Calculation of electron transport in branched semiconductor nanostructures using quantum network model

Electron transport in branched semiconductor nanostructures provides many possibilities for creating fundamentally new devices. We solve the problem of its calculation using a quantum network model. The proposed scheme consists of three computational parts: S-matrix of the network junction, S-matrix of the network in terms of its junctions' S-matrices, electric currents through the network based on its S-matrix. To calculate the S-matrix of the network junction, we propose scattering boundary conditions in a clear integro-differential form. As an alternative, we also consider the Dirichlet-to-Neumann and Neumann-to-Dirichlet map methods. To calculate the S-matrix of the network in terms of its junctions' S-matrices, we obtain a network combining formula. We find electrical currents through the network in the framework of the Landauer$-$B\"uttiker formalism. Everywhere for calculations, we use extended scattering matrices, which allows taking into account correctly the contribution of tunnel effects between junctions. We demonstrate the proposed calculation scheme by modeling nanostructure based on two-dimensional electron gas. For this purpose we offer a model of a network formed by smooth junctions with one, two and three adjacent branches. We calculate the electrical properties of such a network (by the example of GaAs), formed by four junctions, depending on the temperature.


Introduction
An electron transport in branched semiconductor nanostructures provides many possibilities for creating fundamentally new devices [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Modern experiments have shown that it is always a combination of diffusion transport and ballistic one [5][6][7]. Ballistic transport corresponds to the coherent motion of electrons and, therefore, relates to quantum effects in the system. At the same time, it also occurs in structures whose size is an order of magnitude greater than the free length of charge carriers [5,6], and in materials with their low mobility [7]. Therefore, ballistic transport is relevant for a much wider range of structures than "purely" quantum ones, and its accounting is fundamentally important for their correct modeling.
A mathematical model describing ballistic electron transport in branched semiconductor nanostructures is a quantum network [15,16]. Electrical currents through the network are calculated in the framework of the Landauer-Büttiker formalism [17,18] based on its transparency. In turn, the transparency is determined by the scattering matrix of network.
The scattering matrix of the quantum network junction can be found by solving the Schrödinger equation in it with scattering boundary conditions (SBC) at the boundaries with branches [19][20][21]. SBC provide matching of wave function in junction with incident and scattered waves in branches. Therefore, this method is universal and is also used for quantum self-consistent calculation of current through nanostructures [21]. In the literature, the form of SBC is usually defined by a concrete problem. Such approach reduces their clearness. Also, their "non-standard" form makes them difficult to use in computing packages.
It is more convenient to apply the methods based on Dirichlet and Neumann boundary conditions: Dirichlet-to-Neumann and Neumann-to-Dirichlet map (DN-and ND-map) methods [15,16,22,23]. To calculate the scattering matrix of semiconductor nanostructures, the formalism of the R-matrix [24][25][26] is also widespread. This method is similar to the ND-map method, since the definition of the R-matrix is identical to the definition of the ND-map, and the expression for the scattering matrix has the same form [26]. The disadvantage of the DN-and ND-map methods is their low clearness, since additional mathematical considerations are required to solve the initial scattering problem.
Any quantum network can be considered as one junction with a complex structure, and one of the mentioned above methods [15][16][17][18][19][20][21][22][23][24][25][26] can be used to calculate its scattering matrix. However, for calculations, it may be more convenient to find a scattering matrix of network using the rule of combining scattering matrices of its junctions [17,27]. This approach becomes particularly effective when the network contains the same junctions. In its use, as a rule, the boundaries of the junctions are selected so that branches connecting junctions are excluded from consideration. This formal technique can simplify calculations, but makes it difficult to study the transport properties of a quantum network depending on the lengths of the branches. The disadvantage of this method is also the absence of an explicit formula for the scattering matrix of the entire network in terms of its junctions' scattering matrices.
In this work, we will develop the methods discussed above and propose on their basis a scheme for calculating electron transport in branched semiconductor nanostructures. The scheme will be formulated in terms of an extended scattering matrix of the quantum network [27, p. 155]. This approach will allow correctly taking into account tunnel effects in system. For an arbitrary network junction, we will obtain scattering boundary conditions in an integro-differential form. Using them, we will write the procedure for calculating its extended scattering matrix, as well as the calculation scheme using the DN-and ND-map methods. We will obtain a formula for an extended scattering matrix of a quantum network in terms of extended scattering matrices of its junctions, taking into account the lengths of the branches connecting them. Finally, the Landauer-Büttiker formalism, along with all other methods, will be written in the framework of a single notation system. We will demonstrate the effectiveness of the proposed calculation scheme by modeling a nanostructure based on twodimensional electron gas using a model of a quantum network of smooth junctions with one, two and three adjacent branches.
2 Scattering of charge carrier in quantum network 2.1 Agreements and notation

