# The Influence of AlGaN/GaN Heteroepitaxial Structure Fractal Geometry on Size Effects in Microwave Characteristics of AlGaN/GaN HEMTs

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{T}.

## 1. Introduction

_{k}

_{,i}, measurement of manufactured transistors’ HF characteristics H

_{21}(f), G

_{max}(f), f

_{T}, and f

_{max}, extraction of the parameters P

_{k}

_{,i}of their linear models and a verification and physical interpretation of the results obtained. The listed actions are taken in the context of the application in semiconductor materials science and exemplified by heterostructural AlGaN/GaN field-effect transistors with high electron mobility (hereinafter referred to as HEMTs).

## 2. The Basic Concepts of Fractal Geometry in the Context of Its Application for the Description of AlGaN/GaN HEMT Structures

_{k}

_{,i}of the nano- and microstates of the objects listed above often have the properties of chaotic systems [8,9,10,11,12] including: (1) a strong dependence of the system state on initial condition (small changes of environmental condition lead to significant changes of the state); (2) relativity and indefiniteness of the measurement processes, i.e., dependence upon the choice of reference coordinates and the measurement scale; (3) existence of topological mixing properties—when components do not overlap each other [7]. As reported in [4], all of these properties can be used to improve the characteristics of the developed semiconductor devices.

#### 2.1. Using Linear Functional Spaces for a Description of Semiconductor Object Electrophysical Characteristics

_{T}is the limit of local approximation L—the value of linear scale. When the measurement scale l of any linear size of a semiconductor object goes beyond the local approximation limit L (i.e., where l > L), the Hausdorff–Bezikovich dimension D

_{H}of this object becomes equal to its topological dimension: D

_{H}= D

_{T}. This is an applicable scope of Euclidian geometry. When the linear size l of the object is less than or equal to L (l ≤ L), the Hausdorff–Bezikovich dimension D

_{H}of the object becomes less than its topological dimension D

_{T}: D

_{H}< D

_{T}. In this scope the D

_{H}value can be fractional.

_{i}}, in which the distance ρ between any pair of elements is defined with the triangle, symmetry and zero distance axioms [17,19]. In this case, a single-valued, non-negative real function ρ = ρ(r) must be defined for any r (r is a radius-vector) from the X set.

_{k}

_{,i}. Unlike metric spaces R, the measures of functional spaces (functionals Φ) can also have negative values, e.g., the measures of electrical charge, resistance, current density, etc. Specifically, integrals of electrical charge, current or mass density functions can be considered as the measures revealing the capacitive, conductive and mass-dimensional properties of line wire, two-dimensional surfaces or three-dimensional objects [20,21,22,23].

_{k}(x,y,z)—functional Φ

_{k}—as the measure in the functional space M

_{k}= Φ

_{k}:

_{k}

_{,i}from the linear functional space M

_{k}= {Φ

_{k}

_{,i}}—the set of functions describing measures Φ

_{k}

_{,I}= M

_{k}

_{,i}of the k

^{th}additive (integral) physical characteristic of the semiconductor object, where k = 1, 2, … K is a number of such measure, characterizing for example, M

_{1,I}= R

_{gs}

_{,i}, M

_{2}= R

_{ds}

_{,i}, M

_{3}= C

_{gs}

_{,i}, M

_{4}= C

_{gd}

_{,i}, M

_{5}= C

_{ds}

_{,i}, M

_{6}= g

_{max}

_{,i}, and many others, and I = 1, 2, … , … N is a number of the element with the k

^{th}measure. At this, all arguments of the functions F

_{k}

_{,i}from the linear functional spaces M

_{k}must belong to the metric space set R.

_{k}

_{,i}(functional) from the functional space, describing through the arguments belonging to the metric space R some additive (integral) electrophysical characteristic.

_{k}forms the so-called “metric space with a measure” S

_{M}

_{,k}= {R, M

_{k}}, that is essentially an assembly of the elements of this metric space R, on which some measure set M

_{k}

_{,i}from the linear functional space M

_{k}[12,17] depends (through the arguments). In this case the dimension of both metric spaces R and functional spaces M

_{k}can be represented by any one of the real numbers from 0 to 3.

_{k}

_{,i}} belongs to functional spaces. Thus, we obtain the metric space with a measure S

_{M,k}= {R, M

_{k}} that is essentially an assembly of elements of the transistor metric space R, to which parts some set M

_{k}of measures M

_{k,i}of the transistor electrophysical and device characteristics is assigned [12].

#### 2.2. Measures of Semiconductor Objects’ Physical Parameters

^{D}, can be filled with some measuring (gauge) object described by the function $l(\delta )=\gamma ({D}_{{}_{T}}^{}){\delta}_{}^{{D}_{M}^{}}$, where γ is a normalizing index, δ is a dimensionless scale, D

_{M}is a Minkovsky dimension (the dimension of the limited set in metric space). Then, according to [7], $N=1/{\delta}_{}^{{D}_{M}^{}}$. As in the given paper we are going to work only with a subset of metric spaces and functional subspaces defined on them, let us use the Hausdorff–Bezikovich dimension D

_{H}(hereinafter referred to as the Hausdorff dimension) instead of the Minkovsky dimension D

_{M}, because these notions are close enough D

_{H}≈ D

_{M}[7]. Therefore, the dimension D

_{H}of a limited set in metric space can be represented in the following way:

_{H}relates to the fractal dimension D

_{f}as D

_{f}= D

_{T}+ D

_{H}, where D

_{T}= 1, 2, 3 is an integer topological dimension.

_{S}= D

_{H}.

_{0}of some semiconductor object under investigation is taken as a normalizing index γ(D) in Equation (4), the expression can be written in the following way:

_{f}represented through the object measure M:

_{f}, scaling indexes ζ and η, the law of affine transformation x

_{i}= Ax

_{i−1}[7]. Recall that at l ≥ L, D

_{f}= D

_{T}.

_{f}= D

_{f}(l)—the so-called multifractal object. In such cases it is often convenient to use an average value of fractal dimension <D

_{f}> which is obtained by means of averaging out fractal dimensions determined for different surface sections [24].

_{f}:

_{i+}

_{1}of the following similarity level:

_{T}= D

_{f}, the relation of measures M follows the standard law:

_{f}= 2.36, leads to the change of its measure M

_{d}(e.g., surface electric charge), not in ${3}_{}^{{D}_{T}^{}}={3}_{}^{2}=9$ times, but much less, to be exact, just in 3

^{4−Df}= 3

^{4−2.36}= 3

^{1.64}≈ 6.06 times (7). Simultaneously, a relative value of some variable M

_{d}(d), say electrical charge density σ

_{d}= M

_{d}(d), is fully defined by the linear size d of the section under measurement:

_{0}is a linear size of the projection onto the (x,y) plane.

_{f}= D

_{T}, the expression (1) is transformed into the conventional, invariant with d, expression:

_{0}must be represented by the highest or the lowest value of the functional at the functional space under investigation, corresponding to the smallest l = l

_{0}or to the largest l = l

_{max}value of our measurement scale. Assuming η = Φ

_{k}

_{,i}/Φ

_{k}

_{,0}and ζ = l

_{i}/l

_{0}, from Equation (3) it is possible to define the values D

_{S}of functional spaces, e.g., layer resistances [12].

