CN106339562B - Energy interband non-local quantum tunneling analogy method with electric current conservation property - Google Patents
Energy interband non-local quantum tunneling analogy method with electric current conservation property Download PDFInfo
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Abstract
The present invention discloses a kind of energy interband non-local quantum tunneling analogy method with electric current conservation property, includes: discrete semiconductor devices physical region is generated using the nodal value increment of solved physical descriptor as the system of linear equations of variable；Handle non-local quantum tunneling, determine public energy section definition first while and when second；Point-to-point tunneling current density caused by interpolation and energy integral volume carry out weight distribution；The coefficient matrix of system of linear equations is divided into additional coefficient matrix caused by the discrete caused principal matrix of the usual differential equation and non-local quantum tunneling with electric current conservation property and stores；Coefficient matrix is solved using disappear former method of Gauss.The present invention guarantees non-local quantum tunneling electric current conservativeness, eliminates simulation error brought by non-conservation and uncertainty；Using principal matrix with the method that combines of auxiliary incidence matrix can the coefficient system of linear equations that linearizes of rapid solving Nonlinear System of Equations, improve solution efficiency.
Description
Technical field
The present invention relates to semiconductor device simulation methods, and in particular to a kind of non-office of energy interband with electric current conservation property
Domain quantum tunneling analogy method.
Background technique
Occur in the non-local quantum tunneling of different energy interbands to be a kind of very extensive physics of compound semiconductor device
Effect is common in a variety of semiconductor devices, such as multijunction solar cell, tunneling field-effect transistor etc..How accurate simulation this
One physical phenomenon is semiconductor devices physics and the content that numerical mathematics is concerned about always, it is common practice to point-to-point tunnelling electricity
Current density is converted into the generation compound term of various continuity equations.It is more commonly used that be is Kane Local Model (E.O.Kane, "
Zener tunneling in semiconductors",J.Phys.Chem.Solids,vol.12,pp.181-188,
1959), in this model, energy interband non-local quantum tunneling is processed into an only amount relevant to local electric field strength, uses
This processing mode is that software has CROSSLIGHT.Hurkx etc. further joined defect auxiliary tunnelling enhancement effect, by the effect
Should be showed (G.A.M.Hurkx, D.B.M.Klaassen and in the form of carrier lifetime attenuation coefficient
M.P.G.Knuvers,"Anew recombination model for device simulation including
Tunneling ", IEEE Trans.Electron Devices, vol.39, no.2, pp.331-338,1992), using the mould
The software of type has wxAMPS.The business softwares such as Silvaco, Synopsys use so-called non-local quantum tunneling model
(Ieong MK,Solomon PM,Laux SE,Wong HSP,Chidambarrao D.Comparison of raised and
Schottky source/drain MOSFETs using a novel tunneling contact
model.International Electron Devices Meeting SanFransisco,California,USA,
1998,733-736.), which handles tunnelling, the carrier tool on one side by the way of carrier tunnelling in quantum mechanics
Have certain probability be tunneled through potential barrier that material band gap is constituted to it is other on one side, tunnelling current size additionally depends on both sides load
The distribution situation of son is flowed, such non-local is distributed in the different point in space with current density caused by ship, which has applied
(M.Hermle, G.Letay, " Numerical simulation of tunnel diodes in multijunction solar cell
multi‐junction solar cells”,Progress in Photovoltaics Research and
Applications Vol.16,No.5,pp.409-418,2008)。
However above-mentioned non-local model processing when still point-to-point tunnelling current as carrier continuity side
Point in journey generates compound term to handle, and there is no the conservativeness for considering both sides totality tunnelling current, i.e., from one side quantum tunneling
Current density to other one side should be equal.Such as the current density meeting on above-mentioned non-local model both sides is found in practice
One times of difference is big, this brings certain error to device architecture numerical analysis.
Summary of the invention
The present invention provides a kind of energy interband non-local quantum tunneling analogy method with electric current conservation property, eliminates, non-
Simulation error brought by conservation and uncertain, raising solution efficiency.
