NOISE AND NONLINEARITY OF CVCs

OF METAL FILMS AND CONTACTS

It is thought that 1/f noise in solids results from fluctuations in conductance. These fluctuations may also correspond to the absence of current in a specimen. Their equilibrium nature was shown by Voss and Clark [8] who made direct thermal-noise measurements on InSb films and Nb film islands in the absence of current. The noise was found to have a 1/f spectral density that remained unchanged on passing of current through the specimen, irrespective of the time pattern of the current (constant, sinusoidal, or pulsed). This is evidence that

the noise is due to equilibrium fluctuations in conductance. Let us call this type of 1/f γ fluctuation equilibrium flicker noise or equilibrium 1/f noise. It is characterized by a = 2 and, as a rule, γ ≈ 1.

Conducting films and contacts also display another type of noise, with a 1/f γ spectral density, if subjected to electric current, ionizing irradiation, deformation, etc. This is termed nonequilibrium flicker noise or nonequilibrium 1/f γ noise. Its power spectral density may vary as I2 , and γ is 2 or larger. An in-depth analysis of this type of noise was given by Zhigal’skii [9]. Equilibrium and nonequilibrium flicker noise can coexist.

Nonequilibrium flicker noise is more informative than equilibrium flicker noise because it can give clues about lattice disturbances and is very sensitive to latent defects. In many cases, it is the only source of such information. As an example, the noise can be useful for predicting the electromigration immunity of thin-film interconnections [4, 10, 11].

This section examines major types and mechanisms of equilibrium and nonequilibrium 1/f noise in metal films and contacts.

2A. Equilibrium 1/f Noise Due to Phonon Scattering

For equilibrium 1/f noise in metal films, Hooge and Hoppenbrouwers [12] proposed this empirical formula

where V is the voltage across the specimen, R is the specimen resistance, SV( f ) and SR( f ) are the respective power spectral densities of fluctuations in V and R, αH = 2 × 10–3 is the Hooge constant, and N0 is the total number of carriers in the specimen. Furthermore,

where nc is the carrier density and Veff is the effective volume of the specimen.

Equation (1) applies to structurally uniform films with a low density of mobile and fixed defects.

The noise described by Eq. (1) is commonly attributed to fluctuations in carrier mobility due to phonon scattering, whether or not a current flows through the specimen. In general, the phonon-scattering mechanism of equilibrium 1/f noise is supported by the fact that the noise level falls with a decreasing contribution of phonon scattering to the overall specimen resistance, as observed in semiconductor films doped with impurities of high ionization potential [13]. Equation (1) can be modified to apply to such semiconductors by replacing αH with

where µ is the mobility influenced by carrier scattering from phonons, lattice defects, impurity atoms, and the

surface [13, 14]. Phonon scattering and other factors can be characterized by their respective mobilities µlat and µdef such that

The mobility-fluctuation model presumes that of the two constituent mobilities only µlat fluctuates. Accordingly,

from which Eq. (3) follows [14]. Furthermore, αH = αlat = 2 × 10–3. The respective power spectral densities

Slat and S of fluctuations in µlat and µ obey the following relations [14]:

Hooge parameter αH was evaluated experimentally for many semiconductor materials and metals, the values lying in the range 10–8 < α < 10–1 [6, 13].

More specifically, the phonon-scattering mechanism is presumed to consist in fluctuations in the cross section of acoustic-phonon scattering. Experiments by Musha [15] have shown that the intensity of light scattered by acoustic phonons in quartz indeed displays fluctuations with a 1/f spectral density. The number of acoustic phonons per mode is seen as the quantity that fluctuates. However, no model that yields an analytical expression for the phonon noise yet exists. Note also that the equilibrium 1/f noise described by Eq. (1) is inherent in solids and solid-state devices, as are thermal and shot noise.

The phonon-scattering noise component varies slowly with temperature, mainly owing to µlat being a function of temperature [12].

When the noise power spectral density varies as V2 and γ is close to unity, the dimensionless parameter

is often used to evaluate the noise level [6]. The parameter enables one to compare the level of 1/f noise observed in an experiment with that of the noise to which Eq. (1) applies.

Phonon-scattering dissipation leads to a departure from Ohm’s law in metal films. If the sinusoidal current I(t) = I1 sin2πf1t is passed through a film, the resulting third voltage harmonic is given by

where RT is the thermal resistance; K = L/bh, with L, b, and h being the film length, width, and thickness,

respectively; ρ0 is the temperature-independent component of film resistivity; ρf(T) is the film resistivity as a function of absolute temperature T; and αf is the temperature coefficient of resistance for the film [6].

The equilibrium 1/f noise and nonlinear CVC associated with phonon scattering cannot indicate the presence of lattice defects and, thus, are unsuitable for nondestructive quality control.

2B. Equilibrium 1/f Noise Due to Vacancies

Zhigal’skii [6] developed a model of equilibrium 1/f noise that relates fluctuations in metal-film conductance or contact resistance to fluctuations in the number of quasi-equilibrium vacancies on the grounds that a relatively small amount of energy is required for their creation and migration. Again the mechanism works whether or not a current flows through the specimen.