_{M,k}= {R, M

_{k}}, defined depending on n and d in some area of metric space R = {X, ρ(n, d)} of these transistors’ structural elements. The SS-model in a form of equivalent circuit (EC) is essentially an assembly of k = 1…K interacting equivalent elements P

_{k,i}[25,26] belonging to some S

_{M,k}. The used field-effect transistor EC rather accurately describes the behavior of HEMT small-signal S-parameters in a frequency range up to 60 GHz [27].

## 3. The Connection between the Electrophysical Parameters of the AlGaN/GaN HEMT Heteroepitaxial Structure and Its Geometry

_{30}Ga

_{70}N spacer is 4 nm; the thicknesses of the high-alloy (N

_{D}> 10

^{18}cm

^{−3}) Al

_{x}Ga

_{1−x}N:Si n

^{+}layer and encapsulating unalloyed GaN layer are 14 nm and 2 nm, correspondingly. Electron density of two-dimension (2D) electron gas N

_{S}= 1.36 × 10

^{13}cm

^{−2}, mobility μ (at T = 300 K) = 1200 cm

^{2}/(V × s), and the dislocation density is ~10

^{8}cm

^{−2}[22,28].

_{ds}has been formed on one part of the structure. The elements have the form of active rectangular areas with thickness d insulated from each other by means of reactive ion etching.

_{ds}between any two of them (Figure 1a), formed by means of explosive lithography, TiAlNiAu deposition and subsequent annealing in nitrogen atmosphere at the temperature of 780 °C during 30 s.

_{S}definition by calculating the number of closed contours [7] encircling surface irregularities in the (x,y) plane, or lateral irregularities of electrostatic potential. After the tracing we calculated the number N

_{0}of closed contours obtained at the given measurement scale l

_{0}. At the next stage, scale l

_{i}was decreased to obtain an integer number of closed contours N

_{i}> N

_{0}, and so on, until l

_{i}was equal to the minimum distance between pixels. With the use of expression (3), taking into account that η = N

_{i}

_{−1}/N

_{i}and ζ = l

_{i}

_{−1}/l

_{i}, we defined the metric space similarity dimension D

_{S}for the surface under investigation. A graphical representation of the closed contour number N

_{i}vs. l

_{i}is called “freaking stairs”, due to different height and width of its steps (Figure 2a,b).

_{k}

_{,i}and estimation of HEMT HES surface relief shape h(x,y) (Figure 3a) were conducted with the use of the Solver-HV atomic force microscope (AFM)—made by NT-MDT (Zelenograd, Moscow)—in semi-contact mode [29]. The contact-potential difference (CPD) Δφ(x,y) between cantilever needle point and HEMT HES (Figure 3b) was measured by means of the AFM Kelvin probe technique in semi-contact mode with the use of the following expression:

_{p}was a surface potential of the cantilever needle point plating material (p for probe), and φ

_{S}was a potential of the surface (s for surface) under investigation. Therefore, being aware of the work function of the cantilever needle point (qφ

_{p}), from the expression (13) it was possible to find out the distribution of potential φ

_{S}(x,y) (or work function qφ

_{S}) throughout the surface area under investigation. AFM measurements were conducted in air in semi-contact mode with the use of the a double-pass technique with the resolution of 65,536 pixels (NN = 256 points in vertical and horizontal sweeps). The maximum size of the AFM scan area was equal to 100 × 100 µm

^{2}. In order to provide semi-contact mode, NSG10 cantilevers were used, which needles had tungsten carbide W

_{2}C plating with electron work function of ≈4.9 eV. The instrument influence of the other structural elements of the cantilever at the contact point was not more than 25%.

_{i}= a

_{i}/NN < L through the reduction of the studied area side length a leads to a monotonous decrease of the surface irregularities and roughness average values <h> and <R

_{a}>, correspondingly, in a row: for the area of 100 × 100 µm (l = 390 nm) <h> = 15.41 nm and <R

_{a}> = 3.24 nm, for the area of 50 × 50 µm (l = 195 nm) <h> = 8.47 nm and <R

_{a}> = 1.99 nm, for the area of 30 × 30 µm (l = 117 nm) <h> = 6.59 nm and <R

_{a}> = 1.32 nm, for the area of 10 × 10 µm (l = 39 nm) <h> = 1.84 nm and <R

_{a}> = 0.21 nm and for the area of 5 × 5 µm (l = 19.5 nm) <h> = 0.65 nm and <R

_{a}> = 0.11 nm (Figure 3a(III)).

_{a}> grow: at N = 1256 (l = 50/1024 = 0.049 µm = 49 nm) <h> = 15.91 nm and <R

_{a}> = 3.51 nm, at N = 512 (l = 98 nm) <h> = 9.13 nm and <R

_{a}> = 1.59 nm, at N = 256 (l = 195 nm) <h> = 6.89 nm and <R

_{a}> = 1.44 nm, at N = 128 (l = 391 nm) <h> = 2.73 nm and <R

_{a}> = 0.51 nm, and at N = 64 (l = 781 nm) <h> = 2.68 nm and <R

_{a}> = 0.31 nm. These data reveal indefiniteness and relativity of the AlGaN/GaN HES surface relief estimation results (e.g., <h> and <R

_{a}>) in the local limit at l < L, and this is one of the basic attributes of chaotic systems [6,7].

_{i}/l

_{i}

_{+1}-fold decrease of the measurement scale l we obtain a discrete sequence of the embedded sets of non-overlapping contours (Figure 3a(II)) with a step-like increase of their number N(l)—“freaking stairs” (Figure 2a). Figure 3a(II) shows that shapes of the embedded lateral contours of relief irregularities are not self-similar in a literal sense. It is only possible to speak about some statistical self-similarity. The existence of the self-similarity properties of the HES surface area under investigation is revealed by the linear dependence of ln(N(l)) on ln(l

_{0}/l) (Figure 2c), which makes it possible to consider the observed objects as fractal, or (at the presence of several kinks in linear dependence of ln(N(l)) on ln(l

_{0}/l)) multifractal ones. These results are indicative of the topological mixing properties of the HES under investigation that is a third attribute of chaotic systems.

_{3}= 50 µm, l

_{2}= 100 µm, l

_{1}= 150 µm (l

_{3}< l

_{2}< l

_{1}) reveal classic dependence: as the measurement scale l decreases, dispersion decreases too (Figure 3a(IV)). In global approximation, l is defined by a step of an AFM objective table along the axes x and y.

_{0}/l) (Figure 2d). As Figure 2b shows, the “freaking stairs” can be well approximated with the exponent.

_{ds}was conducted on the probing station M-150 (Cascade Microtech, Beaverton, Oregon, USA) with the use of the B-1500 semiconductor device parameter analyzer (Keysight Technologies, Santa Rosa, California, USA). To exclude serial resistance of the probe-to-surface contact, the four probes Kelvin technique was used. The resistance measurement error in the bias range from −5 to +5 V was not more than 0.04 Ω.