To achieve the above object, the present invention provides a kind of energy interband non-local quantum tunneling mould with electric current conservation property
Quasi- method, its main feature is that, this method includes:
S1, discrete semiconductor devices physical region, using Numerical Methods of PDE, such as finite volume method, finite difference
Point-score and FInite Element etc., the discrete semiconductor differential equation and boundary condition are generated with the nodal value increment of solved physical descriptor
For the system of linear equations of variable；
S2, processing non-local quantum tunneling, determine public energy section definition first while and when second；First side determines
The point-to-point tunneling current density and integral energy list of corresponding discrete node non-local quantum tunneling in public energy section
Member；The carrier quasi-Fermi level on the second side is obtained using the discrete corresponding interpolation method of electric current；
Point-to-point tunneling current density caused by S3, interpolation and energy integral volume carry out weight distribution；
S4, system of linear equations coefficient matrix be divided into the discrete caused principal matrix of the usual differential equation and kept with electric current
Additional coefficient matrix caused by the non-local quantum tunneling of permanent characteristic simultaneously stores；
S5, Nonlinear System of Equations is solved using Gaussian elimination method.
Above-mentioned S1 includes:
Discrete semiconductor devices physical region, using Numerical Methods of PDE, such as finite volume method, finite difference calculus
With FInite Element etc., it is change that the discrete semiconductor differential equation and boundary condition, which are generated with the nodal value increment of solved physical descriptor,
The system of linear equations of amount；
In above-mentioned S2, the numerical model of non-local quantum tunneling is for node I, electronics, the net that energy integral volume is dE
Such as formula of current density caused by non-local quantum tunneling (1) at lattice point i:
Wherein, N is the band edge density of states, νthIt is the hot rate of carrier, T (E) is the tunnelling probability for the carrier that energy is E,Be energy be EiThe quasi- Fermi potential of both sides carrier, with quasi-Fermi level existRelationship.
In above-mentioned S2, when handling non-local quantum tunneling, define first first while and when second, be determined to that non-office occurs
The public energy section of domain quantum tunneling.
In above-mentioned S2, the first side determines the point of corresponding discrete node non-local quantum tunneling in public energy section
To current density and integral energy unit.
In above-mentioned S2, the carrier quasi-Fermi level on the second side is obtained using the discrete corresponding interpolation method of current density
, then by discrete node coefficient amendment and its derivative component be added on related point value corresponding to coefficient matrix；
Wherein, the current density obtained with hole is discrete such as formula (2):
In formula (2), 0 and 1 mark discrete direction on some grid line segment on first while with second node, μ be sky
Cave mobility, N are the hole band edge density of states, φp, EvIt is the quasi- Fermi potential, valence-band edge energy and electrostatic potential in hole, h respectively with V
It is line segment length, χpIt is Fermi-Dirac statistical distribution and Bose-Einstein statistical distribution ratio；
The quasi- Fermi potential and tunnelling energy of quantum tunneling occur for certain point in some grid cell that transformation for mula (2) obtains
Incidence relation such as formula (3):
In above-mentioned S3, second while according to first point-to-point tunneling current density and energy integral volume and the second side node
The overlapping degree of energy integral volume carries out weight distribution, by after distribution the first side energy integral volume and its derivative be added to and be
In the junction associated value of the second side of matrix number.
Above-mentioned principal matrix has three diagonal forms, and taking row bandwidth is the column storage form of 2*kl+ku+1.
Above-mentioned additional coefficient matrix storage includes:
A composite variable composed by integer variable and real variable is defined to store the position of additional association matrix element
It sets and numerical value；
One is defined by the row vector for being not fixed length that the complex data forms to store produced by additional association matrix
Except principal matrix member in addition to every row supplementary variable；
Increase the energy integral volume in direction for the coding of fixed edge i node, if containing multiple other one side energy
Integrate volume, the second side node k and equal energy Interpolation Nodes pairThere are associations, ifThen existWithTwo column are lower to introduce the added martix member being located at below diagonal element, line number fromTo k；If node k withOrPhase
Deng principal matrix covers additional association matrix, and additional association matrix only stores the element that principal matrix does not store, for upper biserial
Correlation form, additional association matrix only store additional two column matrix member numerical value and its column index；
The energy integral volume in direction is reduced for the coding of fixed edge i node, if containing multiple other one side energy
Integrate volume, the first side node k and equal energy Interpolation Nodes pairThere are associations, ifThen existWith
Two column on introduce be located at opposite angle matrix element above added martix member, line number from k toIf node k withOrIt is equal,
Principal matrix covers additional association matrix, and additional association matrix only stores the element that principal matrix does not store, and upper biserial is closed
Connection form, additional association matrix in addition to storing additional two column matrix member numerical value and its column index, also store fromArrive k's
Matrix element position, and it is initialized as 0.
