The present disclosure generally relates to quantum structures, such as quantum dots and quantum wires. More specifically, the present disclosure is directed to systems and methods for quantum structure modeling, design, simulation, and manufacture.
Quantum structures, such as self-assembled quantum dots (QDs) have found an increasing range of applications in commercial and research settings, including optoelectronics, medical imaging, quantum computing, and so forth. For example, QDs have been used in light emitting diodes (LED), laser diodes, photodetectors, solar cells, displays, sensors, quantum bits (i.e. qubits), and so on. Due to crystal symmetry, self-assembled QDs tend to form highly symmetric shapes with the dot facet in a specific crystal plane, such as pyramids. With certain doping techniques, other symmetric shapes can also be produced, such as lenses and half-ellipsoids.
A quantum structures experiences strain effects that originate from the lattice mismatch between the quantum structure and a contacted matrix. Several approaches have been introduced to investigate strain-related effects on quantum structures, including atomistic models and analytical models. Atomistic models may provide accurate results, but they demand expensive computational resources and require a large number of difficult to obtain atomic input parameters.
Analytical models, on the other hand, are easier to implement and more computationally efficient. However, analytical approaches are limited in their range of application and accuracy. For example, it is almost impossible to address complicated geometric shapes and anisotropic effects analytically. Recently, it has been reported that analytical models can be calibrated based on accurate atomistic simulation results. However, such calibrated analytical models can only predict strain effects at the center of a QD, and not the entire strain profile in the QD and matrix.
Strain, stress, material displacement, and other factors, play important roles in the formation and ultimate properties of QDs (e.g., carrier confinement, optical properties). Therefore, there is a need for improved approaches that can efficiently and accurately characterize quantum structures to enable rational quantum structure design and fabrication.
The present disclosure provides a computational platform for the characterization, rational design, and fabrication for quantum structures. The presently disclosed platform overcomes the shortcomings of previous methods, providing improved accuracy and computational efficiency. Features and advantages of the present invention will become apparent from the following description.
In one aspect of the present disclosure, a computer-readable medium embedded with instructions executable by a processor of a computational platform is provided. The instructions include steps of receiving, using an input of a computational platform, parameter information corresponding to a quantum structure, and generating, using a processor of the computational platform, a quantum structure model based on the parameter information received. The instructions include steps of determining, using the processor, at least one property of the quantum structure, and generating a report indicative of the at least one property determined.
In another aspect of the present disclosure, a computational platform for a quantum structure is provided. The computational platform includes an input module configured to receive parameter information corresponding to a quantum structure, and a processor programmed to carry out instructions stored in a computer-readable medium. The instructions include receiving, using the input module, the parameter information corresponding to the quantum structure, and generating a quantum structure model based on the parameter information received. The instructions also include determining at least one property of the quantum structure, and generating a report indicative of the at least one property determined. The computational platform also includes an output module for providing the report.
In another aspect of the present disclosure, a method for the fabrication of quantum structure is provided. The method includes receiving, using an input of a computational platform, parameter information corresponding to a quantum structure and generating, using a Strain solver module, at least one mesh based on the parameter information received, obtaining, using the strain solver module, a displacement vector u and a strain tensor
The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee.
The present invention will hereafter be described with reference to the accompanying drawings, wherein like reference numerals denote like elements.
Provided herein is a computation platform for quantum structures, computer-readable medium embedded with instructions executable by a processor of a computational platform, and methods for using the platform for the fabrication of a quantum structure. The presently disclosed technology is capable of providing accurate and efficient methods that enable the rational design and fabrication of quantum structures.
Referring particularly to
In some implementations, the computational platform 100 may be any general-purpose computing system or device, such as a personal computer, workstation, cellular phone, smartphone, laptop, tablet, or the like. In this regard, the computational platform 100 may be a system designed to integrate a variety of software, hardware, capabilities, and functionalities. Alternatively, and by way of particular configurations and programming, the computational platform 100 may be a special-purpose system or device.