_{ds}> 60 µm, sheet resistance (resistance per square) ρ

_{▯}= (d × R

_{ds})/l

_{ds}≈ Const (Figure 1b) within the scope of the admissible values, and channel resistance R

_{ds}and drain saturation current I

_{ds}have virtually linear dependence on d and l

_{ds}. If d and l

_{ds}are equal to or less than given values (d ≤ 30 µm, l

_{ds}≤ 60 µm), the value ρ

_{▯}= ρ

_{▯}(d, l

_{ds}) becomes a function of d and l

_{ds}(Figure 1b), and there is nonlinear dependence of values R

_{ds}= R

_{ds}(d, l

_{ds}) and I

_{ds}= I

_{ds}(d, l

_{ds}). As the result, it is possible to conclude that the given HES sheet resistance ρ

_{▯}= ρ

_{▯}(d, l

_{ds}) depends largely on channel length l

_{ds}and width d in the range of practically valuable measurement scales (linear sizes)—that is size effect.

_{ds}, and consequently to the growth of specific values R

_{sg}and R

_{dg}.

_{D+}, which concentration in the material is equal to 2 × 10

^{18}cm

^{−3}. In order to study the geometry of the irregularity distribution N

_{D}

_{+}(x,y) by the Kelvin method with the use of AFM, the geometry of the CPD irregularity distribution Δφ(x,y) was investigated. It is possible to estimate that at an average level of the doping of the AlGaN epitaxial layer N

_{D}

_{+}= 2 × 10

^{18}cm

^{−3}, the Debye shielding distance (space charge region, SCR) is equal to 17–20 nm that is comparable with the HES thickness. This means that SCR irregularities born by the unequal (fractal) distribution of the ionized shallow-level donor impurity N

_{D}

_{+}come to the surface, forming there surface potential lateral irregularities distributed in a fractal way. It follows that positive ions of shallow-level donor impurity are also distributed in this fractal way throughout the area of the two-dimensional electron gas. At this, as it is shown in Figure 3b, the lateral sizes of such irregularities can exceed 50 µm (limited by the sizes of the area under investigation). The investigations of the 100 × 100 µm surface areas reveal that in this case L can be significantly more than 100 µm. The more precise definition of L is restricted by the technical capabilities of the used AFM. It is clear that the fractal distribution of the ionized shallow-level donor impurity forming heteroepitaxial level capacity and consisting of the scattering centers has great influence virtually on all parameters P

_{k,i}of the internal transistors of the EC linear model.

## 4. AlGaN/GaN HEMT Manufacturing

_{ds}(the distance between the drain and the source) was equal to 5 µm. After that the whole structure was covered with SiO

_{2}dielectric film of the first mask having the thickness of 0.15 µm, and then with the resist. The openings were formed in the resist mask by means of lithography methods. The edges situated closer to the drain corresponded to the position of the Schottky gates of the fabricated transistors. Then, according to the resist mask, openings in the first dielectric mask were formed by means of chemical etching. The etching time was estimated by vision—until the total elimination of dielectric layer from the surface of the structure. After the resist elimination, the second lithography was conducted to create openings, surrounding the edges of the dielectric mask corresponding to the gate positions. The resist mask edge was situated closer to the drain located on the dielectric surface, and the second one was located on the surface of the channel active area. As a result, there was a gap with the length of ~0.15 µm on the HES surface between the dielectric and the resist masks. Therefore, the opening length in the resist mask defined the size of the gate cap, and the offset of the mask relative to the dielectric edge defined the gate length l

_{g}. After NiAu deposition and explosion we obtained the L-shaped “Field-plate” gate with the length l

_{g}= 0.15 µm, which cap was shifted to the drain (Figure 4b). At final stage, the glassivation of the structure active area was performed using an SiO

_{2}dielectric. Then, the openings in this dielectric over the pads were created in order to connect the sources of all transistor sections by air bridges and to perform the galvanic thickening of the drain, source and gate pads.

_{g}= 150 nm was formed. HEMTs differ from each other, not only by section number n = 1, 2, 4, 8 and 12, but also by channel width d

_{ds}= 50, 80, 100 and 150 µm (Figure 4a).

## 5. The Methods of Measurement and Extraction of AlGaN/GaN HEMT Linear Models

_{k,i}) of the equivalent circuit components P

_{k,i}of the SS model for all transistors in the matrix was performed.

_{ds}= 20 ± 1 V. The insignificant spread of U

_{ds}is due to the optimization of current operating points to achieve maximum values of H

_{21}and G

_{max}. The measurement was carried out with the use of the PNA-X N5245A 2-port vector network analyzer (Keysight Technologies, Santa Rosa, California, USA) at the Summit 12,000 semi-automatic probe station (Cascade Microtech, Beaverton, Oregon, USA). The calibration of measuring paths of the ports 1 and 2 of the vector network analyzer was performed by the Line-Reflect-Reflect-Match (LRRM) method [35] with measuring S-parameters of the gage structures: (1) microstrip line, per passing through (Thru); (2) 50 Ω load (Load); (3) open-circuit test element (Open); and (4) short-circuit test element (Short).

_{max}(f) (Figure 5a) and current H

_{21}(f) (Figure 5b) amplification factors of the manufactured HEMTs point out that in average they are compliant with the world analogs.

_{k,i}of the SS model EC components by the measured S-parameters was carried out in accordance with the combined technique [40,41] incorporating advantages of analytic [26] and optimization [39] methods of the extraction.

_{g}, L

_{s}, L

_{d}, R

_{g}, R

_{s}and R

_{d}of the transistor equivalent circuit (Figure 6).

_{gs}is gate-source resistance; R

_{ds}is drain-source resistance of the depletion region; C

_{gs}is gate-source capacity; C

_{gd}is gate-drain capacity; C

_{ds}is drain-source capacity; I

_{ds}is drain current source controlled by the gate-source voltage; g

_{max}is transconductance; τ is a time constant of the frequency dependence of the current source) and external parts of the field-effect transistor small signal model EC.

_{k}of the parameters P

_{k}obtained after extraction of the internal transistor EC components on the number of sections n and their width d, was carried out using the notion of Hausdorff–Bezikovich non-integral dimension D

_{H}(fractal D

_{f}) [7] (see Section 2).

_{21}at short-circuit termination, maximum power amplification factor G

_{max}≡ MUG (maximum unilateral gain) in the absence of the feed-back inside the transistor, current-amplification cutoff frequency f

_{T}, and power-amplification cutoff frequency f

_{max}.

_{21}and G

_{max}are defined through the parameters of scattering matrix (S-matrix) using the following known expressions:

_{T}is specified by the time of electron transmission through the distance between source and drain l

_{ds}with the group saturation velocity v

_{sat}:

_{max}characterizes the maximum oscillation frequency up to which the transistor acts as the active component. At the extrapolation of this frequency the transistor begins to act as the passive one.

_{max}and f

_{T}are determined from the following clause: amplification factors [G

_{max}] = dB and [H

_{21}] = dB are equal to the one.

