Often, it is useful to know the number of states per unit energy, called the density of states (or DOS), D. In any number of dimensions, we can use

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1 Densit of States The hamiltonian for a electron in an isotropic electronic band of a material depends on the band-edge energ of E and an effective mass m ˆ pˆ H E m In an number of dimensions, the dispersion relation is E E m The electronic band is described as "parabolic". For a particular energ either of two values: E E, the wavenumber can be m E E E Assuming, the wavenumber changes with energ as d de m EE Often, it is useful to now the number of states per unit energ, called the densit of states (or DOS), D. In an number of dimensions, we can use d dn m dn D EE de d d where N is the number of states having wavevector with length less than than. This will depend on the number of macroscopic dimensions in which the electron is free. -D case The eigenfunctions in an macroscopic dimensions are plane waves. So in -D x A e ix

2 where an real value of is allowed. It is useful to assume periodic boundar conditions within a box of sie L, so that the wavefunctions in the macroscopic dimension x satisf xle il x x We then have e il, which tells us that n L n, with n. We can choose to normalie these wave functions over the sie of the box, i.e. n x L e inx L Apparentl, the separation in reciprocal space between adjacent states is n n L, so we have one state per L in reciprocal space. In -D, the number of states for which n is N D L L The number of states per unit length with wavevector less than is ND L Assuming, the number of states per unit length of wave vector is dnd L d So the DOS per unit length is m D D E EE, E E L, E E This can be simplified using the step function ; D DE m EE EE L -D case In a -D box with area S Lx L, the wave functions are

3 3 i,, e x x i e x x x Lx L The length squared of wave vector is origin of radius is, so the number of states within a circle about the x N D S 4 L x L The rate of increase of the number of states per unit area with increasing wavevector is dnd S d The DOS per unit area is D DE m EE S 3-D case In a 3-D box with volumev Lx L L, we can write the wavefunction as r V e i r The number of states inside a sphere of radius is N V 6 V 3D 3 3 The rate of increase of the number of states per unit volume with increasing wavevector is

4 4 dn3d V d The DOS per unit volume is 3 D3 DE m E E 3 EE V DOS for real-world nanostructures Real nanostructures exist in a 3-D world, so we will have quantum confinement in some dimensions and no confinement in others. We can describe these as follows. Quantum well In the real, 3-D world, a quantum well has one nanoscale dimension and two bul dimensions. Assuming particle-in-a-box confinement in the -direction onl, the wavenumbers in that direction will be limited to discrete values. The dispersion relation within this subband is E E n m where n edge n E. We can appl the analsis of the -D case above to each subband. The DOS above the n of each subband is constant. For example Dn E m EE n S So the total DOS for all subbands is DE m EE n n S Quantum wire A real quantum wire has two nanoscale dimensions and one bul dimension. Assuming the cross-section of the wire is rectangular, the eigenvalue problem is separable in cartesian coordinates. We can have particle-in-a-box confinement in x and directions, and -D dispersion in the direction, so that the dispersion within a subband is E E m nx, n nx, n We have defined nx, n n x x n nx, n. The previous -D analsis applies to each subband. So the total DOS per unit length of quantum wire is and DE m EE,,, n nx n E E nx n x n L Quantum dot A quantum dot has confinement in all three nanoscale dimensions, with ero bul dimensions. We would find solutions of the form

5 5 E E nx, n, n nx, n, n x. Since we have no macroscopic dimensions, there is no dispersion within these subbands, so that total DOS is where n, n, n 3 n, n, n D E E E Electron oncentration n,, x x n n To specif the electron concentration, we need to now not onl the DOS, but the probabilit that each state is occupied, as given b the fermi function. This can be approached from a microscopic perspective if we now all of the energ eigenstates. Assume some set of quantum numbers specifies each eigenstate, which have wavefunctions r and energ eigenvalues. Then nr r f where the initial factor of is for the possible spin states. We assume the energ eigenstates are formed from the electronic band described previousl. If the region of interest is macroscopic in one or more dimensions, we can appl periodic boundar conditions and, the allowed energies form a continuous spectrum. But if the region is nanoscopic, we will impose infinite barriers at the boundaries, causing the spectrum to become discrete. 3-D (bul) material Since we have no nanoscale dimensions, we can appl periodic boundar conditions in all dimensions. Thus, there is no phsical boundar, and the magnitude-squared of the wave function is constant r V We previousl found the number of states per unit radius of wavevector, per unit volume, so the sum of states becomes an integral dn d 3D d Now we have dn V d m 3D n d f E Specificall d n e e Defining E BT mbt, so mt B mt B and d B mt

6 6 We can write this as 4 d nn E BT e e We define the function 4 d 3D x x e e We have used the "effective" conduction-band DOS 3 m N N m where, at T.6 ev, B 3 mt B 9 3 N.5 cm One can condense further using where n f E 3D E f3de N 3D T B Quantum well In this case, we will have onl a -dependence on the magnitude squared of the wave function r S n We cannot eliminate the sum in n, but we can still replace the sums in x and Now dn d D d n dnd n n d f n E n S d m This gives Using d n n e e n E n BT mbt

7 7, so mt B We can write this as mt B mt and d B d 3 d n n N n En BT e e using the same definition of x D D x x ln e ln e If we also define x e e d e d N as in the 3-D case. Now we can define e e x e x 3 E fden D T we can further condense to n f E n n D n B Quantum-wire In this case, we will have both x and -dependence on the magnitude squared of the wave function r n,, x n x L We change the sum to an integral Now dn d D d nx, n dnd n,,, nx n x d f n E nx, n x n L d m which becomes d n n,, x n x En, n BT e e Using nx, n x mbt

8 8, so mt B mt B and d B mt d we can write this as 3 d n,,, nx n x N n x n E nx, n BT e e Let's define and Finall, d D x x e e 3 E fden D T nx, n B n x, f E nx, n D nx, n The functions f x D x, D x, and 3D x are plotted below. Whereas the fermi function is limited to the range f, the functions we have defined have no upper limits. 4 3 f F D F D F 3D x Quantum dot A quantum dot has confinement in all three nanoscale dimensions, with ero bul dimensions. Assuming all faces of the box are rectangular, the eigenvalue problem is separable in cartesian coordinates, and we would find solutions r for n, n, n 3. The equilibrium electron concentration is nx, n, n f x

9 9 r n, n, nr n, n, n n f E n,, x x x n n Non-degenerate cases The preceding analsis is sufficientl general that it would appl regardless of the relative position of the chemical potential with respect to the conduction-band edge. If the chemical potential lies close to the band edge or within the band, the electron concentration is called "degenerate". In semiconductors, we usuall have the situation where the chemical potential is well below the conduction band edge, such that E T. This case is called "non-degenerate". For the functions defined above, this corresponds to B x x x the limit where e e, so +e e e e, which allows simplification of the expressions we obtained. Bul material In the 3D case, 4 x 3D xe d e The integral is easil found as follows: Sa we want to now So I de I de The square of this is x I dxe de x x dx d e x We can switch to polar coordinates So I d drre I r dr r e r e r r r r I

10 Now we want to find I d We can observe that I I 4 e 3 For x, we have, so 3D n N E BT e x e x. In this limit 3D Quantum well The expression for the quantum well is easil simplified. For example, we can repeat the integral in this limit xe de e D x x Now we can write the electron concentration as 3 n N e En BT n n 3 E T N e n e n Quantum wire In this case, we again have B n BT D xe de e leading to x x 3 n x, N e Enx, n BT n, x, x n n n 3 E BT N e,, e n, n nx n x x nx, n BT