The computational platform 100 may operate autonomously or semi-autonomously based on user input, feedback, or instructions. In some implementations, the computational platform 100 may operate as part of, or in collaboration with, various computers, systems, devices, machines, mainframes, networks, and servers. For instance, as shown in FIG. 1, the computational platform 100 may communicate with one or more server 110 or database 112, by way of a wired or wireless connection. Optionally, the computational platform 100 may also communicate with various devices, hardware, and computers of an assembly line 114. For instance, the assembly line 114 may include various fabrication, processing, or process control systems, such as deposition systems, reactors, lasers, electrochemical systems, and so forth.
The I/O modules 102 of the computational platform 100 may include various input elements, such as a mouse, keyboard, touchpad, touchscreen, buttons, microphone, and the like, for receiving various selections and operational instructions from a user. The I/O modules 102 may also include various drives and receptacles, such as flash-drives, USB drives, CD/DVD drives, and other computer-readable medium receptacles, for receiving various data and information. To this end, I/O modules 102 may also include a number of communication ports and modules capable of providing communication via Ethernet, Bluetooth, or WiFi, to exchange data and information with various external computers, systems, devices, machines, mainframes, servers, networks, and the like. In addition, the I/O modules 102 may also include various output elements, such as displays, screens, speakers, LCDs, and others.
The processing unit(s) 104 may include any suitable hardware and components designed or capable of carrying out a variety of processing tasks, including steps implementing the present framework for quantum structure simulation. To do so, the processing unit(s) 104 may access or receive a variety of information, including parameter information corresponding to a quantum structure or quantum structure array/assembly, and generate a quantum structure model, as will be described. The parameter information may be stored or tabulated in the memory 106, in the storage server(s), in the database(s) 112, or elsewhere. In addition, such information may be provided by a user via the I/O modules 102, or selected based on user input.
In some configurations, the processing unit(s) 104 may include a programmable processor or combination of programmable processors, such as central processing units (CPUs), graphics processing units (GPUs), and the like. In some implementations, the processing unit(s) 104 may be configured to execute instructions stored in a non-transitory computer readable-media 116 of the memory 106. Although the non-transitory computer-readable media 116 is shown in
In some embodiments, a non-transitory computer-readable medium is embedded with, or includes, instructions for receiving, using an input of the computational platform 100, parameter information corresponding to a quantum structure, and generating, using a processor or processing unit(s) 104 of the computational platform 100, a quantum structure model based on the parameter information received. The medium may also include instructions for determining, using the processor or processing unit(s) 104, at least one property of the quantum structure, and generating a report indicative of the at least one property determined. In one non-limiting example, at least some of the instructions for carrying out methods, in accordance with the present disclosure, may be programmed into, executed using, or part of, a commercial software package, such as a COMSOL, or Matlab.
In some configurations, the processing unit(s) 104 may include one or more dedicated processing units or modules configured (e.g. hardwired, or pre-programmed) to carry out steps, in accordance with aspects of the present disclosure. For instance, as shown in
Solver modules of the processing unit(s) 104 may operate independently, or in cooperation with one another. In the latter case, the modules can exchange information and data, allowing for more efficient computation, and thereby improvement in the overall processing by the processing unit(s) 104. For instance, the Strain solver module 118 may generate a finite element method (FEM) mesh, according to accuracy requirements specified by a user via I/O modules 102, for example, and solve coupled partial differential equations to obtain displacement vectors, and strain tensors therefrom.
Strain calculation results (e.g. computed strain tensors) may then be provided to the Piezoelectric solver module 120, which may then use the strain calculation results to calculate piezoelectric information (e.g. charge distribution). In addition, the Strain solver module 118 may also provide the generated FEM mesh to the Piezoelectric solver module 120, which may then use it to calculate piezoelectric potential energy based on the Poisson equation.
Strain calculation results (e.g. computed strain tensors) and Piezoelectric solver results may then be provided to the Schrödinger solver module 122, which may then use the strain calculation results and Piezoelectric calculation results to obtain quantum mechanical information (e.g. Hamiltonians, wavefuctions, eigenstates, or eigen-energies). In addition, the Strain solver module 118 may also provide the generated FEM mesh to the Schrödinger solver module 122 or the Schrödinger solver module 122 may generate a second, different finite element method (FEM) mesh, according to accuracy requirements specified by a user via I/O modules 102, which may then use it to calculate quantum mechanical results.