_{max}can be both higher and lower than f

_{T}, which is defined by transistor structure peculiarities in every individual case.

## 6. Experimental Results (Size Effects in AlGaN/GaN HEMT HF Characteristics)

_{21}, G

_{max}, f

_{T}, and f

_{max}do not explicitly depend upon n and d, as these expressions describe the behavioral structureless model of the field-effect transistor.

_{max}(f) (Figure 5a) and H

_{21}(f) (Figure 5b), as well as f

_{T}and f

_{max}(Table 1), have non-monotonous dependence on the number n and the width d of the transistor sections. The comparison of the measured values of current- and power-amplification cutoff frequencies f

_{T}and f

_{max}for all n and d shows that they cannot be uniquely described by the known expressions (17) and (18).

_{max}and H

_{21}at the frequency f = 10 GHz was carried out for all test structures. The experimental dependences ${H}_{21}^{}={H}_{21}^{}(n,d)|{}_{f=10\begin{array}{cc}& GHz\end{array}}^{}$(Figure 5c) and ${G}_{\mathit{max}}^{}={G}_{\mathit{max}}^{}(n,d)|{}_{f=10\begin{array}{cc}& GHz\end{array}}^{}$(Figure 5d) are non-monotonous and have maxima over both n and d. Thus, for G

_{max}the maximum is observed at d = 80–100 µm and n = 2, and for H

_{21}at n = 4 at the same values of d, which agrees with the results stated in [34].

_{21}(f) and G

_{max}(f), and correspondingly, f

_{T}and f

_{max}, on n and d, let us compare the internal transistor linear model parameter values P

_{k}

_{,i}reconstructed in accordance with the technique described above. In the general case, the increase of n and d leads to the proportional (linear) increase of the total width of the transistor channel W = n × d. In the context of transistor EC, it is identical to a parallel connection of elementary resistances and capacitances in the internal transistor model. In accordance with Ohm’s law, it should result in a linear change of the resulting EC resistances and capacitances, i.e., the measures M

_{k}of transistor parameter values P

_{k}

_{,i}. However, as it is shown below, in real transistors this situation takes places only in particular cases.

_{k}

_{,i}) of internal transistor EC components, grouped by a common feature—the width of the individual section d of the transistor. Thus, data selection 1-1 includes the transistors with d = 50 µm (n = 4, 8 and 12), data selection 1-2—the transistors with d = 80 µm (n = 2, 4, 8 and 12), data selection 1-3—the transistors with d = 100 µm (n = 2, 4, 8 and 12), and data selection 1-4—the transistors with d = 150 µm (n = 2, 4, 8 and 12).

_{k}

_{,i}

^{*}= P

_{k,i}/W

_{i}on section number n: ${P}_{k,i}^{*}={P}_{k,i}^{*}(n)$. In each selection the increase of n leads to a slight decrease of specific capacity C/W and transconductance g

_{m}/W and to a significant decrease of specific resistance R/W. The dimensions D

_{H}(C

_{ds}) and D

_{H}(C

_{gd}) show the same behavior—their values decrease as n increases. Vice versa, the values D

_{H}(C

_{gs}) and D

_{H}(R

_{ds}) increase as n increases. With the n growth the value D

_{H}(R

_{gs}) increases for the selection m

_{1}, decreases for the selection m

_{2}, increases for the selection m

_{3}and decreases again for the selection m

_{4}. The comparability of the dimension values D

_{H}(C

_{gs}) with D

_{H}(R

_{ds}), as well as D

_{H}(C

_{gd}) with D

_{H}(R

_{gs}) and D

_{H}(g

_{max}) means that the physical mechanisms of the operation of internal transistor parameters C

_{gs}and R

_{ds}, as well as C

_{gd}, R

_{gs}and g

_{max}are defined by similar geometric dependences.

^{*}

_{m}

_{,max}in data selection 2-1, leads to the growth of g

^{*}

_{m}

_{,max}values in data selections 2-2 and 2-3, results in slight change of g

^{*}

_{m}

_{,max}in data selection 2-4. The dimensions D

_{H}(C

_{gd}) and D

_{H}(C

_{ds}) have almost the equal values slightly changing at the change of d. The values D

_{H}(C

_{gs}) change inappreciably for data selections 2-1 and 2-2, and grow with d growth for data selections 2-3 and 2-4. The D

_{H}(g

_{m}) values decrease as d increase.

_{k}

_{,i}of the internal transistors with the same W must be equal or at least close to each other. However, as it follows from the Table 4, for the selections 3-1, 3-3 and 3-4 (W = 200, 600 and 1200 µm) the decrease of d leads to the monotonous increase of R

_{ds}. For the selection 3-2 (W = 400 µm) the decrease of d results in a decrease of R

_{ds}. In the general case, changing R

_{ds}can exceed tens of percent.

_{i}≡ R

_{gs}in most cases (for W = 200, 400 and 1200 µm, except W = 600 µm) decrease as d decreases. Here, the changing of R

_{gd}can be up to 40 percent.

_{gd}and C

_{ds}increase with the decrease of d at all values W = 200, 400, 600 and 1200 µm, and the value C

_{gs}have such behavior at W = 400, 600 and 1200 µm (except W = 200 µm).

_{max}, which corresponds to the results in [28,32].

## 7. Discussion of the Experimental Results

_{s}and R

_{d}of the external transistor EC (Table 2). With these corrections, according to [13], the expressions for f

_{T}and f

_{max}have the following form:

_{T}and f

_{max}for the manufactured transistors with the ones calculated according to the expressions (19) and (20) (Table 1) with the use of the external transistor EC parameters (parasitic parameters) R

_{s}, R

_{d}and R

_{g}(Table 2). It should be pointed out that external parasitic parameters are commonly believed to directly depend on linear sizes, so they cannot explain any non-linear effects observed.

_{max}= G

_{max}(n, d) and H

_{21}= H

_{21}(n, d), as well as the ones of cutoff frequencies f

_{T}= f

_{T}(n, d) and f

_{max}= f

_{max}(n, d) depending on n and d have no necessary physical explanation. In our opinion, the reason for it consists in a failure to take into account the relation of the internal transistor linear model parameters P

_{k}

_{,i}and non-linear—fractal in particular—parameters of HES electrophysical properties. It is possible to assume, that the reason of the divergences of the experimental and calculated results observed in [13,14,15,16,33] is associated with the fact that (19) and (20) do not take into account such important parameters P

_{k}

_{,i}of the internal transistor linear model as C

_{ds}and C

_{gd}, C

_{gs}, g

_{max}, R

_{gs}= R

_{i}and R

_{ds}, which in local approximation are totally defined by non-linear (fractal) HES electrophysical parameters p

_{k}

_{,i}depending on n and d.

_{k}

_{,i}and parameters P

_{k}

_{,i}do not immediately (directly) reveal size effects by themselves (they are not fractal). The fractality occurs only after taking into account the dependence of these functions upon spatial coordinates (x,y,z), which gives the opportunity to do a quantitative description of the observed size effects.