Agreements
We use the following agreements to simplify the presentation.
I. Computations. In computations, we use the symbols of implication ⇒ and equivalence ⇔, equality := and equivalence :⇔ by definition. The ellipsis symbol is used to exclude obvious computations in the implication chain; the semicolon symbol is used to write multi-level computations to a string: where A, B, C and D are statements or references to them (definitions, formulas, etc.). In some cases, we use references to statements without naming them II. Indices and ranges. The ranges of the superscripts and subscripts values in the enumerations and sums are set at the appropriate levels, for example: {d l n } l∈A n∈B , ∑ l∈A n∈B d l n . The implicitly defined range of index values is determined by its location relative to the letter and its semantics. By default, any enumeration in this work is orderedtuple [28, p. 33].
III. Vectors and matrices. Enumeration by one index (for example, d = {d l } l ) is a column vector, by two indexes at the same level (for example, D = {D kl } kl ) is a matrix: the first index is a row number, the second one is a column number. Objects with tuple indexes outside square brackets are enumerations by all tuple elements. For example, VI. Differentiation. ∂ is the operator of partial differentiation, where is the name of the variable (named differentiation) or the argument number (numbered differentiation) of the corresponding function. Reduction: VII. Iverson notation [29, p. 24]. The brackets with the statement are 1 if it is true and 0 if it is false.
VIII. Line over the symbol. The semantics of the line over the symbol is determined by the semantics of the symbol (Table 1).
The reverse precedence rule sets the index reduction rule. Arguments in brackets at functions are also taken last, when they are partially represented, ellipses are placed: f (x, ...).
X. Reduction of arguments and identifiers. If it is necessary to specify which set the reduced arguments belong to, the words "in" and "on" (usually for domain boundaries) are used. For example, When reducing the identifier in section, also uses the word "in". For example, the reduction for symbol with the top identifier is: XI. Bra-ket symbolism. The semantics of objects in the bra-ket symbolism is determined by the semantics of objects in brackets. In particular, the scalar product for the functions of a discrete and continuous argument has the form respectively, where D( ) is the function domain.
XII. Integration. When integrating, the interval in the Iverson brackets is the 1D analogue of the oriented curve that defines the integration direction:

Notation
Physically, a quantum network consists of quantum dots connected to each other by quantum wires. Since the charge carrier can scatter both in the point and in the wire, it is convenient to introduce a functional definition: a quantum network is a set of junctions and branches. An internal junction is a network element in which the charge carrier scatters, an external junction is a source or drain of charge carriers, a branch is a network element in which the charge carrier does not scatter, an internal branch connects two internal junctions, an external branch connects the internal junction to the external one. Channels are energy levels that arose due to size-quantization across the branch, with the associated movement of the charge carrier along the branch.
To unambiguously identify the elements of the network, we number its branches. In notation, the superscript is the branch number, the subscript is the channel number, the superscript in square brackets is the structural identifier of the junction, the contents of the brackets is the tuple of the numbers of the branches adjacent to the junction (Fig. 1). In this work, we also use the notations: I := ( A∈N A)\E is tuple of internal branches' numbers, E := ⊖ A∈N A is tuple of external branches' numbers (A ⊖ B := (A\B) (B\A) is symmetric difference of tuples A and B), N is tuple of structural identifiers of internal junctions, containing information about all connections of junctions in the network. The concept of a junction is conditional: depending on its structural identifier A, it can be either any section of the network or the entire network if A = E.  In this work, we also use a type identifier. It is placed in angle brackets and can be a number, letter, or word. A typical identifier is convenient, in particular, when specifying the structure of the network by the type of its internal junctions, for example: where Ω A is a junction of type "A" (Ajunction), a A , b A , ... are parameters of A-junction, , ...} K∈A are values of parameters of A-junctions in the network, A is a tuple of structural identifiers of A-junctions.

Charge carrier in network
Let us consider a three-dimensional quantum network. The results obtained for it are easy to use for two-dimensional and one-dimensional networks. Without loss of generality, for simplicity of presentation, we consider an isotropic semiconductor. The motion of the charge carrier is described by the Schrödinger equation: whereh is the Planck constant, m is the effective mass of the charge carrier, ∆ = ∂ 2 1 + ∂ 2 2 + ∂ 2 3 is the Laplace operator, V is the potential, E is the energy of the charge carrier. We write it in dimensionless form: depending on the problem dimension, we have 1D : Ψ (r) := L 1/2 ϕ(Lr) where L is the characteristic length (one can choose any for convenience in a particular problem). Then the motion of the charge carrier is described by boundary problems for dimensionless Schrödinger equations in the branches where Γ k is the boundary of the branch Ω k with adjacent junctions, in internal junctions of network and matching conditions on the boundaries of the internal junctions with the branches: Equations (15), (16) and conditions (17) are written in the global coordinate frame (GCF) -the coordinate frame associated with the network. Its location one can set in a specific problem for convenience reasons.

Charge carrier in branch
To solve the problems (15), we introduce the concept of the local coordinate frame (LCF) at the boundary of the junction with the branch (Fig. 2 Since branches are quantum wires, they have a simple geometry: where a k is the length of the branch Ω k , and boundaries of the junction with branches have the form: The motion of the charge carrier in the domain Ω k in GCF is described by equation (15). To separate variables in problem (15), we change GCF to LCF: and assume that the potential υ k does not change along the branch. Since the branches have a simple geometry (19), we obtain the following analogue of problem (15) in LCF: We are looking for a solution of equation (23) in the form: Substituting expression (24) into problem (23), we get From here follows Everywhere in this work, we suppose that functions h [A] are orthonormal: The expression (27) is also conveniently written in terms of the wavenumber operator K [A] in two equivalent representations: We group the incident and scattered waves in the expression (27): To interpret expressions, mnemonic rules are convenient: , c [A]⊲"to" "to" , S [A]"to""from" "to""from" Taking into account our notation, everywhere in this work, we call the extended scattering matrix briefly: S-matrix. We can formulate two problems: the incoming scattering problem is the search for scattered waves from known incident waves, the outgoing scattering problem is the search for incident waves from known scattered waves. By solving the incoming and outgoing scattering problems, one can find S [A] and (S [A] ) −1 matrices, respectively. As a rule, only the first problem is considered in the literature. It is also the main one for this work.