As appreciated from the above, having specialized solver modules allows multiple calculations to be performed simultaneously or in substantial coordination, thereby increasing processing speed. In addition, sharing data and information between the different solver modules can prevent duplication of time-consuming processing and computations, thereby increasing overall processing efficiency.
In some implementations, the processing unit(s) 104 may also generate various instructions, design information, or control signals for designing or manufacturing a quantum structure, or quantum structure array/assembly, in accordance with computations performed. For example, based on computed deformations, strain, polarization charge distribution, and other information or quantities computed, the processing unit(s) 104 may identify and provide an optimal method for designing, manufacturing, or adapting design or manufacture of, a quantum structure, or quantum structure array/assembly.
The processing unit(s) 104 may also be configured to generate a report and provide it via the I/O modules 102. The report may be in any form and provide various information. For instance, the report may include various numerical values, text, graphs, maps, images, illustrations, and other renderings of information and data. In particular, the report may provide various strain, piezoelectric, and other information or properties generated by the processing unit(s) 104 for one or more quantum structures. The report may also include various instructions, design information, or control signals for manufacturing a quantum structure, or quantum structure array/assembly. To this end, the report may be provided to a user, or directed via the communication network 108 to an assembly line 114 or various hardware, computers or machines therein.
Referring now to
As shown, the process 200 may begin at process block 202 with receiving, using an input of a computational platform, various parameter information corresponding to a quantum structure, such as a QD as shown in
Based on the parameter information received, at process block 204, a quantum structure model may be generated using at least one processor of a computational platform 100. As will be described, the quantum structure model relies on a framework that includes the generation of a mesh representing the quantum structure suitably allowing one to obtain one or more quantum structure properties by solving a set of partial differential equations implementable using a finite element method (FEM). Using the quantum structure model, at least one property of the quantum structure may be determined, as indicated by process block 206. Exemplary properties of the quantum structure may include electrical properties, transport properties, optical properties, elastic properties, and other properties described herein.
In one non-limiting implementation of process blocks 202-206, material and geometric parameters for a quantum structure and surrounding matrix are provided. The parameters are then used to generate a quantum structure model that includes coupled partial differential equations. To solve the model, a FEM mesh is generated, according to accuracy requirements specified by a user. The model may then be solved using a Strain solver module to generate a displacement vector , which in turn may be used to calculate a strain tensor
The strain tensor
Strain and piezoelectric potential distributions from the Strain solver and piezoelectric solver may then be inputted into a Schrödinger (equation) solver. A new mesh may be generated, if necessary. The Schrödinger solver may then use the properties determined by the Strain solver and Piezoelectric solver to obtain quantum mechanical properties. Suitably, the Schrödinger solver may use k·p theory to obtain wavefunctions or states corresponding to the quantum structure, as well as associated energies, coefficients, and the like. When k·p theory is used, the Schrödinger solver may obtain a solutions for 8 coupled partial differential equations using FEM and evaluate symmetries of solutions using group theory. By evaluating the symmetries of the solutions, the Schrödinger solver may be used to identify spurious solutions or reduce the computational load.