_{ds}and C

_{gd}, C

_{gs}, g

_{max}, R

_{gs}= R

_{i}and R

_{ds}of the parameters P

_{k}

_{,i}(Table 2) and consequently H

_{21}, G

_{max}, f

_{T}and f

_{max}(Table 1), become functions of the parameters n and d, which in accordance with that which is stated above makes it possible to operate with these values as with subspace elements S

_{M}.

_{M}formed by the measures M

_{k}

_{,i}of the parameters P

_{k}

_{,i}, i.e., in order to find out the dependences of the measures M

_{k}

_{,i}on n and d with the use of (8) and (9), it is necessary to determine Hausdorff dimensions D

_{H}of the subspaces M

_{k}. It is worth to remember that functional spaces M

_{k}defined at the metric space R have all its geometrical properties, which allow using the expression (8) to determine their D

_{H}.

_{k,i}of the parameters P

_{k,i}in the expression (8), which correspond to different total widths of the transistor channel W = n × d. For example, if the initial value M

_{0}in every such subspace S

_{M}(M

_{k}

_{,i}) of the transistor parameter values P

_{k}

_{,i}is equal to the parameter value P

_{k,}

_{0}corresponding to the maximum channel total width W

_{0}= 1800 µm, then η = P

_{k}

_{,i}(W

_{i})/P

_{k}

_{,0}(W

_{0}), and ζ = W

_{i}/W

_{0}. The dependences ln(P

_{k}

_{,i}/P

_{k}

_{,0}) on ln(W

_{0}/W

_{i}) written in log–log scale are well approximated by the straight lines (Figure 7), which in accordance with that stated above (see Section 3) point out the presence of self-similarity properties of subspaces S

_{M}. This make it possible to operate with the studied subspaces of the measures of the parameter P

_{k}

_{,i}as with the fractal or multifractal objects, and use the expression (8) to define their D

_{H}(Table 2). The values D

_{H}

_{(f)}of the resistances R

_{i}and R

_{ds}are calculated by their inverse values—conductivities G

_{i}= 1/R

_{i}and G

_{ds}= 1/R

_{ds}. Such substitution does not change the absolute values |D

_{f}|, but keeps its positive sign and dimensional subsequence.

_{M}(M

_{k}

_{,i}) of the selections of the parameters P

_{k}

_{,i}has its own value D

_{H}

_{(f)}in the defined ranges. This points out the multifractality of the subspaces S

_{M}(M

_{k}

_{,i}), characterized by several values D

_{H}

_{(f)}depending on linear sizes (in our case on n and d). For instance, for the selection 1-1, the range of the Hausdorff dimension D

_{H}values of the subspaces of capacities C

_{ds}and C

_{gd}measures is 1.79–1.86, of the subspace of capacity C

_{gs}measures it is 2.23–2.59, of the subspace of resistance R

_{ds}measures this is 2.15–2.30, and so on. As it is reported in Section 2.2 and below, in this case it is convenient to use average values of the dimensions <D

_{H}> for each selection (Table 2, Table 3 and Table 4).

_{H}> = 1.82, 1.87, 1.78 and 1.89 for C

_{ds}(Figure 7a) and C

_{gd}(Figure 7b) capacity parameters subspace—close to 2—mean that in spite of the linearity of the channel the law of change of their values is close to the square one. The proximity of the spatial dimension values <D

_{H}>(C

_{ds}) and <D

_{H}>(C

_{gd}) to the spatial dimension value D

_{H}= 1.85 of the lateral irregularities of the subsurface potential point out the connection between the geometry of the functional space of the used HES electrophysical characteristics p

_{k}

_{,i}and the device characteristics of the transistor linear model EC components. The fractal values <D

_{H}>(g

_{max}) and <D

_{H}>(R

_{i}) of parameter measure subspaces g

_{max}(Figure 7d) and R

_{i}(Figure 7e), which values are also close to <D

_{H}> = 1.87, 1.79 and 1.78, behave in the similar way. In such cases, the values <D

_{H}> close to 2 are probably defined by the assembly of one-dimensional transistor channels distributed in HES two-dimensional surface—the more n, the closer to two values D

_{H}.

_{gs}parameter subspaces (Figure 7c) exceeding 2—<D

_{H}>(C

_{gs}) = 2.42, 2.52, 2.37 and 2.39—point out the fact that their physical nature is defined not only by two-dimensional effects, but in some degree also by three-dimensional ones (Table 2). The resistive parameter measure subspaces behave the similar way—<D

_{H}>(R

_{ds}) = 2.23, 2.26 and 2.43 (Figure 7f). This means that their behavior is also strongly influenced by the third dimension. The D

_{H}values proximity to three can be explained by the fact that the R

_{ds}and C

_{gs}parameters are defined by flowing 2D-electrons of the HEMT structure channel, not through the 2-dimensional, but through the 3-dimensional (D

_{T}= 3) plane, which corresponds to the results represented in [12,30].

_{k,i}of the studied HES electrophysical characteristics in the local approximation have non-linear dependence on linear sizes l and d (e.g., Figure 1b). Subsequently, the parameter values P

_{k,i}= P

_{k,i}(n,d) of the internal transistor EC components defined by them have non-linear dependence on n and d with all that it implies.

_{k,i}= P

_{k,i}(W) on the way of W = n × d obtaining—the variant of comparison—leads to the fact that every such variant of comparison belongs to the individual subspace S

_{M}, each of which can be characterized by the own average value <D

_{H}> (Table 2, Table 3 and Table 4). The usability of one or another comparison variant depends on the individual case.

_{f(H)}(n,d) of the subspace S

_{M}and the known values of the measures M

_{k,}

_{0}of the parameters P

_{k,}

_{0}of the initial transistor, from (9) it is possible to reconstruct the values M

_{k}

_{,i}belonging to this subspace of the parameters P

_{k,i}of another transistor with some predefined n and d. For example, for the first comparison variant (d = Const):

_{f}is fractal dimension of the metric subspace with the measure of the parameter P

_{k,i}. The similar expressions can be written for other comparison variants too. In the expression (21) the relation of electrophysical parameters P

_{k,i}of the HES with the values (measures) of the EC small-signal model parameters P

_{k,i}is realized through the HES fractal dimension (in general case, Hausdorff dimension).

_{M}= {R, M

_{k}}) of their electrophysical characteristic measures, which in the general case is not obvious. At this, the similarity dimensions D

_{S}(R) and D

_{S}(S

_{M}) of subspaces R and S

_{M}have the values close enough. Probably, it can reveal the existence of some transformation

**Π**between HES metric space R and the defined on it metric space with the measure S

_{M}(p

_{k,i}):

^{*}which associates the metric space S

_{M}(P

_{k,i}) of linear (or non-linear) transistor model parameters P

_{k,i}with metric space S

_{M}(p

_{k,i}) of electrophysical parameters p

_{k,i}of the transistor’s materials and structural elements (HES structure, ohmic contacts, gates, passivating coatings, connections, etc.):

^{*}can be represented in the form of a physical or mathematical law. Thus, being aware of the spatial dimension of the semiconductor structure geometry and its additive electrophysical characteristics depending on it in local approximation, it is possible to reconstruct, for the individual n and d from the set R, the parameter values of the linear model internal transistor EC components through the transformations Π and Π

^{*}. The further investigations are necessary to find out and analyze Π and Π

^{*}.