Above-mentioned S5 includes:
Assuming that current column index is i；
Judge that the i-th column diagonal matrix member is following with the presence or absence of the matrix element bigger than opposite angle matrix element absolute value；
If then exchanging the row with the i-th row, the i-th column Gauss vector is then calculated, it is following for the column diagonal matrix member
The column vector that matrix element is formed divided by the later opposite angle matrix element of column selection pivot, such as formula (4):
Forward steps are that each matrix element subtracts Gaussian vectors element and Gauss vector corresponding to row where matrix element
The product of the column element of corresponding row, such as formula (5):
Compared to the prior art the present invention has the energy interband non-local quantum tunneling analogy method of electric current conservation property,
The advantage is that method proposed by the invention can guarantee the conservativeness of non-local quantum tunneling electric current, meet on quantum mechanics
All be the requirement of conservation in entire space about carrier tunnelling, thus eliminate simulation error brought by non-conservation with not really
It is qualitative；
Method proposed by the invention using principal matrix with the method that combines of auxiliary incidence matrix can rapid solving it is non-
The coefficient system of linear equations that system of linear equations linearizes, improves solution efficiency.
Detailed description of the invention
Fig. 1 is the flow chart for the energy interband non-local quantum tunneling analogy method that the present invention has electric current conservation property；
Fig. 2 is the flow chart of semiconductor device analog；
Fig. 3 is node variable Discrete geometry relational graph；
Fig. 4 is non-local quantum tunneling schematic diagram；
Fig. 5 is correspondence diagram of the tunnelling both sides spatial distribution of nodes on energy；
Fig. 6 is that the coding of fixed edge i node increases the schematic diagram in direction；
Fig. 7 is that the coding of fixed edge i node reduces the schematic diagram in direction；
Fig. 8 is the structural schematic diagram of embodiment two；
Fig. 9 is the structural schematic diagram of embodiment three.
Specific embodiment
Below in conjunction with attached drawing, specific embodiments of the present invention are further illustrated.
As shown in Figure 1, the invention discloses a kind of, the energy interband non-local quantum tunneling with electric current conservation property is simulated
The embodiment one of method, this method comprising the following steps:
S1, discrete semiconductor devices physical region, using Numerical Methods of PDE, such as finite volume method, finite difference
Point-score and FInite Element etc., the discrete semiconductor differential equation and boundary condition are generated with the nodal value increment of solved physical descriptor
For the system of linear equations of variable.
As shown in Fig. 2, the process of usually semiconductor device analog includes: being carried out first according to the geometry of device architecture
Grid generates.Secondly on mesh point according to partial differential equation discrete method by Poisson's equation, electrons and holes continuity equation
It is separated into using node variable as nonlinearity in parameters equation group, discrete method usually has finite difference, limited bulk and limited
Member etc., the generation compound term in continuity equation usually assume that on a certain small discrete section as constant.
As shown in figure 3, being certainly not limited to finite volume method here by taking finite volume method as an example.Assuming that the function on node i
Value is fi, and the space integral volume that definition belongs to node i is from left side interval midpoint to the section of the right interval midpoint, this section of area
Between geometric volume be xVol=0.5* (h1+h2).The energy integral volume that definition simultaneously belongs to node i is from left side interval midpoint
The difference of energy band value and the right interval midpoint energy band value, value are EVol=0.5* (Ei+1-Ei-1).If it is assumed that on the section
F all with fi come approximate, then be xVol*fi to whole contribution in this section of section.Using node variable as nonlinearity in parameters side
Journey group is usually solved in the method for iteration, and basic process is unfolded by appropriate stage-number using node Delta as the linear of coefficient
Equation group, the solution of system of linear equations usually have the direct elimination and iterative solution, and system of linear equations usually has some special
Structure, for example three is diagonal or block three is diagonal, can be by the Gaussian elimination method of some rapid solving algorithms such as column selection pivot
Deng.Continued to modify node variate-value according to obtained Delta, until node Delta or Nonlinear System of Equations function
Until value meets certain requirements.Result output and subsequent processing are carried out later.