Scattering boundary conditions
Let us write a direct method for solving the scattering problems (section 2.2.3). To do this, one should obtain scattering boundary conditions [19][20][21] for an arbitrary junction of a quantum network (Fig. 1).
The incoming scattering problem is a problem (16) supplemented by incoming scattering boundary conditions (incoming SBC): The outgoing scattering problem is a problem (16) supplemented by outgoing scattering boundary conditions (outgoing SBC): PROOF 1. We rewrite the matching conditions (17) in LCF taking into account the form (21): Here, the branch index is reduced, and all expressions with vector and matrix objects (W , ψ, ω, γ, K etc.) are vector (section 2.1.1). Using the expression (35), we get a derivative of the function in the branches From the formulas (31), (44) and (45) follows Excluding from expressions (46) functions ψ ⊲ , we have incoming SBC (42). Excluding from expressions (46) functions ψ ⊳ , we have outgoing SBC (43). Solving the problem (16) with incoming SBC (42), it is possible to find scattered waves ψ ⊲ according to the conditions (44) by known incident ones ψ ⊳ : and hence the matrix S (38). Solving the problem (16) with outgoing SBC (43), it is possible to find incident waves ψ ⊳ according to the conditions (44) by known scattered ones ψ ⊲ : and hence the matrix S −1 (39).
SBC is also convenient to write in GCF. For incoming SBC (42) and outgoing SBC (43) we have: respectively, where the operator K acts on functions in the GCF as follows: Here Ψ ⊳ :=W ψ ⊳ , Ψ ⊲ :=W ψ ⊲ , ∂ n is a derivative along the outer normal to the boundary. The expressions (47) and (48) take the form: The boundary conditions (42), (43), (49) and (50) are integro-differential because they contain both the differentiation operators ∂ 1 , ∂ n and the integral operator K (34). At the same time, in terms of SBC, the physical meaning of the operator K is isolation of junction (Appendix A). Therefore, these boundary conditions are clear: they provide an intuitive and concise formulation of the scattering problem at the junction of the quantum network.

Calculation of S-matrix by SBC method
In all scattering problems, the S-matrix of a junction can be found element by element using SBC. At the same time, it is convenient to write SBC in GCF for implementation in computing packages. Proposition 2. Any element of the S-matrix of a junction Ω is written as where the function Ψ p q is the solution to the scattering problem Obviously, to calculate one column of the S-matrix p q , it is enough to solve only one boundary problem of the form (55). This is an advantage of SBC in the numerical study of individual S-matrix elements.
In some scattering problems, using SBC one can write the S-matrix of a junction explicitly.

Calculation of S-matrix by DN-and ND-map methods
The SBC method is the most clear when calculating the Smatrix of a quantum network junction, because provides the wave function in a junction during scattering. However, SBC are integro-differential boundary conditions that are not supported by all compute packages. In the framework of the DN-and ND-map [15,16,22,23] method, it is enough to solve the Dirichlet and Neumann problems, respectively. It is acceptable in cases where the wave function in the junction during scattering is not essential. We specify the DNand ND-map methods for the notation system used.
Proposition 4. The S-matrix of the junction Ω is written using the DN-map operator D: Proposition 5. The S-matrix of the junction Ω is written using the ND-map operator N: Nχ := WΨ on γ, Expressions (61) and (63) are equivalent taking into account the property: D = N −1 . By analogy with (55) the matrices D and N can be found element by element from (222), (224) and (30): To find any element of the S-matrix, according to the expressions (61) and (63), one need to find all elements of the D or N operator. DN-and ND-map are singular if ε is the eigenvalue of the problem with zero Dirichlet and Neumann boundary conditions, respectively.

S-matrix of quantum network in terms of its junctions' S-matrices
Based on the agreements and notation introduced in section 2.1, we will obtain an expression for the S-matrix of an arbitrary quantum network in terms of its junctions' S-matrices.

Elementary section of network
Let us consider the scattering of the charge carrier in a quantum network. Since the network consists of junctions and branches connecting them, its elementary section is the connection of two junctions (Fig. 3). Any quantum network can be represented as a set of such connections. Therefore, the In expression (69), let us separate the first sum as ∑ l = ∑ l∈J + ∑ l∈K and the second one as ∑ l = ∑ l∈K + ∑ l∈L . Taking into account this separation and the agreements of section 2.1.1, from expression (69) follows According to expressions (27) Since LCFs are related to each other by translation and rotation transformations so that X [J,K]k X [K,L]k , according to condition (20), we have At the same time, one can shows that where U [J,K] and U [K,L] are the operators providing changing coordinate frame (72). For them from (72) we havē Taking into account property (80) and agreements in section 2.1.1, equations (81) take the form where K KK and A KK are the operators of wavenumbers and branch lengths, respectively (73). In the system (82), we separate the sums into two parts: in the first equation as ∑ l = ∑ l∈K + ∑ l∈L , in the second one as ∑ l = ∑ l∈J + ∑ l∈K , therefore The equality (83) When executing the propertiesW [J,K]k h the definition (72), it also follows that these operators are diagonal: The combining formula (71) is universal. When external branches absent, for example, on the right in If there is no connection between junctions (K = ∅, , the formula (71) is written as Thus, due to the agreements of section 2.1.1, formula (71) is applicable for combining network junctions in all possible cases.

Network combining formula
Based on the combining formula (71), assuming in it that for S-matrices with tuple identifiers A and B we define a combining operation: Then according to expressions (73), (71) and (90) we have There is no element exp −iK KK A KK in formula (91). Therefore, in the case of a "branchless" network (90), the combining operation (89) is simplified. Formally, it is convenient. However, branch lengths are parameters that affect the S-matrix of the network. Their explicit accounting is fundamentally important when designing networks with predetermined transport properties.
Using definition (89) we write the network combining formula (NCF) -an expression for the S-matrix of an arbitrary quantum network in terms of its junctions' S-matrices: Here, the tuple N also sets the order of combining network junctions by formula (89). Since formula (92) is written in terms of extended scattering matrices, it allows correctly taking into account closed channels in the network and, therefore, tunnel effects between junctions. Due to the agreements and notation of section 2.1, formula (92) is convenient both in analytical and numerical calculations. In the particular case, the result obtained using it coincides with the previously one obtained in the literature (Appendix D.2).