_{T}and f

_{ma}

_{x}depending on n and d, even without using compact models. As it is shown in Table 1, the values f

_{T}and f

_{max}calculated from (21) have the best correspondence to the experimental ones. In this case, the cycle of high-frequency oscillations T = 1/f is used as the measure, for it is additive value. Thus, the increase of channel length leads to the increase of the time τ during which electrons pass through it and, consequently, to the growth of the value T. At this, the average values of fractal dimensions <D

_{H}(g

_{max})> are used to calculate f

_{T}: 1.87 for the selection 1-1, 1.79 for the selection 1-2, 1.78 for the selection 1-3, 1.46 for the selection 1-4. The average values <D

_{H}(C

_{gs})> are used to calculate f

_{max}: 2.42 for the selection 1-1, 2.52 for the selection 1-2, 2.37 for the selection 1-3, 2.39 for the selection 1-4. The initial values M

_{0}are equal to the experimental values of the oscillation cycles T corresponding to the maximum value W.

_{T}= 1), the values <D

_{H}> used for cutoff frequency f

_{T}and maximum frequency f

_{max}calculation have the average dimension values <D

_{H}(R

_{ds})> and <D

_{H}(C

_{gs})> close to two and three, correspondingly (Table 2, Table 3 and Table 4). This fact reveals the strong influence of the two-dimensional effects on f

_{T}values (defined by the pattern of linear channels’ distribution in the two-dimensional plane of the transistor) and three-dimensional effects on f

_{max}values (the peculiarities of the scattering 2D-electrons during their motion in three-dimensional plane), that is not obvious.

_{H}> (Table 2 and Table 3), and by means of selecting the values d and W—the other average values (Table 3 and Table 4). This fact can probably point out the existence of some superior attribute of chaotic systems.

## 8. Conclusions

_{f}

_{(H)}and initial values of the measures M

_{k,}

_{0}of the parameters P

_{k,}

_{0}for the initial transistor, it is possible to find out the values M

_{k}

_{,i}of the parameters P

_{k,i}of any other transistor made from the same material (with the same value D

_{f}

_{(H)}) and with predefined n and d. The proposed approach makes it possible, not only to define the sizes of the desired transistor structural elements, but to also reconstruct simultaneously the parameters of its linear (and may be non-linear) model, which significantly speeds up the development, optimization and approbation processes for the HEMT with predefined device characteristics.

_{T}and f

_{max}(and probably H

_{21}and G

_{max}) depending on n and d.

_{T}= D

_{f}, the behavior of HES electrophysical parameters, and consequently, AlGaN/GaN HEMT internal transistor equivalent circuit parameters, as well as their HF characteristics, have linear (classic) dependences upon n and d, which are peculiar to the spaces of integer topological dimensions D

_{T}.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The optical image of the matrix of TiAlNiAu ohmic contacts of the test elements with different values of channel length l

_{ds}and width d, formed on the AlGaN/GaN high-electron-mobility transistor (HEMT) heteroepitaxial structure—(

**a**). The dependence of the channel resistance per unit length ρ on the length l

_{ds}and width d—(

**b**).

**Figure 2.**The “freaking stairs” for the 50 × 50 µm areas of the relief surfaces of the AlGaN/GaN HEMT heteroepitaxial structure (polyline 1), gallium nitride (GaN) buffer layer (polyline 2), and their exponential approximation (curve 3)—(

**a**); the “freaking stairs” for the contact potential difference (CPD) Δφ(x;y) of the AlGaN/GaN heteroepitaxial structure (polyline 1) and its exponential approximation (curve 2)—(

**b**). The linear dependences of ln(N) on ln(l

_{0}/l) for the surface areas of AlGaN/GaN HES structure (line 1) and GaN buffer layer (line 2)—(

**c**); the same dependence for the contact potential difference (CPD) Δφ(x;y) of AlGaN/GaN heteroepitaxial structure—(

**d**).

**Figure 3.**The atomic force microscopy (AFM) images of 50 × 50 µm surface areas (column I): (

**a**) the relief h(x;y) of the AlGaN/GaN HEMT structure; (

**b**) irregularities of surface potential Δφ(x;y); (

**c**) the relief h(x;y) of the GaN buffer layer—with the corresponding contour images (column II) and distribution histograms in local (column III) and global (column IV) approximations.

**Figure 4.**The electron microscopy images: the 5 × 4 matrix of AlGaN/GaN HEMTs with section number n = 1, 2, 4, 8 and 12 and section width d = 50, 80, 100 and 150 µm—(

**a**); the AlGaN/GaN HEMT crystal 4 × 100 µm with the L-shape gate length of 0.15 µm (insertion)—(

**b**).

**Figure 5.**The frequency dependences of maximum values of the power and current amplification factors G

_{max}(f)—(

**a**) and H

_{21}(f)—(

**b**), correspondingly. The functional dependences G

_{max}= G

_{max}(d)—(

**c**) and H

_{21}= H

_{21}(d)—(

**d**), measured at different values of n at frequency f = 10 GHz.

**Figure 6.**The equivalent circuit of the linear model of field-effect transistor. The components of the internal transistor are circled with a dotted line.

**Figure 7.**The linear dependences on ln(W

_{0}/W) written in log–log scale: ln(C

_{ds}/C

_{ds}

_{,0})—(

**a**); ln(C

_{gd}/C

_{gd}

_{,0})—(

**b**); ln(C

_{gs}/C

_{gs}

_{,0})—(

**c**); ln(g

_{max}/g

_{max}

_{,0})—(

**d**); ln(R

_{i}/R

_{i}

_{,0})—(

**e**); ln(R

_{ds}/R

_{ds}

_{,0})—(

**f**).

**Table 1.**The cutoff frequencies f

_{T}and f

_{max}of the manufactured AlGaN/GaN HEMTs with the gate length l

_{g}= 0.15 μm, at various n and d.

The Total Gate Width: n × d = W, μm | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

