Definition : material systems with length scale of ~ 1-100 nm in at least one dimension.
Classification :
2-D: quantum wells, atomic layers
1-D: quantum wires, nanowires
0-D: quantum dots, macro-molecules
Mesoscopic structures : Between individual atoms/molecules and macroscopic bulk materials.
Two approaches : bottom-up and top-down
Integrated circuits : various layers of semiconductor, insulators and metal interconnects
Unique Properties of Nanostructures :
Compact desity, high speed and efficiency
Quantum confinement:quantization and reducecd
dimensionality of electroic states
Quantum coherence
Surface/interface states
Meta-stable phases,adjustable atomic structure, size and shape - lead to properties tunable
Various functionalities from same material
Special properties quite different from bulks
Unique and novel functions
Sensitive
Plenty room at the bottom
Time-independent Schrödinger equation :
− h 2 2 m ∇ 2 ψ ( r ) + V ( r ) ψ ( r ) = E ψ ( r ) -\frac{\mathrm{h}^2}{2 m} \nabla^2 \psi(\boldsymbol{r})+V(\boldsymbol{r}) \psi(\boldsymbol{r})=E \psi(\boldsymbol{r})
− 2 m h 2 ∇ 2 ψ ( r ) + V ( r ) ψ ( r ) = E ψ ( r )
Free particle [with V ( r ) = 0 V(r)=0 V ( r ) = 0
Ψ ( r , t ) = A exp ( i k ⋅ r − i E t / ℏ ) \Psi(\boldsymbol{r}, t)=A \exp (i \mathbf{k} \cdot \boldsymbol{r}-i E t / \hbar)
Ψ ( r , t ) = A exp ( i k ⋅ r − i E t / ℏ )
Energy and momentum of the particle :
E = ℏ ω = ℏ 2 k 2 / ( 2 m ) = ℏ 2 ( k x 2 + k y 2 + k z 2 ) / ( 2 m ) = ε ( k ) p = ℏ k \begin{aligned}
& E=\hbar \omega=\hbar^2 k^2 /(2m)=\hbar^2\left(k_x^2+k_y^2+k_z^2\right) /(2m)=\varepsilon(k) \\
& \boldsymbol{p}=\hbar \boldsymbol{k}
\end{aligned}
E = ℏ ω = ℏ 2 k 2 / ( 2 m ) = ℏ 2 ( k x 2 + k y 2 + k z 2 ) / ( 2 m ) = ε ( k ) p = ℏ k
de Broglie wavelength : λ = h / p \lambda=h / p λ = h / p Probability of finding the particle at r \boldsymbol{r} r P ( r , t ) = ∣ Ψ ( r , t ) ∣ 2 \mathrm{P}(\boldsymbol{r}, t)=|\Psi(\boldsymbol{r}, t)|^2 P ( r , t ) = ∣ Ψ ( r , t ) ∣ 2
For a free particle, the probability is the same everywhe
1D potential well of infinite depth :
V ( x ) = { 0 , if 0 ≤ x ≤ a ∞ , otherwise V(x)= \begin{cases}0, & \text { if } 0 \leq \mathrm{x} \leq \mathrm{a} \\ \infty, & \text { otherwise }\end{cases}
V ( x ) = { 0 , ∞ , if 0 ≤ x ≤ a otherwise
ψ n ( x ) = { A n sin ( n π a x ) , for 0 ≤ x ≤ a 0 , otherwise ε n = h 2 k n 2 2 m = h 2 π 2 n 2 2 m a 2 = ε ( k n ) \psi_n(x)=\left\{\begin{array}{ll}A_{\mathrm{n}} \sin \left(\frac{n \pi}{a} \mathrm{x}\right), & \text { for } 0 \leq \mathrm{x} \leq \mathrm{a} \\ 0, & \text { otherwise }\end{array} \quad \varepsilon_n=\frac{\mathrm{h}^2 k_n^2}{2 m}=\frac{\mathrm{h}^2 \pi^2 n^2}{2 m a^2}=\varepsilon\left(k_n\right)\right. ψ n ( x ) = { A n sin ( a n π x ) , 0 , for 0 ≤ x ≤ a otherwise ε n = 2 m h 2 k n 2 = 2 m a 2 h 2 π 2 n 2 = ε ( k n ) n = 1 , 2 , 3 … \mathrm{n}=1,2,3 \ldots n = 1 , 2 , 3 …
Ground-state ( n = 1 \mathrm{n}=1 n = 1 = h 2 / ( 8 m a 2 ) =\mathrm{h}^2 /\left(8 \mathrm{~m} a^2\right) = h 2 / ( 8 m a 2 )
Potential wells of finite depth :
V ( x ) = { − V 0 for − a / 2 ≤ x ≤ a / 2 0 otherwise V(x)=\left\{\begin{array}{cc}
-\mathrm{V}_0 & \text { for }-a / 2 \leq \mathrm{x} \leq \mathrm{a} / 2 \\
0 & \text { otherwise }
\end{array}\right.