Referring again to
The present framework will now be introduced for 3-dimensional QDs. Specifically, for a QD system with a QD embedded in a matrix, the strain tensor
In Eqn. (1), the superscript T denotes the transpose operation; the displacement vector
=ux+uy+uz
is from the originally matched position; and
where aM and aQ are lattice constants in the matrix and QD, respectively. From Eqn. (1), it may be shown that εij=εji. The stress tensor, which is also of rank 2, may be then calculated using the following equation:
or
σij=Cijklεkl (3)
In Eqn. (3), C(4) is the fourth rank elasticity tensor, “:” denotes the dyadic tensor product, and the Einstein summation notation has been used. For common semiconductor materials with a cubic crystalline lattice (e.g. InAs, and GaAs), the symmetric relations reduce the coefficients of C(4) to three independent terms, namely C11, C12 and C44. Therefore, using σij=σji, the stress tensor in Eqn. (3) can be simplified as follows:
In addition, one can write
where, the external force can be set to zero. Substituting Eqns. (5) and (2) into Eqn. (6), three coupled second-order partial differential equations may be obtained for . The boundary conditions at the interface between the QD and the matrix are
·
In Eqn. (7), is the normal directional unit vector of the interface. At the outer boundary of the matrix, far away from the QD, Dirichlet boundary can be applied, as follows:
|Outer Boundary=0. (8)
The displacement vector be then be obtained by solving the second order anisotropic partial differential equations using Eqns. (5) and (6) with boundary conditions from Eqns. (7) and (8). In turn, the strain tensor
δEhyC=ac(εxx+εyy+εzz), (9)
where ac is the conduction band hydrostatic deformation potential energy. For the valence band, both hydrostatic and biaxial deformations exist, and the biaxial deformation causes the heavy hole and light hole bands to split.
where av and bv are valence band hydrostatic and biaxial deformation energies, respectively. Due to strain-induced electric polarization, the piezoelectric effect can also modify the potential energy profile of the QD system. The resulting linear and quadratic polarization can then be expressed as:
linear=2e14(εyz+εxz+εxy)
quad=2B114(εxxεyz+εyyεxz+εzzεxy)+2B124[εyz(εyy+εzz)+εxz(εxx+εzz)+εxy(εxx+εyy)]+4B156(εxyεxz+εxyεyz+εxzεyz)
=linear+quad (11)
The polarization charge, piezoelectric potential, and resulting piezo-induced energy terms can then be written as:
−∇·=σP
−∇·ε0εr∇Vp=σP
δEPiezo=−eVP (12)
In Eqn. (12), σP is the induced polarization charge density; ε0 and εr are the vacuum and semiconductor relative permittivity constants, respectively. VP is the resulting piezo-potential and e is the elementary charge.
Finally, the deformed edges of the conduction and valence bands due to strain and piezoelectric effects can be written as
EC=EC0+δEhyC+δEPiezo
EHH=EV0−(δEhyV+δEbiV)+δEPiezo
ELH=EV0−(δEhyV−δEbiV)+δEPiezo
EC0=Eg+EV0
EV0=Ev,ave+⅓Δ (13)
In Eqn. (13), ECO and EVO are bulk material conduction band and valence band edge energies, respectively. A is the spin-orbit coupling energy, and Eg is the band gap of the bulk semiconductor material.
Unlike prior techniques reported, the above-described framework based on partial differential equations may be readily implemented using a standard FEM solver, and may be naturally reduced to 2D equations for quantum wires and 1D equations for quantum wells. Additionally, if an external force or stress is applied to the QD system, as common in sensor applications, for example, such external effects may be readily included using the force term of Eqn. (6).
Generally, carrier bound state energies and wavefunctions in a quantum dot (QD) are determined by the potential profile as a whole. Numerical continuum elasticity models, treating the QD and surrounding matrix as continuous materials, provide a good compromise between the computation complexity and accuracy. In contrast to atomistic models, numerical continuum elasticity models are much more efficient and provide results that are accurate enough for most applications. Moreover, unlike the atomistic models, the model inputs, semiconductor elasticity coefficients and band parameters, are easy to obtain.
Therefore, the present disclosure provides a framework based on a continuum elasticity model approach. Using a finite element method (FEM), the potential symmetry reduction of a QD system due to strain and piezoelectric effects is systematically investigated. Compared with other numerical approaches, such as finite volume methods, FEM is more versatile and suitable for various QD geometric shapes, and the boundary conditions can be naturally implemented and satisfied without any specific treatment. Additionally, a search for the minimum strain energy configuration is not required in the present framework.