4 × 50 = 200 | 8 × 50 = 400 | 12 × 50 = 600 | 2 × 80 = 160 | 4 × 80 = 320 | 8 × 80 = 640 | 12 × 80 = 960 | 2 × 100 = 200 | 4 × 100 = 400 | 8 × 100 = 800 | 12× 100 = 1200 | 2 × 150 = 300 | 4 × 150 = 600 | 8 × 150 = 1200 | 12 × 150 = 1800 | |

f_{T}, GHz calc. (17) | 60.27 | 61.69 | 54.99 | 72.32 | 71.03 | 81.49 | 56.15 | 84.59 | 79.82 | 65.08 | 54.51 | 77.71 | 109.13 | 82.08 | 62.74 |

f_{T}, GHz calc. (21) | 81.58 | 82.69 | 73.71 | 132.95 | 129.88 | 75.56 | 86.51 | 144.06 | 119.14 | 96.35 | 78.15 | 115.06 | 65.65 | 99.26 | 61.17 |

f_{max}, GHz calc. (22) | 69.46 | 70.28 | 57.16 | 108.57 | 103.60 | 50.08 | 65.41 | 95.83 | 86.01 | 67.89 | 63.13 | 80.68 | 46.88 | 42.46 | 43.80 |

f_{T}, GHz experiment | 23.32 | 24.98 | 24.45 | 27.01 | 25.56 | 24.79 | 20.01 | 30.01 | 25.51 | 22.34 | 19.98 | 27.48 | 27.19 | 18.61 | 12.47 |

f_{T}, GHz calc. (10) | 28.20 | 25.77 | 24.44 | 29.15 | 25.20 | 21.79 | 20.01 | 29.63 | 25.44 | 21.84 | 19.98 | 32.81 | 22.57 | 15.52 | 12.46 |

f_{max}, GHz experiment | 58.01 | 45.20 | 35.03 | 63.40 | 56.37 | 32.11 | 25,50 | 62.03 | 55.08 | 30,00 | 25.13 | 51.00 | 50.03 | 28.10 | 20.01 |

f_{max}, GHz calc. (10) | 71.53 | 47.85 | 37.82 | 63.91 | 45.82 | 32.85 | 27.04 | 79.84 | 51.59 | 33.34 | 25.82 | 58.92 | 38.79 | 25.54 | 20.00 |

**Table 2.**The parameters P

_{k,i}of the AlGaN/GaN HEMT internal transistor EC components, grouped by channel width d = Const.

Group № | n | d, μm | W = n × d, μm | C_{ds}, pF | D_{H}(C_{ds}) | C_{gd}, pF | D_{H}(C_{gd}) | C_{gs}, pF | D_{H}(C_{gs}) | g_{max}, A/V | D_{H}(g_{max}) | R_{gs}, Ω | D_{H}(R_{gs}) | R_{ds}, Ω | D_{H}(R_{ds}) | R_{s}, Ω | R_{d}, Ω | R_{g}, Ω |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1-1 | 4 | 50 | 200 | 0.107 | 1.81 | 0.0267 | 1.81 | 0.202 | 2.23 | 0.0405 | 1.83 | 6.42 | 1.85 | 1300 | 2.24 | 3.95 | 9.10 | 1.60 |

8 | 50 | 400 | 0.175 | 1.86 | 0.0437 | 1.86 | 0.338 | 2.45 | 0.0678 | 1.87 | 2.85 | 1.69 | 482 | 2.15 | 2.85 | 4.98 | 0.91 | |

12 | 50 | 600 | 0.267 | 1.79 | 0.0667 | 1.79 | 0.523 | 2.59 | 0.0922 | 1.90 | 3.48 | 2.14 | 359 | 2.30 | 1.8 | 4.21 | 1.39 | |

<D_{H}> | 1.82 | 1.82 | 2.42 | 1.87 | 1.89 | 2.23 | ||||||||||||

1-2 | 2 | 80 | 160 | 0.0676 | 1.92 | 0.0169 | 1.92 | 0.155 | 2.22 | 0.0307 | 1.87 | 6.78 | 1.79 | 1200 | 2.09 | 1.65 | 7.85 | 3.04 |

4 | 80 | 320 | 0.128 | 1.93 | 0.0319 | 1.93 | 0.291 | 2.35 | 0.0571 | 1.85 | 4.07 | 1.81 | 385 | 1.87 | 0.72 | 4.52 | 2.12 | |

8 | 80 | 640 | 0.254 | 1.88 | 0.0635 | 1.88 | 0.571 | 2.60 | 0.130 | 1.63 | 1.44 | 1.35 | 278 | 2.14 | 2.06 | 4.05 | 5.21 | |

12 | 80 | 960 | 0.397 | 1.74 | 0.0992 | 1.76 | 0.835 | 2.91 | 0.140 | 1.81 | 4.00 | 1.29 | 295 | 2.95 | 0.23 | 3.01 | 0.81 | |

<D_{H}> | 1.87 | 1.87 | 2.52 | 1.79 | 1.56 | 2.26 | ||||||||||||

1-3 | 2 | 100 | 200 | 0.0736 | 1.98 | 0.0184 | 1.98 | 0.183 | 2.27 | 0.0391 | 1.84 | 6.82 | 1.87 | 696 | 1.95 | 1.94 | 5.53 | 3.47 |

4 | 100 | 400 | 0.160 | 1.91 | 0.0400 | 1.91 | 0.383 | 2.37 | 0.0802 | 1.75 | 4.36 | 1.98 | 557 | 2.24 | 0.75 | 4.13 | 2.55 | |

8 | 100 | 800 | 0.345 | 1.75 | 0.0863 | 1.75 | 0.618 | 2.94 | 0.141 | 1.70 | 3.54 | 2.56 | 301 | 2.55 | 0.30 | 2.88 | 1.65 | |

12 | 100 | 1200 | 0.520 | 1.48 | 0.130 | 1.48 | 1.09 | 1.88 | 0.178 | 1.83 | 2.04 | 2.76 | 230 | 2.96 | 0.10 | 2.75 | 1.95 | |

<D_{H}> | 1.78 | 1.78 | 2.37 | 1.78 | 2.29 | 2.43 | ||||||||||||

1-4 | 2 | 150 | 300 | 0.117 | 1.94 | 0.0293 | 1.94 | 0.269 | 2.34 | 0.0571 | 1.82 | 4.14 | 1.79 | 510 | 1.99 | 1.61 | 5.47 | 5.01 |

4 | 150 | 600 | 0.232 | 1.91 | 0.0579 | 1.91 | 0.791 | 2.21 | 0.159 | 1.41 | 1.12 | 1.10 | 298 | 2.13 | 3.71 | 4.35 | 3.29 | |

8 | 150 | 1200 | 0.452 | 1.83 | 0.113 | 1.83 | 1.54 | 2.64 | 0.233 | 1.16 | 3.48 | 1.19 | 164 | 1.60 | 1.22 | 1.10 | 1.20 | |

12 | 150 | 1800 | 0.632 | --- | 0.158 | --- | 2.99 | --- | 0.249 | --- | 1.00 | --- | 85.8 | --- | 1.24 | 2.00 | 1.56 | |

<D_{H}> | 1.89 | 1.89 | 2.39 | 1.46 | 1.36 | 1.91 |

**Table 3.**The parameters P

_{k,i}of the AlGaN/GaN HEMT internal transistor EC components, grouped by section number n = Const.