Electron transport in quantum network
The approach described in section 2 enables efficient calculation of the scattering properties of semiconductor nanostructures using quantum network model. In section 3, we will show how the Landauer-Büttiker formalism [17,18] is used to find electric currents through the network as part of a calculation scheme suggested in this work.

Probability currents and S-matrix
The Landauer-Büttiker formalism uses the scattering probabilities of charge carriers. The association of scattering probabilities with the S-matrix of the quantum network can be established based on expressions for probability currents in the branches.

Probability currents
In terms of the stationary problem (11), dimensionless incident and scattered currents in the branch-channel k m take the form By rewriting the expressions (37) as from definitions (94), we get Assuming that the currents in the closed channels are zero, for the total current in the branch-channel k m we have When calculating the currents in the branch Ω k , one can show that the currents in the channels are additive. Then taking into account the expressions (96) for incident and scattered currents we have and the total current takes a form

Structure of operator K
The channels separated on open and closed ones (29) sets the structure for operators introduced in section 2. In particular, the operator K in the representation (32) takes the form The operator K can also be written as a sum of Hermitian and anti-Hermitian operators: According to definitions (102), in the representation (34) we get K kl Then accurate to the representation and expansion from expressions (100), (101), (103) follows which is completely compliance with the notation used in the literature At that the expression (100) takes the form

Current conservation and S-matrix
Based on the structure of the form (106) set for operators, we can write the law of current conservation in terms of the S-matrix of the quantum network.
Proposition 7. The S-matrix of the quantum network has the propertȳ where S ++ is the S-matrix submatrix that relates wave amplitudes in open channels with each other: PROOF 7. With elastic scattering, the total current (99) is zero: From expressions (96), (97), (109), (106) follows According to notation (105), the structure of the S-matrix (38) is similar to the structure (106) and can be written as (108). Taking into account agreements IV and VIII, we havē where is a complex matrix. Then from expressions (110), (108) and (111) we get Since the S-matrix does not depend on the incident waves, we set c ⊳ − = 0. Then, according to the formula (112), the property (107) is executed for the submatrix S ++ .

Extended current scattering matrix
Property (107) can be formulated in terms of a unitary matrix. We write expressions (96) as where d ⊳ and d ⊲ are incident and scattered current amplitudes respectively. The relation between them has the same form as between wave amplitudes (38): where C is an extended current scattering matrix (combining the concepts of extended [27, p. 155] and current [25] scattering matrices). From expressions (113) and (114) From expressions (113), (114) and (38), we can find the relation of matrix C with matrix S: The distinction between the concepts of a matrix relating wave amplitudes and a matrix relating current amplitudes avoids ambiguity in the interpretation of analytical and numerical calculations. Taking into account structure (106) from formula (116) we have According to expression (117), the property (107) takes the form Thus, the unitary is a submatrix of an extended current scattering matrix that relates current amplitudes in open channels with each other -the current scattering matrix C ++ . According to property (118), the squares of the modules of its elements have a probabilistic interpretation.

Electrical properties
Based on the scattering probabilities of charge carriers in a quantum network, it is possible to find its electrical properties in the framework of the Landauer-Büttiker formalism. For this purpose, we will calculate electric currents in terms of average values from the ensemble of charge carriers in the external branches of the network. We will assume that they are smoothly connected to the reservoirs of charge carriers. Then when calculating, we can only take into account the scattering properties of the network (section 2).

Currents in branch-channels
The difference in voltages on the reservoirs leads to the appearance of currents through the quantum network. Since the external branches of the network are long, we neglect the effects of barrier tunneling in them. This means taking into account only open channels. For current in the k-th branch we have: where J ⊳k is current incident on the network from the k-th branch, J ⊲k is current scattered by the network to the k-th branch. According to formula (98), the currents in the channels are additive: where J ⊳k m is current incident on the network from the k m -th branch-channel, J ⊲k m is current scattered by the network to the k m -th branch-channel. Proposition 8. The incident and scattered currents in the k m -th branch-channel are calculated as where e is the charge of the charge carrier, f k is the distribution function in the k-th branch (reservoir), E k F is the Fermi level in the k-th branch (reservoir), P = {P kl mn } kl mn is the matrix of scattering probabilities. PROOF 8. Outside the domains of connection with reservoirs, branches are translational invariant. Since structurally the branch is a quantum wire, the dispersion law of charge carriers in the energy frame of section 2.2 takes the form: where M = {snp} is a multi-index (set of quantum numbers), uniquely identifying the state, s = 1, 2 is a spin quantum number, n is a quantum number corresponding to sizequantization, p is a quasi-momentum, E k ⊥n is the dimensional energy of the k n -th branch-channel, p x is a projection of the quasi-momentum along the branch. The density of single-particle states in the k-th branch is written as where δ is the Dirac delta function. Replacing in definition (128) from summation by p x to integration taking into account formula (126), we have where g k n is density of states in the k n -th branch-channel, L k x is length of the k-th branch. Based on g k n one can find the average value of the physical quantity Q in the k n -th branchchannel: where N k n is the number of particles in the k n -th branchchannel. We can see from definition (124), that the distribution functions in the branches differ in the positions of their Fermi levels. Fermi level E k F is written as where E F is the Fermi level of the quantum network in the absence of voltages on the reservoirs, U k is the voltage applied to the k-th reservoir. We find incident and scattered currents in the k m -th external branch-channel as the average values (131):