By way of example, the following description illustrates how the present framework can be applied to compute strain and resulting band energy changes for various QD systems. Although the focus of the following is on pyramidal QDs, other QD shapes may be possible. As such, in some instances, comparisons are made with circular lens-shaped QDs or half-ellipsoidal QDs. These different QD shapes are shown in
As described below, strain deformation and piezoelectric potential profiles inside a QD need not be homogeneous. Specifically, results herein show that the minimum bandgap for a symmetric QD, such as a pyramidal QD, is found at the base or the top of the QD, not at its geometric center. Bandgap variations were also investigated for different QD shapes and sizes. In addition, for InAs/GaAs or other III-V group zinc-blende crystal quantum dots, it is shown that maximum symmetry group for the Hamiltonian is C2v double group. This group has two symmetric reflection planes: (110) and (110), which pass through the central axis of the symmetric QD, and one symmetric rotation of 180° along the central axis. Any QD with these three symmetric operations will reach the maximum C2v double group symmetry independent of the QD's original geometric symmetry group. However, if the geometric group only partially has or does not have these three symmetric operations, the Hamiltonian's symmetric group will be reduced to the subgroup of C2, double group or have no symmetry at all.
Referring now to
In some applications, it may be important to identify the potential profile change due to the strain and its consequence to the QD's symmetry reduction. Without strain effect, the potential energy symmetry of a QD system is determined by its geometric shape. For example, for the pyramidal QD shown
Referring now to
δEhyV=−av(εxx+εyy+εzz).
are shown. For conduction band, the symmetric profile of
δEhyC
is expected to be similar as appreciated from Eqns. (9) and (10). Specifically,
−δEhyV
profiles for the pyramid QD.
Referring particularly to
is shown for the pyramidal and lens shape QDs. The potential symmetric features are identical to the hydrostatic cases, i.e., C4v. However, in the vertical z direction, the biaxial deformation change is more dramatic. Specifically, the biaxial potential may have different signs within the QD. In the pyramid case, as shown in
−δEbiV
changes from being positive at the bottom of the pyramid, zero near middle of central axis, and negative at the top.
The polarization charge distribution in the pyramidal QD system is shown in
δEPiezo=−eVP
profiles (in eV) for a pyramid, a circular-based lens, and half-ellipsoid QDs. As appreciated from
With consideration to deformations, the transition energy from the edges of conduction band to heavy hole band (C-HH) and light hole band (C-LH) can be calculated as follows,
EC-HH=EC−EHH
EC-LH=EC−ELH (14)
In
EC-HH
isosurface of 0.75 eV, 0.95 eV, and 1.15 eV for a pyramidal QD is shown. It is observed that the bandgap between the conduction band to the heavy hole bands is smallest at the pyramidal base and largest at the top of the pyramid. For
EC-LH,
which is not shown in the figure, the situation is inverted, namely the largest band gap is around the base, and the smallest band gap is at the top.
The effects due to QD size and shape variations were also investigated. Specifically, the band gap changes were analyzed as a function of pyramidal QD shape and size. First, the pyramid height was fixed to a constant, namely h=5 nm, and the base side-length d was varied between 5 nm up to 12 nm, i.e., the aspect radio R=h/d varied from 1 to 0.42. As seen in
However, piezoelectric potential itself has no effect on the bandgap. Specifically,
The above described model and its numerical implementation provide a practical and efficient computation platform for the characterization and rational design of QWR materials.