S2, processing non-local quantum tunneling, determine public energy section definition first while and when second；First side determines
The point-to-point tunneling current density and integral energy list of corresponding discrete node non-local quantum tunneling in public energy section
Member；The carrier quasi-Fermi level on the second side is obtained using the discrete corresponding interpolation method of electric current.
S2.1, processing non-local quantum tunneling, define first first while and when second, be determined to that non-local quantum occurs
The public energy section of tunnelling.
The numerical model of non-local quantum tunneling for node I, energy integral volume be the electronics of dE, at mesh point i it is non-
Such as formula of current density caused by local quantum tunneling (1):
Wherein, N is the band edge density of states, νthIt is the hot rate of carrier, T (E) is the tunnelling probability for the carrier that energy is E,Be energy be EiThe quasi- Fermi potential of both sides carrier, with quasi-Fermi level existRelationship.Due to electricity
Current density is the integral function about carrier energy, according to our description above, also can be converted space integral function
Form.
As shown in figure 4, being non-local quantum tunneling schematic diagram, the line of bottom indicates the distribution of material both sides mesh point.It is non-
The basic physical process of local quantum tunneling, while also having indicated the both sides mesh point distribution situation of transmitting quantum tunneling, from Fig. 3
It is real since the discrete node in both sides may not be overlapped in energy if can be seen that using the first side (left side) discrete node as fixed point
This point is also impossible to accomplish in the operation of border, because carrier energy is also one of the variable for needing to solve, but can affirm
Ground is each node i (corresponding tunnelling ENERGY E qt) for certain one side of tunnelling occurs, and the energy of node i falls in other one side
WithWithIn the grid cell of position endpoint, i.e., all exist on one side etc. can Interpolation Nodes pair otherTherefore
Interpolation is just needed to obtain the quasi- Fermi potential of energy points carriers such as the second side (the right).
S2.2, the first side determine the point-to-point of corresponding discrete node non-local quantum tunneling in public energy section
Current density and integral energy unit.
S2.3, the second side carrier quasi-Fermi level obtained using the discrete corresponding interpolation method of current density, so
Afterwards by discrete node coefficient amendment and its derivative component be added on related point value corresponding to coefficient matrix.
The invention proposes a kind of inserting according to energy points carrier quasi-Fermi levels such as current density discrete way acquisitions
Value method, this method are obtained by converting the discrete scheme of carrier current density equation.Usual current density is in two nodes
Between using being considered fixed or linear change it is assumed that also being thought simultaneously with boundary values to be linear change, here it is
The discrete mechanism of Sharfetter-Gummel, the current density obtained with hole are discrete such as formula (2):
In formula (2), 0 and 1 mark discrete direction on some grid line segment on first while with second node, μ be sky
Cave mobility, N are the hole band edge density of states, φp, EvIt is the quasi- Fermi potential, valence-band edge energy and electrostatic potential in hole, h respectively with V
It is line segment length, χpIt is Fermi-Dirac statistical distribution and Bose-Einstein statistical distribution ratio.
If current density uses the discrete mechanism of Sharfetter-Gummel, the quasi- Fermi of carrier is actually just established
The variation relation of gesture, (both sides grid point 0 and 1) certain point occurrence quantity in some grid cell obtained by transformation for mula (2)
The quasi- Fermi potential of sub- tunnelling and the incidence relation such as formula (3) of tunnelling energy:
From formula (3) as can be seen that the quasi- Fermi potential of carrier is neither constant nor linear change, but it is presented one
Kind distribution relevant to grid endpoint value difference.
Point-to-point tunneling current density caused by S3, interpolation and energy integral volume carry out weight distribution.Second side root
According to first when point-to-point tunneling current density and energy integral volume are with second node energy integral volume overlapping degree into
Row weight distribution, by first after distribution when energy integral volume and its derivative are added to coefficient matrix the second junction associated value
On.
The invention proposes the weights of a kind of point-to-point tunnelling current with electric current conservation property and energy integral volume
Distribution method.In order to ensure both sides tunneling current density is equal, Σ is needediJ[Ei]EVoli=ΣkJ[Ek]EVolk, to meet
Formula can only fix point-to-point tunneling current density and energy integral volume on one side, the weight of volume is integrated according to both sides node energy
In addition folded degree progress is carrying out weight distribution on one side.