V ( x ) = { − V 0 0 for − a / 2 ≤ x ≤ a / 2 otherwise
For E < 0 E < 0 E < 0
For E > 0 E > 0 E > 0
Contribution to electric conductivity : Confind(localized) states are insulating, free states are conductive
Quantum well : particle confined by a 1-D potential well, but free in other 2-D, quantum states labeled by n , k x n, k_x n , k x k y k_y k y
ε ( n , k x , k y ) = h 2 2 m ( π 2 n 2 a 2 + k x 2 + k y 2 ) Each n represents a branch or subband \begin{aligned}
&\varepsilon\left(n, k_x, k_y\right)=\frac{\mathrm{h}^2}{2 m}\left(\frac{\pi^2 n^2}{a^2}+k_x^2+k_y^2\right)\\
&\text { Each } n \text { represents a branch or subband }
\end{aligned}
ε ( n , k x , k y ) = 2 m h 2 ( a 2 π 2 n 2 + k x 2 + k y 2 ) Each n represents a branch or subband
Quantum wire : particle confined by 2-D potential wells, free only in 1-D (1-D free particle), quantum states labeled by n 1 , n 2 n_1, n_2 n 1 , n 2 k z k_z k z
ε ( n 1 , n 2 , k z ) = h 2 2 m ( π 2 n 1 2 a 2 + π 2 n 2 2 b 2 + k z 2 ) \varepsilon\left(n_1, n_2, k_z\right)=\frac{h^2}{2 m}\left(\frac{\pi^2 n_1^2}{a^2}+\frac{\pi^2 n_2^2}{b^2}+k_z^2\right)
ε ( n 1 , n 2 , k z ) = 2 m h 2 ( a 2 π 2 n 1 2 + b 2 π 2 n 2 2 + k z 2 )
Quantum dot: particle confined by potential wells in 3-D, quantum states labeled n 1 , n 2 n_1, n_2 n 1 , n 2 n 3 n_3 n 3
ε ( n 1 , n 2 , n 3 ) = h 2 π 2 2 m ( n 1 2 a 2 + n 2 2 b 2 + n 3 2 c 2 ) \varepsilon\left(n_1, n_2, n_3\right)=\frac{h^2 \pi^2}{2 m}\left(\frac{n_1^2}{a^2}+\frac{n_2^2}{b^2}+\frac{n_3^2}{c^2}\right)
ε ( n 1 , n 2 , n 3 ) = 2 m h 2 π 2 ( a 2 n 1 2 + b 2 n 2 2 + c 2 n 3 2 )
Count states in k-space : Allowed states are separated by a spacing 2 π / L 2\pi/L 2 π / L
DOS in k k k N ( k ) : N 1 D ( k ) δ k = 2 L 2 π δ k N(k): \quad N_{1 D}(k) \delta k=2 \frac{L}{2 \pi} \delta k \quad N ( k ) : N 1 D ( k ) δ k = 2 2 π L δ k
For a unit length: n 1 D ( k ) = N 1 D ( k ) / L = 1 / π n_{1 \mathrm{D}}(k)=N_{1 \mathrm{D}}(k) / L=1 / \pi n 1 D ( k ) = N 1 D ( k ) / L = 1 / π L L L
n 1 D ( E ) = 2 n 1 D ( k ) / ( d ε / d k ) = 2 π / ( h 2 k m ) = 1 π h 2 m E n 1 D ( E ) diverges as E − 1 2 when E → 0 , van Hove singularity \begin{aligned}