Currents in branches and conductivity
Using expressions for currents in the branch-channels (122) it is possible to find currents in the branches (120).
Proposition 9. Current in the k-th branch is calculated as where {P kl mn } l =k mn are matrix probabilities of transmission into the k-th branch (transparencies).
Theoretically, it should be zero due to law of charge conservation. Let us write the expression for currents (135) in dimensionless form. It is convenient in calculations together with the proposed calculation scheme of the S-matrix of the quantum network (sections 2.3 and 2.4). From expressions (135) and (125) taking into account definitions (13) we have where F −1 is the Fermi-Dirac integral of the order −1, and the relations of dimensionless quantities with dimensional ones is given by the formulas Proposition 10. In case of low temperatures current in k-th branch is calculated as where we use agreement XII.

PROOF 10. See Appendix E.3.
Taking into account the definition (125), expression (142) has a simple physical interpretation: the electric current in the k-th branch causes the branch-channels whose energies {E k ⊥m } k m are located between the Fermi level of the k-th branch and the Fermi levels of all other branches.
whereσ := {σ kl } kl is the approximate conductivity of the network, U lk is the voltage between the reservoirs l and k (bias voltage).

PROOF 11. See Appendix E.2.
We can see from definition (146), that the approximate conductivity of branched semiconductor nanostructures does not depend on bias voltages. This is a convenient quantity for analyzing their volt-ampere characteristics at low bias voltages.
Proposition 12. At low voltages on reservoirs (143) with the same (144) low (141) temperature, the approximate conductivity is written as Thus, the material of section 3.2 is the last part of the scheme for calculating electronic transport in branched semiconductor nanostructures using a quantum network model.

Two-dimensional quantum network of Q-, I-and Y-junctions
Due to the high level of development of planar technology, nanoelectronic devices based on two-dimensional electron gas are of interest. Their simplest model is a twodimensional quantum network. In section 4, we will demonstrate the approach proposed in sections 2 and 3 by calculations for a two-dimensional quantum network of smooth junctions with one, two and three adjacent branches.

Geometry
Let us consider a two-dimensional quantum network of smooth Q-, I-and Y-junctions -QIY-network (Fig. 4). The absence of corners in the geometry of the junctions (their smoothness) corresponds to the typical nano-devices studied in experiment [14]. Due to its structure, the QIY network is suitable for modeling many two-dimensional nanostructures, as well as for demonstrating calculations using the scheme proposed in sections 2 and 3. Fig. 4 Junctions of smooth QIY-network: a -Q-junction, b -Ijunction, c -symmetric Y-junction.
During calculating we will use NCF (92), which is based on the combining formula (71). Formula (71) contains operators relates to changing LCFs on the branch boundaries: U [J,K] and U [K,L] (72). We choose such an LCF location when they are relates to each other by rotation and translation (Fig. 5). It is universal, since it is convenient during calculating the S-matrix of any two-dimensional network using the formula (92).
In this work, we consider a QIY-network with branches of the same width B = : The quantity B = is assumed to be the characteristic length during the transition (13) and (14) to the dimensionless Schrödinger equation (12): Since dimensionless widths are defined as {b k := B k /L} k∈I E , taking into account equalities (149), assuming zero potential in all branches, we have: According to the chosen LCFs location (Fig. 5), for operators providing their changing in the combining formula (71), from expressions (72), (30) and (152) Using the expression (154) in calculations according to NCF (92), one can find the S-matrix of the QIY-network based on the S-matrices of its junctions (Fig. 4) and information about its structure.

Parameters
The structure of arbitrary QIY-network has the form where Q, I, Y are tuples of Q-, I-, and Y-junction structural identifiers, respectively. For smoothness of the network, there must be no corners on the boundaries of the junctions with branches. Therefore, according to equalities (151) b Q = b I = b Y = 1 and the arch radius r Q = 1/2. We also assume r Y = 1/2 that provides the network small size. Without loss of generality of the approach, the electric field in the I-junction ε I ⊥ will be considered isotropic and Fermi level of network transverse. As a result, based on example (10), we have the following structure specificity of the QIY-network: For a two-dimensional electron gas in all formulas m = m e , e = −e 0 , where m e is the effective mass of an electron, e 0 is an elementary charge. Then the dimensionless network parameters are written as a table 2.
The values and ranges of dimensionless geometric parameters of the network are determined by their dimensional analogues (Table 2), which are expressed using a characteristic length L = B = . Let us choose L = 10 nm. Then, according to equalities (149) and table 2, in terms of the dimensional parameters of the network we have These values are much larger than the lattice constant of III-V semiconductors [30]. Therefore, when modeling such a nanostructure, one can use the zone theory of a solid. Due to small sizes, the effects of size-quantization here will be especially significant.
Ballistic electron transport will dominate in a structure that is smaller than the free length of electrons ℓ e . It has been experimentally shown that for a two-dimensional electron gas formed in heterostructures based on III-V semiconductors at room temperatures ℓ e ∼ 100 nm [2].
As the maximum permissible intensity of electric field in the I-junctions of the network, we assume a value 10 8 V/m. This is an order of magnitude less than autoemission intensities: ∼ 10 9 V/m [31]. Typical for calculations in the work are bias voltages ∼ 10 mV [3,4].