As shown in figure 5, if fixed left side tunneling current density, there are a kind of situation, the energy product of the multiple nodes in the right
Partial volume has been fallen in the energy integral volume of one, left side node, i.e. J [Ei]EVoliWith the multiple node EVol in the rightkOverlapping,
The weight for the left side i node tunneling current density that each k node is assigned on the right of in this way is J [Ei]EVoli∩EVolk.According to preceding
Face narration, the point-to-point tunneling current density J [E of left side node ii] contain the right interpolation pointThe quasi- Fermi of carrier
Gesture, in this way the right node k and nodeJust it is associated with.
S4, system of linear equations coefficient matrix be divided into the discrete caused principal matrix of the usual differential equation and kept with electric current
Additional coefficient matrix caused by the non-local quantum tunneling of permanent characteristic simultaneously stores.
The storage of coefficient matrix is divided into two parts, and a part is that the usual differential equation is discrete caused, referred to as principal matrix,
Usually there are three diagonal forms, another part is to claim caused by the non-local quantum tunneling with electric current conservation property
For additional coefficient matrix, usually there is row form or column form, but do not have both forms concurrently simultaneously, in order to improve speed, be
The solution of matrix number uses the modified Gaussian elimination method with column selection member.
Principal matrix storage: most of since energy interband non-local quantum tunneling generally only occurs in very little device area
System of linear equations coefficient matrix is the discrete caused principal matrix of usual carrier continuous renalreplacementtherapy, diagonal with three
Form, taking row bandwidth is the column storage form of 2*kl+ku+1, facilitates the Gaussian elimination method using column selection pivot.
The storage of additional coefficient matrix:
A composite variable composed by integer variable and real variable is defined to store the position of additional association matrix element
It sets and numerical value.
One is defined by the row vector for being not fixed length that the complex data forms to store produced by additional association matrix
Except principal matrix member in addition to every row supplementary variable.
Incidence matrix for being present in upper triangle and lower triangle is adopted according to the Gaussian elimination method basic principle of column selection pivot
Take relatively different storing modes.
As shown in fig. 6, increasing the energy integral volume in direction for the coding of fixed edge i node, if containing multiple another
Outer one side energy integral volume, the second side node k and equal energy Interpolation Nodes pairThere are associations, ifThen existWithTwo column are lower to introduce the added martix member being located at below diagonal element, line number fromTo k；If knot
Point k withOrEqual, principal matrix covers additional association matrix, and additional association matrix only stores principal matrix and do not store
Element, for upper biserial correlation form, additional association matrix only stores additional two column matrix member numerical value and its column index；
As shown in fig. 7, the energy integral volume in direction is reduced for the coding of fixed edge i node, if containing multiple another
Outer one side energy integral volume, the first side node k and equal energy Interpolation Nodes pairThere are associations, ifThen
?WithTwo column on introduce be located at opposite angle matrix element above added martix member, line number from k toIf node k withOrEqual, principal matrix covers additional association matrix, and additional association matrix only stores the element that principal matrix does not store, right
In upper biserial correlation form, additional association matrix in addition to storing additional two column matrix member numerical value and its column index, also store fromTo the matrix element position of k, and it is initialized as 0.
In view of the not plyability of left side node energy integral volume, additional association matrix element can only have row form or column
Form, but both forms cannot be had concurrently simultaneously.
S5, system of linear equations is solved using Gaussian elimination method.
The invention discloses a kind of Gaussian elimination method for solving coefficient matrix, this method combination principal matrix and added martix,
By the way of column selection pivot, rapid solving coefficient equation saves the calculation times for directlying adopt iterative algorithm.According to column selection
The Gaussian elimination method basic operation process of pivot:
Assuming that current column index is i.
Judge that the i-th column diagonal matrix member is following with the presence or absence of the matrix element bigger than opposite angle matrix element absolute value；
If then exchanging the row with the i-th row, the i-th column Gauss vector is then calculated, it is following for the column diagonal matrix member
The column vector that matrix element is formed divided by the later opposite angle matrix element of column selection pivot, such as formula (4):
Forward steps are that each matrix element subtracts Gaussian vectors element and Gauss vector corresponding to row where matrix element
The product of the column element of corresponding row, such as formula (5):
According to above-mentioned thought, the specific practice of the present embodiment is as follows:
A), for upper biserial correlation form, no matter column selection pivot or forward steps, will not all generate additional matrix
Member, the method for the present invention directly exchange and disappear member for this situation；
B), for lower biserial correlation form, no matter column selection pivot or forward steps, can all generate additional matrix
Member, the method for the present invention generated row index for this situation before this and areTo k+kl column indexTo k and each matrix element is institute in S4
The submatrix of the complex data of statement, directly exchanges and the member that disappears in the submatrix；
The present invention obtains the left side and the right quantum tunneling by fixed point-to-point tunnelling current and energy integral unit on one side
The conservation of electric current meets the requirement of quantum mechanics tunnelling.This method can be applied to be analyzed in the component values containing tunnel junctions, all
Such as tie efficient solar battery, electron electric power power device more.