Shear strain profiles and their symmetry were also determined for pyramidal QDs. Specifically, in the calculation, shear strain terms εxy, εxz, and εyz appear in the 4×4, 6×6 and 8×8 Hamiltonians of the k·p theory method, relating to the band mixing effect. As second order tensors, shear strain terms εxy, εxz, and εyz have intrinsic transformation properties. But it should be noted that the symmetric properties of these tensors rely on the quantum structure geometry and material properties. For example, the hydrostatic potentials in Eqns. (9) and (10) have C4v symmetry instead of spherical symmetry for the pyramid QD as demonstrated in
Herein, symmetry reduction of potential profile in the quantum structure and surrounding matrix was shown. Hydrostatic and biaxial deformations intrinsically displayed C4v symmetry because of semiconductor cubic crystal structure. Normally, these two deformations are localized within or near the QD. Piezoelectric effects, however, induced an accumulation of polarization charges along the edges of the pyramidal QD; and consequently, a long-range electric potential is built up, expanding far into the matrix. The piezoelectric potential had an intrinsic C2v symmetry with the following symmetric operations: two vertical reflection planes (110) and (1
However, for the half-ellipsoidal QD with geometric C2v symmetry as shown in
The present framework will now be introduced in 2 dimensions for QWRs. In
Any vector V, such as the material position r=(x, y, z)T, displacement u=(ux, uy, uz)T due to the strain, and the piezoelectric polarization P=(Px, Py, Pz)T, where the superscript T means transpose, is transformed between the two coordinate systems as follows,
The semiconductor strain is expressed as a rank two symmetric tensor in the following form
In Eqn. (16), aM and aQ are the lattice constants of the matrix and the QWR, respectively, and εMM0 is the intrinsic strain due to the lattice mismatch. The stress σij is also a rank two symmetric tensor. For these rank two tensors, they can be expressed as a 3×3 symmetric matrices,
The transformation of rank two tensors between the two coordinate systems can be expressed as
or in the matrix form
Because of symmetry, the strain and stress tensors only have six independent components. We rewrite Eqn. (19) as
The elastic constitutive relation in the (x, y, z) system,
where C(6x6) is the 6×6 elasticity matrix with three independent coefficients C11, C12 and C44 can be transformed into the (x′, y′, z) system by left multiplication of the {circumflex over (K)} on both sides of Eqn. (20). We obtain
Considering that the QWR physical properties have no x′ dependence in the (x′, y′, z) system, we have
Substituting Eqn. (23) into (22), we get the stress tensor in the (x′, y′, z) system
Finally, we obtain the coupled partial differential equations for the three dependent variables u′=(u′x′, u′y′, u′z) in the 2D QWR cross-sectional plane (y′, z) by applying
∇′·
where f′ is the external force density, which should be set to zero for our cases.
From Eqn. (20), it is easy to show that hydrostatic and biaxial deformations are invariant under z axis rotation. For the conduction band, we can write the band hydrostatic deformation as
δEhyC=aC(ε′x′x′+ε′y′y′+ε′zz), (26)
where ac is the conduction band hydrostatic deformation energy coefficient. For the valence band, the hydrostatic and the biaxial deformation energies can be written as
δEhyV=−av(ε′x′x′+ε′y′y′+ε′zz)
δEbiV=−bv/2(ε′x′x′+ε′y′y′−2ε′zz), (27)
where av and bv are the valance band hydrostatic and biaxial deformation energy coefficients, respectively.
From Eqns. (15) and (20), the linear and quadratic electric polarizations induced by the piezoelectric effects can be expressed in the (x′, y′, z) system as
In Eqn. (28), e14 is the linear piezo effect coefficient; and in Eqn. (29), B114, B156, and B124 are quadratic piezo effect coefficients. From the polarizations expressed in Eqns. (28) and (29), we can calculate the polarized charge density σ′P in the (y′, z) plane. The 2D Poisson equation may be solved to obtain the electric potential V′P and the piezoelectric energy change δEPiezo, as follows
−∇′·(P′linear+P′quad)=σ′P
−∇′·ε0εr∇′V′P=σ′P
δEPiezo=−eV′P, (30)
where ε0 and εr are the vacuum and semiconductor relative permittivity constants, respectively, and e is the elementary charge.
This model is readily implemented with a FEM with the following steps: (i). Solve the 2D (y′, z) coupled partial differential equations with Eqns. (24) and (25) to obtain the displacement vector u′=(u′x′, u′y′, u′z); (ii). Calculate the strain tensors as a function of y′ and z using Eqn. (9); (iii). Obtain the hydrostatic and biaxial strain energy changes δEhyC, δEhyv, and δEbiv by Eqns. (26) and (27), respectively; (iv). Calculate the piezo induced polarizations and charge density in the (y′, z) plane by Eqns. (28), (29) and (30); and (v). Use Eqn. (30) to solve the 2D Poisson equation to get the polarization induced potential V′p and then the piezoelectric energy change δEPiezo.