Group № | n | d, μm | W = n × d, μm | C_{ds}, pF | D_{H}(C_{ds}) | C_{gd}, pF | D_{H}(C_{gd}) | C_{gs}, pF | D_{H}(C_{gs}) | g_{max}, A/V | D_{H}(g_{max}) | R_{gs}, Ω | D_{H}(R_{gs}) | R_{ds}, Ω | D_{H}(R_{ds}) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

2-1 | 2 | 80 | 160 | 0.0676 | 1.92 | 0.0169 | 1.92 | 0.155 | 2.22 | 0.0307 | 1.87 | 6.78 | 1.79 | 1200 | 2.09 |

2 | 100 | 200 | 0.0736 | 1.98 | 0.0184 | 1.98 | 0.183 | 2.27 | 0.0391 | 1.84 | 6.82 | 1.87 | 696 | 1.95 | |

2 | 150 | 300 | 0.117 | 1.94 | 0.0293 | 1.94 | 0.269 | 2.34 | 0.0571 | 1.82 | 4.14 | 1.79 | 510 | 1.99 | |

<D_{H}> | 1.95 | 1.95 | 2.28 | 1.84 | 1.82 | 2.01 | |||||||||

2-2 | 4 | 50 | 200 | 0.107 | 1.81 | 0.0267 | 1.81 | 0.202 | 2.23 | 0.0405 | 1.83 | 6.42 | 1.85 | 1300 | 2.24 |

4 | 80 | 320 | 0.128 | 1.93 | 0.0319 | 1.93 | 0.291 | 2.35 | 0.0571 | 1.85 | 4.07 | 1.81 | 385 | 1.87 | |

4 | 100 | 400 | 0.160 | 1.91 | 0.0400 | 1.91 | 0.383 | 2.37 | 0.0802 | 1.75 | 4.36 | 1.98 | 557 | 2.24 | |

4 | 150 | 600 | 0.232 | 1.91 | 0.0579 | 1.91 | 0.791 | 2.21 | 0.159 | 1.41 | 1.12 | 1.10 | 298 | 2.13 | |

<D_{H}> | 1.89 | 1.89 | 2.29 | 1.71 | 1.69 | 2.12 | |||||||||

2-3 | 8 | 50 | 400 | 0.175 | 1.86 | 0.0437 | 1.86 | 0.338 | 2.45 | 0.0678 | 1.87 | 2.85 | 1.69 | 482 | 2.15 |

8 | 80 | 640 | 0.254 | 1.88 | 0.0635 | 1.88 | 0.571 | 2.60 | 0.130 | 1.63 | 1.44 | 1.35 | 278 | 2.14 | |

8 | 100 | 800 | 0.345 | 1.75 | 0.0863 | 1.75 | 0.618 | 2.94 | 0.141 | 1.70 | 3.54 | 2.56 | 301 | 2.55 | |

8 | 150 | 1200 | 0.452 | 1.83 | 0.113 | 1.83 | 1.54 | 2.64 | 0.233 | 1.16 | 3.48 | 2.94 | 164 | 2.60 | |

<D_{H}> | 1.83 | 1.83 | 2.66 | 1.59 | 2.14 | 2.36 | |||||||||

2-4 | 12 | 50 | 600 | 0.267 | 1.79 | 0.0667 | 1.79 | 0.523 | 2.59 | 0.0922 | 1.90 | 3.48 | 2.14 | 359 | 2.30 |

12 | 80 | 960 | 0.397 | 1.74 | 0.0992 | 1.76 | 0.835 | 2.91 | 0.140 | 1.81 | 4.00 | 2.90 | 295 | 2.95 | |

12 | 100 | 1200 | 0.520 | 1.48 | 0.130 | 1.48 | 1.09 | --- | 0.178 | 1.83 | 2.04 | 2.76 | 230 | 2.96 | |

12 | 150 | 1800 | 0.632 | --- | 0.158 | --- | 2.99 | --- | 0.249 | --- | 1.00 | --- | 85.8 | --- | |

<D_{H}> | 1.67 | 1.68 | 2.75 | 1.85 | 2.60 | 2.74 |

**Table 4.**The parameters P

_{k,i}of the AlGaN/GaN HEMT internal transistor EC components, grouped by channel total width W = Const.

Group № | n | d, μm | W = n × d, μm | C_{ds}, pF | D_{H}(C_{ds}) | C_{gd}, pF | D_{H}(C_{gd}) | C_{gs}, pF | D_{H}(C_{gs}) | g_{max}, A/V | D_{H}(g_{max}) | R_{gs}, Ω | D_{H}(R_{gs}) | R_{ds}, Ω | D_{H}(R_{ds}) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

3-1 | 4 | 50 | 200 | 0.107 | 1.81 | 0.0267 | 1.81 | 0.202 | 2.13 | 0.0405 | 1.98 | 6.42 | 1.34 | 1300 | 2.16 |

2 | 100 | 200 | 0.0736 | 2.01 | 0.0184 | 2.01 | 0.183 | 2.19 | 0.0391 | 1.99 | 6.82 | 1.38 | 696 | 1.81 | |

<D_{H}> | 1.91 | 1.91 | 2.16 | 1.985 | 1.36 | 1.985 | |||||||||

3-2 | 8 | 50 | 400 | 0.175 | 1.87 | 0.0437 | 1.87 | 0.338 | 2.38 | 0.0678 | 2.12 | 2.85 | 1.18 | 482 | 1.98 |

4 | 100 | 400 | 0.160 | 1.95 | 0.0400 | 1.95 | 0.383 | 2.27 | 0.0802 | 1.97 | 4.36 | 1.21 | 557 | 2.11 | |

<D_{H}> | 1.91 | 1.91 | 2.325 | 2.045 | 1.195 | 2.045 | |||||||||

3-3 | 12 | 50 | 600 | 0.267 | 1.76 | 0.0667 | 1.76 | 0.523 | 2.56 | 0.0922 | 2.34 | 3.48 | 1.03 | 359 | 2.13 |

4 | 150 | 600 | 0.232 | 1.97 | 0.0579 | 1.97 | 0.791 | 1.96 | 0.159 | 1.55 | 1.12 | 2.64 | 298 | 1.86 | |

<D_{H}> | 1.865 | 1.865 | 2.26 | 1.945 | 1.835 | 1.995 | |||||||||

3-4 | 12 | 100 | 1200 | 0.520 | --- | 0.130 | --- | 1.09 | --- | 0.178 | --- | 2.04 | --- | 230 | --- |

8 | 150 | 1200 | 0.452 | --- | 0.113 | --- | 1.54 | --- | 0.233 | --- | 3.48 | --- | 164 | --- |

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## Share and Cite

**MDPI and ACS Style**

Torkhov, N.A.; Babak, L.I.; Kokolov, A.A.
The Influence of AlGaN/GaN Heteroepitaxial Structure Fractal Geometry on Size Effects in Microwave Characteristics of AlGaN/GaN HEMTs. *Symmetry* **2019**, *11*, 1495.
https://doi.org/10.3390/sym11121495

**AMA Style**

Torkhov NA, Babak LI, Kokolov AA.
The Influence of AlGaN/GaN Heteroepitaxial Structure Fractal Geometry on Size Effects in Microwave Characteristics of AlGaN/GaN HEMTs. *Symmetry*. 2019; 11(12):1495.
https://doi.org/10.3390/sym11121495

**Chicago/Turabian Style**

Torkhov, Nikolay A., Leonid I. Babak, and Andrey A. Kokolov.
2019. "The Influence of AlGaN/GaN Heteroepitaxial Structure Fractal Geometry on Size Effects in Microwave Characteristics of AlGaN/GaN HEMTs" *Symmetry* 11, no. 12: 1495.
https://doi.org/10.3390/sym11121495