Metal thin films tend to be structurally inhomogeneous. They contain different types of vacancy sources and sinks, which are distributed nonuniformly over the film. Vacancies arise and disappear under the action of lattice thermal vibrations. The energy of vacancy creation is specific to the type of vacancy source, because the number of bonds that must be broken for a vacancy to arise depends on where the atom is located [16]. The characteristic positions are grain boundaries, dislocations, asperities, pore walls, etc.

In reality, the lattice of a metal film is not at thermal equilibrium. Defects present in the film increase its free energy and act as vacancy sources and sinks. In doing so, they can change their position or size with time; for example, dislocations can migrate and pores can grow. The density of microscopic defects decreases under annealing. Furnace annealing can bring defects to a quasi-equilibrium state in which they are stationary or almost so. Every vacancy source can therefore be characterized by its quasi-equilibrium vacancy density [16]. At quasi-equilibrium, the 1/f noise due to fluctuations in the number of vacancies can be regarded as almost time-invariant. This is referred to as quasi-equilibrium 1/f noise.

A change in external conditions (temperature, current, mechanical stress, etc.) can drive the film and its 1/f noise away from equilibrium [9].

At thermal equilibrium, the respective rates of vacancy creation and annihilation are equal, so the mean vacancy density is time-independent. Vacancy lifetime τvac in a film is given by

where Lvac is the distance between vacancy sinks and Dvac is the vacancy diffusion coefficient [16].

The set of the vacancy lifetimes is related to the distribution of the activation energy for vacancy diffu-

sion [16, 17] and to the distribution of Dvac . The activation energy and Lvac are subject to random variation over the bulk of the film. The random nature of the diffusion activation energy stems from the multiplicity of structural imperfections. In practicable metal films, the spatial variation of vacancy-sink density produces a wide variety of relaxation times, so that the resulting 1/f noise should occupy a wide frequency range [6, 18]. At quasi-equilibrium, vacancy creation is sustained by the internal energy of the crystal; this process is of a statistical nature. As vacancy creation involves defects with widely differing activation energies, the wideband character of the noise can be explained by the simultaneous action of many relaxation stochastic processes [19, 20].

The review by Zhigal’skii [6] demonstrates the existence of a wide range of relevant activation energies in Mo films. It also sets out physical reasons for the wide ranges of activation energies for creation and annihilation of quasi-equilibrium vacancies in metal films. Reasons are also given for widely differing relaxation times. As a result, 1/f noise in the range ≈4 × 10–9 to ≈1010 Hz is explained.

Potemkin et al. [18] found that the noise power spectral density can be written as

Svac0 = KU20∆n2vac0/ f = KU02nvac0/Na f , V2/Hz. (9a)

Here, K is a constant of the metal film and nvac0 is defined as nvac0 = Nvac0/N‡ , where Nvac0 is the tempera- ture-dependent number of quasi-equilibrium vacancies

and N‡ is the number of atoms in the film.

Equation (9a) applies to the noise produced by the simultaneous action of relaxation processes relating to creation and annihilation of quasi-equilibrium vacancies [9, 18]. It is derived by the well-known summation technique for relaxation processes differing in charac-

teristic time [19, 20] and assumes that ( ∆ Nvac0 )2 =

Nvac0 [5]. Consequently, ∆n2vac0 = nvac0/N‡.

According to Ohm’s law the fall in voltage in a film is expressed as U0 = RI0 , where I0 is a direct current and R is the resistance of the film. We substitute U0 = RI0 into Eq. (9a) to obtain

where the preexponential factor A is related to entropy [17] and Evac is the activation energy of vacancy creation. For films, A can be much greater than unity [6].

The vacancy model is supported by a considerable amount of experimental data on the effects of different factors on noise level; these factors include annealing, aging, internal stress, and temperature [6]. An important piece of evidence is a connection between noise level and the third-harmonic distortion, which was found by both measurement [6] and calculation [22, 23].

Equations (9a), (9b), and (10) show that the noise spectral density obeys the Arrhenius equation, in which the activation energy Evac corresponds to vacancy creation at grain boundaries, micropore walls, and other lattice defects. In many experiments, Evac has been found to be equal to the work required for breaking one or two bonds [6].

Dissipation due to quasi-equilibrium vacancy scattering produces the third-harmonic distortion. If the sinusoidal current I(t) = I1 sin2πf1t is passed through a film, the resulting third harmonic of voltage has the amplitude

U3 = 0.25RT K2ρ01ρf( T )

(11)

× [ αf + ( AA1 Evac/kT20ρ01 ) exp( Evac/kT0 ) ] I31,

where A1 is a constant [22, 23].

The vacancy-induced noise and the voltage–current nonlinearity can be useful for nondestructive detection of defects in and damage to metal films and contacts.