S-matrices of junctions
To calculate electric currents through the QIY-network in the framework of the Landauer-Büttiker formalism (137), it is necessary to find its S-matrix using NCF (92). One can do it based on S-matrices of network junctions (Fig. 4) taking into account the specificity of its structure (156). At the same time, we assume that local coordinate frame at junctions are related to each other by rotation and translation (Fig. 5).
We find the S-matrices of Q-and Y-junctions by direct numerical calculation using the ND-map (63), (66) method based on their triangulation. In comparison with DN-map, the ND-map method is more preferable, since in this case the unitary condition (118) is better met for the current scattering matrix C ++ .
Let us find the S-matrix of the I-junction in analytical form using SBC. Structurally, the I-junction is a section of a two-dimensional electron waveguide with applied electric field (Fig. 6). Without loss of generality, consider the I-junction with the potential The transverse linear potential (159) is a rough approximation to the real one. However, it implements the main feature of the control element: the presence of inhomogeneity in it in the form of a variable electric field. At the same time, in experiments for three-terminal ballistic junctions, it was found that their high-temperature electrical properties are qualitatively insensitive to the details of their structure [4]. Therefore, the potential (159) can be used to search for the features of the transport properties of the network that will be present in the experiment. In addition, in this case, one can organize the calculation of the S-matrix of the Ijunction, exceeding in speed by 2-3 orders of magnitude the general algorithm proposed in the Appendix B.2.
We find the S-matrix of the I-junction based on the matrix G ♦ (56), (58). Taking into account expressions (13), (159), (156) and L = B, we will use the following dimensionless quantities: Then the electron scattering problem (57) in the I-junction Ω = (0, a) × (0, 1) when {β k = (0, 1)} k=1,2 takes the form Proposition 13. The operator G ♦ for the scattering problem in the I-junction (161) is written as where coefficients c 1 ⊥m and c 2 ⊥m are the solution of the system of equations PROOF 13. We are looking for a function Ψ in the form: where h m is solution of a problem on eigenfunctions and eigenvalues The solution of the problem (168) is a linear combination of Airy functions (163). Using the boundary conditions in the problem (168), from the expression (163) we obtain the system of equations (164). The system (164) allows to find unknown coefficients in the expression (163), and the condition of its solvability (equality to zero of the determinant of the matrix) will give the equation for λ m (165). Substituting function (167) to the problem (161), we get The solution to the problem (169) is where functions g ♦1 m and g ♦2 m are defined according to expressions (166). From expression (167) taking into account equalities (170) and (153) for the function in the junction one can obtain an expression of the form (58), where The definition (171) is written as By choosing the universal location of the LCF in the network (Fig. 5) for the I-junction we get Based on expressions (172) and (173) we write the operator G ♦ in matrix form When LCFs are related to each other by rotation and translation, transverse functions in the branches when transitioning to GCF (Fig. 6) are transformed as Taking into account equalities (152) and (153) from formulas (175) we get From expressions (174)-(176) we have statement (162).
The formulas (162) and (56) allow a high speed calculation of the S-matrix of the I-junction with transverse potential (161).

Geometry and parameters
As an example, we model a smooth branched nanostructure using a QIY-network (Fig. 7) Ω ex := Ω [0, 4,5] . Its structure (155) is written in the form We consider a system based on a two-dimensional electron gas formed in a heterostructure based on GaAs (m e = 0.063m 0 ) [30]. In addition to the dimensional parameters of the network (157), we assume for the nanostructure According to equalities (157) and (178) linear sizes of the nanostructure < 60 nm (Fig. 7), and the free length of electrons ℓ e at room temperature for it ∼ 100 nm [2]. Therefore, up to room temperatures, ballistic electron transport dominates in the nanostructure, and the proposed scheme can be used to calculate it.
According to the assumed width of the branches (149) and (157), taking into account the equalities (152) and (127) for the dimensional energies of the channels we have: The distance between energies is more than thermal broadening of levels k 0 T at room temperature: 0.026 eV, which indicates their good resolution. For the range of values of the Fermi level of the nanostructure we choose E ex F ∈ (0, 10E = ⊥1 ). Such a range, on the one hand, allows to consider boundary cases E , on the other hand, excludes from consideration higher channels that are difficult to observe in a real experiment. Fermi levels in reservoirs (external junctions) are determined based on external voltages applied in push-pull fashion: where U := U 04 = U 05 is the bias voltage (147). When solving problems about electron transport in low-dimensional systems, typical values of bias voltages ∼ 10 mV [3,4]. We choose a range for calculations U = 0 ÷ 100 mV. We assume that the temperatures of all reservoirs are the same (144). We consider the range T = = 0 ÷ 300 K. Finally,

Expression for S-matrix
According to the approach of sections 2 and 3, the electrical properties of the nanostructure (Fig. 7) are calculated based on its S-matrix. The S-matrix of the nanostructure is calculated using NCF (92). Let us write this procedure step by step. For a QIY network (155) of structure (177), the procedure for combining of junctions' S-matrices according to NCF (92) takes the form [2] ⊛S [1,3] ⊛S [0,1,2] ⊛S [3,4,5] =S [2,1,3]   Using this procedure, one can find an extended scattering matrix S [0, 4,5] of nanostructure (Fig. 7). In section 4.2.2, during matrix operations we consider 6 channels, since their further increase has little effect on the electrical properties calculated in section 4. In our numerical calculation, the written procedure is implemented automatically. It can be programmed for an arbitrary quantum network due to the agreements and notation (section 2.1) used in writing NCF (92).