Using the model explained above, we systematically investigate the strain induced band profile variations as a function of the rotation angle θ. We scan θ from −90° to 90°. The QWR material is InAs; and the matrix material is GaAs. The material parameters used in the calculations are the same as those in Table 1. To consider the geometric effect of the QWRs, two types of QWRs are considered: one is the isosceles-triangle-based QWR with the cross-sectional triangle base length 10 nm along the y′ axis and the triangle height 10 nm along the z axis; the other is the cylindrical QWR with the circle radius of 10 nm in the (y′, z) cross-sectional plane.
Referring to
Referring to
Referring to
Because of hydrostatic, biaxial, and piezoelectric effects, the band gap of the QWRs will be greatly modified when compared with bulk materials. Referring to
The disclosed computational platform may use semiconductor k·p theory to obtain quantum mechanical information. The electronic properties in the crystal may be calculated by a perturbation method with semiconductor conduction and valence bands at Γ point as the basis. The size computational complexity is determined by the interactions included. If only the coupling among heavy-hole and light-hole bands is included, we have a 4-band model. Including of the spin-orbit interaction bands results in a 6-band model. Also including coupling between conduction and valence bands results in an 8-band model. Higher order models may also be evaluated.
The k·p model can reach the expected accuracy for a wide range of applications. Suitably, the 8-band model is very efficient and cost-effective compared with atomistic models and other k·p theory models that utilize a larger band number. Unlike the atomistic models, all input parameters have been extensively studied and well documented for k·p theory methods.
Because of the periodic structure of semiconductor crystals, self-assembled QDs tend to form highly symmetric shapes such as a pyramid or lens. Group theory has been applied to study QD symmetric properties and it is understood that the geometric symmetry of a QD is not identical to the symmetry of the Hamiltonian. For example, in the crystal space, the symmetric group of the pyramidal QD, as shown in
In k·p theory models, this C2v symmetry is obtained when the piezoelectric potential, resulting from strain-induced polarization, is considered. A point group element acts on the crystal space, which defines the geometric symmetry. This action or operation in the crystal space induces a unitary transformation in the wavefunction space or the Hilbert space, which forms a representation of the point group in the Hilbert space. The symmetric group of the Hamiltonian is the largest group whose elements all commute with the Hamiltonian in the Hilbert space. With the symmetric group of the Hamiltonian identified, one can investigate the electronic and optical properties of QD materials. This includes the ability to obtain the symmetry of wavefunctions and selection rules for optical transitions. Additionally, numerical simulations of QDs are very complicated, sometimes even producing spurious solutions. Group theory predictions can be a benchmark to test the correctness of the numerical simulation results and simplify the computations. For example, evaluating that the symmetric group of the Hamiltonian for the QDs in
In a quantum structure, spatial potential and electron spin are strongly coupled. Group elements, such as rotations and reflections, should act on both spatial and spin parts of the wavefunction, not just on the spatial part. Applying the 8-band k·p theory model to zinc-blende type QDs shown in
Turning to the 8×8 Hamiltonian, there are several equivalent forms with different bases. They can be transformed from each other by unitary transformations. For the convenience of our investigation, we start with the following basis
v=[|s⬆>,|X⬆>,|Y⬆>,|Z⬆>,|s⬇>,|X⬇>,|Y⬇>,|Z⬇>]T, (31)
where |s>corresponds to the conduction band s orbital state, and |X>, |Y>, and |Z>, describe the valence band p orbital states. The corresponding Hamiltonian is
Ĥ8×8=Ĥk+Vso+V0+Vstrain. (32)
In Eq. (32), is the kinetic energy part or k operator part of the Hamiltonian.
In Eq. (33), k is an operator defined as ki=kj=−∂/∂j. For convenience, the notation k instead of {circumflex over (k)} will be used. P0 is a material related parameter responsible for the coupling between conduction and valence bands. The other material related parameters are defined as
In Eq. (34), m0 is the free electron mass, mc is electron effective mass in the conduction band, Eg is the bulk semiconductor material band gap, Δ is the spin-orbit coupling energy, and γ1, 2, 3 are normalized Luttinger parameters.