As shown in figure 8, being the embodiment of the present invention two, by taking GaInP/GaAs Double Junction Tandem Solar Cells as an example comprising GaAs
Buffer layer 81, AlGaAs back surface field 82, GaAs active layer 83, AlInP Window layer 84, GaInP n++ doped layer 85, AlGaAs p++
Doped layer 86, AlGaInP back surface field 87, GaInP active layer 88, AlGaInP Window layer 89, GaAs cap layer 810.The structure uses
Low pressure metal organic chemical vapor deposition equipment is grown on N-shaped GaAs substrate, and interband non-local quantum tunneling occurs
Between GaInP n++ doped layer 85 and AlGaAs p++ doped layer 86, in practice it has proved that tunneling characteristics directly restrict entire device
Can, practical devices design can significantly be instructed and develop by accurately simulating tunneling characteristics.
As shown in figure 9, being the embodiment of the present invention three, by taking tunneling field-effect transistor (TFET) as an example comprising buffering
Layer 91, n-type doping layer 92, n++ doped layer 93, p++ doped layer 94.The structure passes through ion implanting or the method system of epitaxial growth
Standby, interband non-local quantum tunneling occurs between p++ doped layer 94 and n++ doped layer 93.Accurate simulation tunneling characteristics energy
It is enough that directive function is played with development to the design of practical devices structure
It is discussed in detail although the contents of the present invention have passed through above preferred embodiment, but it should be appreciated that above-mentioned
Description is not considered as limitation of the present invention.After those skilled in the art have read above content, for of the invention
A variety of modifications and substitutions all will be apparent.Therefore, protection scope of the present invention should be limited to the appended claims.
Claims (10)
1. a kind of energy interband non-local quantum tunneling analogy method with electric current conservation property, which is characterized in that this method packet
Contain:
S1, discrete semiconductor devices physical region, using Numerical Methods of PDE, such as finite volume method, finite difference calculus
With FInite Element etc., it is change that the discrete semiconductor differential equation and boundary condition, which are generated with the nodal value increment of solved physical descriptor,
The system of linear equations of amount；
S2, processing non-local quantum tunneling, determine public energy section definition first while and when second；First side is determined public
The point-to-point tunneling current density and integral energy unit of corresponding discrete node non-local quantum tunneling in energy section；The
The carrier quasi-Fermi level on two sides is obtained using the discrete corresponding interpolation method of electric current；
Point-to-point tunneling current density caused by S3, interpolation and energy integral volume carry out weight distribution；
S4, system of linear equations coefficient matrix be divided into the discrete caused principal matrix of the usual differential equation and have electric current conservation special
Additional coefficient matrix caused by the non-local quantum tunneling of property simultaneously stores；
S5, disappeared former method solving system system of linear equations using Gauss.
2. as described in claim 1 with the energy interband non-local quantum tunneling analogy method of electric current conservation property, feature
It is, the S1 includes:
Discrete semiconductor devices physical region generates grid according to semiconductor devices geometry；
According to common finite volume method, finite difference calculus and the FInite Element discrete semiconductor differential equation and boundary condition；
It generates simultaneously using the nodal value increment of solved physical descriptor as the system of linear equations of variable.
3. as described in claim 1 with the energy interband non-local quantum tunneling analogy method of electric current conservation property, feature
It is, in the S2, the numerical model of non-local quantum tunneling is for node I, electronics, the mesh point that energy integral volume is dE
Such as formula of current density caused by non-local quantum tunneling (1) at i:
Wherein, N is the band edge density of states, νthIt is the hot rate of carrier, T (E) is the tunnelling probability for the carrier that energy is E,
Be energy be EiThe quasi- Fermi potential of both sides carrier, with quasi-Fermi level existRelationship.