Electrical properties
Based on the S-matrix of the nanostructure (Fig. 7) let us study its electrical properties (section 3.2.2). Calculations showed that in section 4.2.3 in the sums (137) and (146) it is enough to take into account only the first 4 of the channel, and one can take the energy of the 5-th channel for an upper limit of integration: At the same time, the lower limit of integration has the values of energies of the first 4 channels {λ = m } m=1..4 . We consider the approximate conductivity of the nanostructure (Fig. 7)  and its temperature T = (Fig. 8). Both graphs show that at T = = 0 K elementsσ [0,4,5]40 andσ [0,4,5]50 have many local extremes. According to expression (148), such conductivity behavior reveals the complex resonant structure of the system (Fig. 7). With rising temperatures, extremes are smoothed out. This is due to the fact that according to the equations (140) and (144), the parameter µ = grows, and in the expression (146), the peak of the integrand {cosh −2 ([ε ex F − ε]/[2µ = ])|ε ∈ R} widens. Physically, this means that with an increase in temperature, the resonant structure of the system (Fig. 7) on conduction graphs (Fig. 8) smoothed due to a blur in the energies of electrons involved in conduction.
In graphs (Fig. 8) it is also seen that as the number of "open" channels increases, the conductivity grows. However, this growth is not monotonous. Note also that at T = = 0 K the concept of "open" channels is conditional. This is due to a blur in the energies of electrons involved in conduction. In particular, therefore, at T = = 0 K a non-zero conductivity is observed in the "zero-channel" mode (ε ex F ≤ λ = 1 ). This effect is clearly visible in two-dimensional graphs of conductivity (Fig. 9). They also show the resonant structure of the system (Fig. 7), which is smoothed with increasing temperature.
On VAC of nanostructures ( Fig. 10) we see the grouping of curves in the "one-channel" and second boundary modes, in the "two-channel" and third boundary modes. Calculations have shown that this is not a threshold phenomenon: with an increase of Fermi level, curves, bending, gradually shift. Such an offset is not monotonous: at the same time, the curve can both rise and fall. This means that with an increase of Fermi levels, electric current, having a global upward trend, can locally decrease. Such behavior of VAC is observed at all temperatures within the range T = = 0 ÷ 300 K.
The dependency of VAC of the nanostructure (Fig. 10) from temperature has the following features. At T = = 0 K curves have many small corners that reveal the complex resonant structure of the system (Fig. 7). At the same time, the curves are strongly deformed. As the temperature increases, the corners smooth out and the VAC approaches linear. This suggests that as the temperature increases, the applicability of the expression for currents at low bias voltages (145) and, therefore, the expression for approximate conductivity (146) increases. At the same time, it becomes apparent that the expression for conductivity based only on the probabil- ity of transmission (148) (low temperatures and small bias voltages) has a very small domain of applicability. Finally, we note that in section 4.2.3, the discussion of the electrical properties of the nanostructure at T = ≈ 0 K is given purely to explain their general trends. To obtain in this case adequate results for the real system, an accurate calculation of the potential in it is necessary. This is due to the fact that even for the simplest systems, only their hightemperature electrical properties are qualitatively insensitive to the details of their structure [4].
Thus, from the study of approximate conductivity ( Fig. 8  and Fig. 9) and VAC (Fig. 10) of four-junctions nanostructure (Fig. 7) in section 4.2.3, we have the following conclusion. To correctly describe the electrical properties of a nanostructure only on the basis of its S-matrix, two conditions must be met: extremely low temperatures (T = ≈ 0 K) and high-precision calculation of the potential in it.

Conclusion
In this work, we considered the calculation of electron transport in branched semiconductor nanostructures. For this purpose, we developed and applied a model of a quantum network. The proposed calculation scheme is based on a special notation system that simplifies numerical implementation.
In the calculation of the S-matrix of the junction, preference is given to the method of scattering boundary conditions. For them, a clear integro-differential form was proposed, and the method of calculating the S-matrix of a junction in an explicit form has also been developed. To calculate the S-matrix of the entire network in terms of its junctions' Smatrices, a network combining formula was proposed. It explicitly takes into account all connections and is a universal calculation algorithm suitable for networks of an arbitrary structure. The calculation of electrical currents through the network performed using the Landauer-Büttiker formalism, adapted to the designation system used.
We demonstrated the calculation scheme proposed in the work by modeling a nanostructure based on twodimensional electron gas. For this purpose, we proposed a model of a network of smooth junctions with one, two and three adjacent branches. We calculated the electrical properties of such a network (using the example of GaAs), consisting of four junctions, depending on the temperature. We have found that the resonance structure of such a system manifests itself in its electrical properties only at extremely low temperatures (≈ 0 K). Under such conditions, it is possible to reduce their calculation to only the S-matrix of the system with a high-precision calculation of the potential in it. With increasing temperature, electron statistics should be taken into account. At the same time, in electrical properties, a significant smoothing of the resonant structure of the system occurs, and its effect on transport properties is averaged.
The prospects for the theme of this work are follows: application of scattering boundary conditions for selfconsistent calculation of currents; generalization in the case of semiconductors with the relativistic dispersion law of charge carriers, taking into account spin-orbital interaction; modeling of semiconductor nanostructures with predefined transport properties.