2C. Nonequilibrium 1/f Noise in Metal Films and Contacts

If direct current I0 is passed through a practicable one-port passive network, circuit resistance R will vary in a random manner with time and in a nonlinear manner with current. Let us express the dependence as

where R0 is independent of I0 and N(I0, t) = 0 when I0 = 0 [24].

The nonlinear term R(I0, t) may be approximated in different ways. With a polynomial approximation containing k + 1 terms, Eq. (12) becomes

n = 1

where Rn are independent of n (n = 1, 2, …, k) [25, 26].

The individual terms of the polynomial represent different mechanisms of carrier transport and scattering. These mechanisms include scattering from phonons and mobile vacancies, which are responsible for the third-order term [6]. The classical and tunnel transport across grain boundaries or contacts are also relevant [1, 27, 28].

It is easily seen that fluctuations in R0 correspond to equilibrium noise (i.e., the one independent of current), whereas fluctuations in Rn (n = 1, 2, 3, …) represent the nonequilibrium noise induced by current I0 .

For industrial applications, it may be advantageous to employ a simpler approximation:

where Z and β are constants that characterize the nonlinear CVC [24].

With both approximations, the relationship between voltage V across the circuit and current I through it can be written as

where N(I, t) is a nonlinear function of I. Note that I consists of a dc and an ac component in the general case.

The magnitude of the nonlinearity may be measured

where N(I) and R0 are obtained by averaging N(I, t) and R0(t), respectively, over time.

As in [24], the nonlinearity will be regarded low if

If N(I, t) and R0(t) are uncorrelated, the power spectral density SV( f ) of voltage flicker noise for a given I0 is in general expanded as

where SRL( f ) and SRN(I0, f ) are the power spectral densities relating to the linear and nonlinear resistance components, respectively. Accordingly, the first and second right-hand summands of Eq. (18) correspond to equilibrium and nonequilibrium resistance fluctuations, respectively.

Under dc conditions the two spectral densities cannot be measured individually, which prevents us from extracting the nonequilibrium noise component. At low I0 , nonequilibrium 1/f noise tends to be buried under equilibrium noise (this is often the case with high-qual- ity metal films when the current density is less than 106 A/cm2).

As metal films and ohmic contacts do not deviate greatly from Ohm’s law, Rn should decrease rapidly with an increase in n; furthermore, we can assume that

so that

∆V ( t) = ∆ R0( t) I0 + ∆ R1( t) I20 + ∆ R2( t) I30, (20)

where ∆V(t) = V(t) – V and ∆Rn(t) = Rn(t) – Rn (n = 0,

1, 2) represent fluctuations in V and Rn from their respective mean values.

If R0(t), R1(t), and R2(t) fluctuate in an uncorrelated manner,

SV ( I0, f ) = SR0( f ) I20 + SR1( f ) I40 + SR2( f ) I60, (21)

where SR0, SR1, and SR2 are the respective power spectral densities of the fluctuations in R0, R1, and R2 (see Eq. (19)).

We see that under nonequilibrium conditions the noise spectral density as a function of current deviates from a square-law relationship, especially at high currents. At low currents one can observe only equilibrium flicker noise corresponding to the first right-hand term of Eq. (21).

Conducting films and contacts display different types of nonequilibrium flicker noise with specific mechanisms [9]. The noise may or may not vary with time, depending on external conditions and the microstructure of the film. Below we shall briefly discuss experimental results concerning nonequilibrium flicker noises in metal films and contacts; more details can be found in [9].

Some types of nonequilibrium flicker noise have γ ≥ 2, whereas other types have γ close to unity. In metal films, current-induced flicker noise can be produced

(i) by vacancy creation due to dissipation throughout the film or at local inhomogeneities or (ii) by migration of metal atoms along grain boundaries or over the whole lattice. The first type of noise does not change with time, whereas the second type is time-dependent. This is known as electromigration noise.

With oxide or nitride inclusions at grain boundaries or contacts, a third type of nonequilibrium flicker noise may arise owing to nonmetallic conduction mechanisms, although these contribute insignificantly to the net resistance. The noise is time-independent. This is observed in Mo films with oxidized grain boundaries, which have been deposited at a low rate [25, 29].

Nonequilibrium 1/f γ noise also occurs in films whose structure is not at thermal equilibrium; furthermore, it can be produced by ionizing irradiation or deformation [9].

Let us now consider the case of vacancy creation by dissipation [9]. As the current is increased, the film temperature and the number of nonequilibrium vacancies increases. Once this exceeds the equilibrium number of vacancies for room temperature, as given by Eq. (10), nonequilibrium flicker noise will become stronger than equilibrium flicker noise [9, 26], so that the power spec-

Fig. 3. Power spectral density of current-induced 1/f noise vs. direct current for a Cr film having an elevated vacancy density, the film being 160 nm thick and 0.37 mm wide [26]. Curves 1 and 2 are obtained at 2 and 5 Hz, respectively.

Fig. 4. Power spectral density of current-induced 1/f noise vs. direct current for Ni–Cr films with R = 500 Ω at T = 300 K [4]. Curves 1 and 2 correspond to n = 2 and 4.5, respectively.