In Eq. (32), Vso is the spin-orbit coupling potential. The associated Hermitian matrix is defined as
In Eq. (32), V0 is related to the bulk material band edge potential energy. It is a diagonal matrix defined as
where Ec and Ev are the bulk material conduction and valence band edge energies, respectively.
In Eq. (32), Vstrain is the potential energy matrix due to strain deformation and strain-induced piezoelectric effects. The strain-induced deformation part, Vdeform, is an 8×8 Hermitian matrix that can be obtained from the matrix by making the following replacements
where ac, av, b, and d are deformation potentials for the conduction and valence bands, and εij is the strain tensor. In addition to the above replacements, the piezoelectric potential effect, which is an 8×8 diagonal matrix, should be added to Vstram. We then have
Vstrain=Vdeform(εij)+δEpiezoI, (38)
where δEpiezo is the piezoelectric potential energy due to strain-induced polarization, and I is the 8×8 unity matrix. The strain tensor, εij, in Eq. (37) and piezoelectric potential, δEpiezo, in Eq. (38) can be calculated based on the QD material, the surrounding matrix material, and the QD geometric shape.
For the QDs shown in
Rz(θ)=e−iĵ
for
n=1,2,3,4. The piezoelectric potential part of the Hamiltonian, δEpiezoI, only commutates with the rotation operator θ=π, 2π. This in turn means that the 8×8 Hamiltonian commute with the same operators. Because of the coupling between spatial potential and spin, a full 2π rotation in the Hilbert space is not the unitary transformation. In other words, these operators do not form a new representation of the C2v group in the Hilbert space. A new element {circumflex over (N)}=(θ=2π) has to be introduced into the C2v group to create a new group called the C2v double group, or C2v(D) that has twice the group element of C2v.
V0, Vdeform(εij) and δEpiezoI depend on the geometric shape of the quantum structure. If the quantum structure has lower geometric symmetry or the orientation changes, the Hamiltonians symmetry may be reduced. Therefore, C2v(D) is the maximum symmetric group of the 8×8 Hamiltonian.
When the 8×8 Hamiltonian has C2v(D) symmetry, the eigenstates are two-fold degenerate. Any eigenfunction of the Hamiltonian should be the basis of one of the IRs of C2v(D). As shown in Table 2, C2v(D) has four 1D IRs: A1, A2, B1, and B2; and one 2D IR, E1/2. However, the eigenfunctions of the 8×8 Hamiltonian have to be in the 2D IR, E1/2. In other words, the eigenstate is two-fold degenerate. If there are accidental degeneracies, the eigenstate of the Hamiltonian can have 2n-fold degeneracy, where n=1, 2, . . .
These transformation invariances can be used as a benchmark to check the correctness of numerical calculations and to reduce the computational load on the computational platform.
In summary, based on a continuum elasticity model, a novel framework relying on a set of coupled partial differential equations (PDEs) was developed, as described. The present approach was used to investigate the changes of InAs/GaAs QD band edge due to strain and piezoelectric effects. Compared with other numerical approaches, such as finite volume or plane wave expansion methods, FEM is more efficient and versatile for various geometric shapes and boundary conditions.
The present invention has been described in terms of one or more preferred embodiments, and it should be appreciated that many equivalents, alternatives, variations, and modifications, aside from those expressly stated, are possible and within the scope of the invention.
This application claims benefit of priority to U.S. Provisional Application No. 62/748,986, filed Oct. 22, 2018, the contents of which is incorporated by reference in its entirety.
Number | Name | Date | Kind |
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20180336302 | Klimeck | Nov 2018 | A1 |
20190157393 | Roberts | May 2019 | A1 |
20210398018 | Dao | Dec 2021 | A1 |
20220019931 | Jiang | Jan 2022 | A1 |
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20200125986 A1 | Apr 2020 | US |
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