A Resonant and bound states
In section 2.3.1, we justified the clearness of the scattering boundary conditions, relying on a problem about resonant and bound states. In Appendix A we will show how it is formulated in terms of SBC.
Resonant and antiresonant, bound and antibond states are determined based on the poles and zeros of the S-matrix [32] (Table 3).
From definitions (36)-(39) we get Due to the criterion for the existence of non-zero solutions of homogeneous system of linear algebraic equations from statements (191) and (192) in GCF, it follows Then according to statements (193) and (194) taking into account expressions (16), (49) and (50) we have It means that the poles of the S-matrix are the eigenvalues of the problem with zero incoming SBC, the zeroes of the Smatrix are the eigenvalues of the problem with zero outgoing SBC. The problem (195) about resonant and bound states Ψ + describes a "emitter": a junction can emit electrons in the absence of incident ones on it. The problem (195) about antiresonant and antibound states Ψ − describes an "absorber": a junction can absorb electrons in the absence of scattered ones by it. Here, the terms "incident" and "scattered" are used conventionally for waves in branch-channels with complex κ ± (28).
The type of κ ± for the waves in the branch-channels of problems (195) depends on the type of eigenvalue ε ± corresponding to it. In turn, the type of ε ± is determined by the amount of isolation of the junction. Let us explain it as follows. The energy structure of the channels in the branches {λ k m } k m creates potential barriers of heights {min m λ k m } k on the boundaries with the junction. The barrier with the lowest height min k m λ k m provides the best connection with the environment. Depending on the type of junction and the value min k m λ k m , among the eigenvalues of the problems (195) there may be real numbers ε ± ≤ min k m λ k m -energies of bound and antibound states. Other eigenvalues with Im ε ± = 0 are interpreted as energies of resonant and antiresonant states ( Table 3).
The min k m λ k m -dependent isolation of a junction is easily interpreted in terms of SBC. With the growth of channels' energies (for example, due to a decrease in the cross section of the branches), the isolation of the junction grows. In particular, when min k m λ k m → ∞ the problem (195) takes the form Problem (196) describes a completely isolated junction. This limit transition is equivalent to the case when, in comparison with the operator K ± , the operator ∂ n becomes negligible. Therefore, in terms of SBC, the physical meaning of the operator K is isolation of junction.

B Scattering boundary conditions in quantum wire
In section 2.3.2, we indicated that in some cases, using the SBC, the S-matrix of the quantum network junction can be written in terms of the operator G ♦ (56). This is possible at least in the problem of an electron scattering in a quantum wire with a compact potential (a consequence of a defect, an external electric field, etc.). In the quantum network model, this problem is equivalent to the scattering problem in the I-junction.
In Appendix B, we will show how to use the scattering boundary conditions (42) to write an expression for the Smatrix of the I-junction in terms of the operator G ♦ (58). For this purpose, we will consider three-, two-, and onedimensional mathematical models of quantum wire.

B.1 Three-dimensional I-junction
The specificity of the quantum wire is that both branches (19) have cross sections β = {β k } k=1,2 equal to the cross section of the wire in GCF B = : Taking into account the property (198), the electron scattering problem (57) in the support of the compact potential (I-junction) Ω = (−a, +a) × B = takes the form: ∑ n −I mn ∂ 2 ∀x ∈ (−a, +a) The solution of system (205) can be written as: where g ♦1 , g ♦2 are matrices of the fundamental system of solutions (FSS) of problem (205), defined as (201).
The operator G ♦ in expression (207) completely complies with the definition (58). It can also be written in matrix form by the branches: The scalar products in expression (208) are determined by the cross section of the wire and the choice of LCFs in a particular problem. In the simplest case, when LCFs relates to each other by reflection relative to the plane Y Z, we havē W 1 h 1 n = h n =W 2 h 2 n , and the expression (208) takes the form (200).
FSS matrices {g ♦k } k=1,2 can be found numerically, for example, by solving Cauchy problems: Thus, the three-dimensional scattering problem on heterogeneity in quantum wire (199) is reduced to a system of ordinary differential equations. Having calculated the matrices of the fundamental system of solutions based on problems (209), one can write the operator G ♦ (207). With it, it is possible to find the decomposition coefficients for the function Ψ in expressions (204), (206) and the S-matrix of the I-junction using formulas (60) and (56), respectively.

B.2 Two-dimensional I-junction
The two-dimensional mathematical model of quantum wire is relevant for structures formed on the basis of twodimensional electron gas. As in the three-dimensional case, the specificity (198) for branches with cross sections in the form of intervals {β k = (0, b)} k=1,2 is preserved here. Also we choose GCF such that B = = (0, b). Then, taking into account the property (198), the electron scattering problem (57) in the support of the compact potential (I-junction) Ω = (−a, +a) × (0, b) takes the form: Proposition 15. The operator G ♦ for the scattering problem in the I-junction (210) is written as Subsequent computations dimensionally accurate are similar to those following decomposition (204). Therefore, the expressions (205)-(209) are also valid here. For two-dimensional systems, it is convenient to use LCFs relates to each other by rotation and translation (Fig. 5). Let us find the operator G ♦ (208) in this case.
Thus, the two-dimensional problem of scattering on inhomogeneity in the quantum wire (210) is reduced to a system of ordinary differential equations. Having calculated the ma-trices of the fundamental system of solutions based on problems (209), one can write the operator G ♦ (207). With it, it is possible to find the decomposition coefficients for the function Ψ in expressions (213), (206) and the S-matrix of the I-junction using formulas (60) and (56), respectively.

B.3 One-dimensional I-junction
The simplest mathematical model of quantum wire is a onedimensional model. It describes the motion of an electron in a single channel whose energy is assumed to be zero. Due to its specificity, this model has a small field of application and gives an idea of the transport properties of quantum wire at a qualitative level. For one-dimensional case S [1] = S [1,2] ⊛ S [2] (226) from formula (226) respectively. In both cases, we will shorten notation: κ 2 = √ ε =: κ, a 2 =: a. Since the values in definitions (230) and (231) are numbers, they commute among themselves. Therefore, formulas (227) and (229) taking into account definitions (73) are simplified: respectively. For a "branchless" network (90), taking into account the specifics of the notation, the expression (233) coincides with the expression obtained earlier [27, p. 49].
E Landauer-Büttiker formalism In section 3.2, we write the Landauer-Büttiker formalism in framework of the notation system used (section 2.1.2). In Appendix E we will give the mathematical justification for it.