4. as described in claim 1 with the energy interband non-local quantum tunneling analogy method of electric current conservation property, feature
Be, in the S2, when handling non-local quantum tunneling, define first first while and when second, be determined to that non-local occurs
The public energy section of quantum tunneling.
5. as claimed in claim 4 with the energy interband non-local quantum tunneling analogy method of electric current conservation property, feature
It is, in the S2, the first side determines the point pair of corresponding discrete node non-local quantum tunneling in public energy section
Point current density and integral energy unit.
6. as claimed in claim 4 with the energy interband non-local quantum tunneling analogy method of electric current conservation property, feature
It is, in the S2, the carrier quasi-Fermi level on the second side is obtained using the discrete corresponding interpolation method of current density, so
Afterwards by discrete node coefficient amendment and its derivative component be added on related point value corresponding to coefficient matrix；
Wherein, the current density obtained with hole is discrete such as formula (2):
In formula (2), 0 and 1 mark discrete direction on some grid line segment on first while with second node, μ be that hole is moved
Shifting rate, N are the hole band edge density of states, φp, EvIt is the quasi- Fermi potential, valence-band edge energy and electrostatic potential in hole respectively with V, h is line
Segment length, χpIt is Fermi-Dirac statistical distribution and Bose-Einstein statistical distribution ratio；
The quasi- Fermi potential of quantum tunneling and being associated with for tunnelling energy occur for certain point in some grid cell that transformation for mula (2) obtains
Relationship such as formula (3):
7. as claimed in claim 4 with the energy interband non-local quantum tunneling analogy method of electric current conservation property, feature
Be, in the S3, second while according to first point-to-point tunneling current density and energy integral volume and the second side node energy
Amount integral volume overlapping degree carry out weight distribution, by after distribution the first side energy integral volume and its derivative be added to coefficient
In the junction associated value of the second side of matrix.
8. as described in claim 1 with the energy interband non-local quantum tunneling analogy method of electric current conservation property, feature
It is, the principal matrix has three diagonal forms, and taking row bandwidth is the column storage form of 2*kl+ku+1.
9. as claimed in claim 4 with the energy interband non-local quantum tunneling analogy method of electric current conservation property, feature
It is, the additional coefficient matrix storage includes:
Define a composite variable composed by integer variable and real variable store the position of additional association matrix element with
Numerical value；
One is defined to be removed caused by additional association matrix by the row vector for being not fixed length that the complex data forms to store
Every row supplementary variable outside principal matrix member；
Increase the energy integral volume in direction for the coding of fixed edge i node, if containing multiple other one side energy integrals
Volume, the second side node k and equal energy Interpolation Nodes pairThere are associations, ifThen existWith
Two column are lower to introduce the added martix member being located at below diagonal element, line number fromTo k；If node k withOrIt is equal, main square
Battle array covers additional association matrix, and additional association matrix only stores the element that principal matrix does not store, and is associated with shape for upper biserial
Formula, additional association matrix only store additional two column matrix member numerical value and its column index；
The energy integral volume in direction is reduced for the coding of fixed edge i node, if containing multiple other one side energy integrals
Volume, the first side node k and equal energy Interpolation Nodes pairThere are associations, ifThen existWithTwo column
It is upper introduce be located at opposite angle matrix element above added martix member, line number from k toIf node k withOrIt is equal, principal matrix
Additional association matrix is covered, additional association matrix only stores the element that principal matrix does not store, for upper biserial correlation form,
Additional association matrix in addition to storing additional two column matrix member numerical value and its column index, also store fromTo the matrix element of k
Position, and it is initialized as 0.
10. as claimed in claim 9 with the energy interband non-local quantum tunneling analogy method of electric current conservation property, feature
It is, the S5 includes:
Assuming that current column index is i；
Judge that the i-th column diagonal matrix member is following with the presence or absence of the matrix element bigger than opposite angle matrix element absolute value；
It is exchanged if then the matrix element bigger than opposite angle matrix element absolute value is expert at the i-th row, then calculates the i-th column Gauss arrow
Amount is the column following matrix element of diagonal matrix member divided by the column vector of the later opposite angle matrix element composition of column selection pivot, such as formula
(4):
Forward steps subtract Gaussian vectors element corresponding to the row where matrix element for each matrix element and Gauss vector is right
The product for the column element that should be gone, such as formula (5):
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