基于格林函数的纳米器件模型分析

1、基于格林函数的纳米器件模型分析一 The Greens function1. 定义A Green's function, G(x, s), of a linear differential operator L = L(x) acting on distributions over a subset of the Euclidean space Rn, at a point s, is any solution of    (1)where is the Dirac delta function. This prop

2、erty of a Green's function can be exploited to solve differential equations of the form    (2)As a side note, the Green's function as used in physics is usually defined with the opposite sign; that is, （3）If the operator is translation invariant, that is when L has consta

3、nt coefficients with respect to x, then the Green's function can be taken to be a convolution operator, that is, （4）In this case, the Green's function is the same as the impulse response of linear time-invariant system theory.2. 推算Loosely speaking, if such a function G can be found for the o

4、perator L, then if we multiply the equation (1) for the Green's function by f(s), and then perform an integration in the s variable, we obtain; （5）The right hand side is now given by the equation (2) to be equal to L u(x), thus: （6）Because the operator L = L(x) is linear and acts on th

5、e variable x alone (not on the variable of integration s), we can take the operator L outside of the integration on the right hand side, obtaining; （7）And this suggests;3. 格林函数在纳米器件中作用 （8） （9）对比方程（1），可知：带入方程（9），可知：其中， 这就是格林函数表示电荷密度和密度矩阵，其中A(E)是谱函数（spectral function）,谱函数A的物理意义：Indeed the di

6、agonal elements of A（E）/2*pi in the real space representation give us the local density of states at different points in space (a quantity that can be measured with scanning probe microscopy)4. 自能矩阵（self-energy）（1） 物理含义The concept of self-energy is used in many-body physics to describe electronelect

7、ron and electronphonon interactions. In the present context, however, we are using this concept to describe something much simpler, namely, the effect of a semi-innite contact.（2） 自能矩阵的推导在考虑了电极时，沟道的总哈密顿量为是电极对沟道的作用矩阵是电极的哈密顿矩阵总的格林函数为从上面的公式可得：其中电极作用的等价表示：This shows that the effect of the coupling to th

8、e reservoir can be accounted for by adding a self-energy matrix to the Hamiltonian H This is a very general concept that allows us to eliminate the huge reservoir and work solely within the device subspace whose dimensions are much smaller自能矩阵的求解：The indices m, n refer to points within the device wh

9、ile refer to points inside the reservoir.表面格林函数：the coupling matrix couples the points within the device to a small number of points on the surface ofthe reservoir, so that we only need for points that are on the surface. （）it should be noted that the periodic boundary conditions merely get rid of e

10、nd effects through the artifact of wrapping the device into a ring while the self-energy method treats the open boundary condition exactly. An open system has a continuous energy spectrum, while a ring has a discrete energy spectrum.It might appear that the self-energy method is just another method

11、for handling boundary effects.（3）自能矩阵的性质与推演Firstly, they are energy dependent.Secondly, they are not Hermitian.自能矩阵性质的影响The point we want to make is that the self-energy terms have two effects. One is to change the Hamiltonian from HL to which changes the eigenstates and their energies. But more imp

12、ortantly, it introduces an imaginary part to the energy determined by the broadening functions （扩展函数）and . The former represents a minor quantitative change（量变）; the latter represents a qualitative change （质变）with conceptual implications.H + has complex eigenvalues and the imaginary part of the eige

13、nvalues both broadens the density of states and gives theeigenstates a nite lifetime.扩展矩阵：We have often made use of the fact that we can simplify our description of a problem by using the eigenstates of the Hamiltonian H as our basis. For open systems we would want to use a representation that diago

14、nalizes H +in our energy range of interest.If the same representation also diagonalizes , then the problem could be viewed simply in terms of many one-level devices in parallel. 本征能量为 where are the corresponding diagonal elements respectively5. 态密度（density of states）和局域态密度(Local density of states)a

15、system with a set of eigenvalues has a density of states given by 从这个公式发现能级的态密度权重为1，但实际发现不同的能级态密度的权值不同，这主要是原来的态密度没有考虑空间态密度分布（spatial distribution of the states）,所以为了知道沟道的局域态密度，我们要乘入属于沟道波函数平方，即If we look at the local density of states in the channel we see a series of energy levels with varying heigh

16、ts, reecting the fraction of the squared wavefunction residing in the channel局域态密度定义：局域态密度更普通的概念：the diagonal element(divided by 2)of the spectral function A(E)同样的电荷密度是密度矩阵的对角元素因为 所以 可证明 二 相干传输（Coherent transport）1.传输系数（Transmission）传输函数：One could view the device as a “semi-permeable membrane” that

17、separates two reservoirs of electrons (source and drain) and the transmission function T (E)asa measure of the permeability of this membrane to electrons with energy E.传输模型：In the transmission formalism (sometimes referred to as the Landauer approach) the channel is assumed to be connected to the co

18、ntacts by two uniform leads that can be viewed as quantum wires with multiple modes or subbands having well-dened Ek relationships传输定理：This allows us to dene an S-matrix for the device analogous to a microwave waveguide where the element tnm of the t-matrix tells us the amplitude for an electron incident in mode m in lead 1 to transmit to a mode n in lead 2 两端器件：Th