&n_{1 \mathrm{D}}(E)=2 n_{1 \mathrm{D}}(k) /(d \varepsilon / d k)=\frac{2}{\pi} /\left(\frac{\mathrm{h}^2 k}{m}\right)=\frac{1}{\pi \mathrm{~h}} \sqrt{\frac{2 m}{E}}\\
&n_{1 D}(E) \text { diverges as } E^{-\frac{1}{2}} \text { when } E \rightarrow 0 \text {, van Hove singularity }
\end{aligned}
n 1 D ( E ) = 2 n 1 D ( k ) / ( d ε / d k ) = π 2 / ( m h 2 k ) = π h 1 E 2 m n 1 D ( E ) diverges as E − 2 1 when E → 0 , van Hove singularity
n 2 D ( E ) = m π h 2 n_{2 \mathrm{D}}(E)=\frac{m}{\pi h^2}
n 2 D ( E ) = π h 2 m
n 3 D ( E ) = m π 2 h 3 2 m E n_{3 \mathrm{D}}(E)=\frac{m}{\pi^2 \mathrm{~h}^3} \sqrt{2 m E}
n 3 D ( E ) = π 2 h 3 m 2 m E
Distributions of Electrons in Energy Space Density of States (DOS)
DOS for a 3-D system: n 3 D ( E ) = m π 2 h 3 2 m E \quad n_{3 \mathrm{D}}(E)=\frac{m}{\pi^2 \mathrm{~h}^3} \sqrt{2 m E} n 3 D ( E ) = π 2 h 3 m 2 m E n 2 D ( E ) = m π h 2 \quad n_{2 \mathrm{D}}(E)=\frac{m}{\pi \mathrm{~h}^2} \quad n 2 D ( E ) = π h 2 m n 1 D ( E ) = 1 π h 2 m E \quad n_{1 \mathrm{D}}(E)=\frac{1}{\pi \mathrm{~h}} \sqrt{\frac{2 m}{E}} \quad n 1 D ( E ) = π h 1 E 2 m E → 0 \mathrm{E} \rightarrow 0 E → 0
Quantum tunneling : A particle can be reflected by or tunnel through a barrier of V 0 > E V_0>E V 0 > E
Define:
k ≡ 2 m E / ℏ κ ≡ 2 m ( V 0 − E ) / ℏ \begin{gathered}
k \equiv \sqrt{2 m E} / \hbar \\
\kappa \equiv \sqrt{2 m\left(V_0-E\right)} / \hbar
\end{gathered}
k ≡ 2 m E / ℏ κ ≡ 2 m ( V 0 − E ) / ℏ
Tunneling probability : T ∼ exp [ − 2 h ∫ a b 2 m ( V ( x ) − E ) d x ] \quad T \sim \exp \left[-\frac{2}{\mathrm{~h}} \int_a^b \sqrt{2 m(V(x)-E)} d x\right] T ∼ exp [ − h 2 ∫ a b 2 m ( V ( x ) − E ) d x ]
For a thick or tall barrier, κ a ≫ 1 \kappa a \gg 1 κ a ≫ 1
T ≈ 16 k 2 κ 2 ( k 2 + κ 2 ) 2 exp ( − 2 κ a ) = 16 E ( V 0 − E ) V 0 2 exp [ − 2 a h 2 m ( V 0 − E ) ] T \approx \frac{16 k^2 \kappa^2}{\left(k^2+\kappa^2\right)^2} \exp (-2 \kappa a)=\frac{16 E\left(V_0-E\right)}{V_0^2} \exp \left[-\frac{2 a}{\mathrm{~h}} \sqrt{2 m\left(V_0-E\right)}\right]
T ≈ ( k 2 + κ 2 ) 2 1 6 k 2 κ 2 exp ( − 2 κ a ) = V 0 2 1 6 E ( V 0 − E ) exp [ − h 2 a 2 m ( V 0 − E ) ]
Electronic states in a perfect crystal: Bloch wave function
ψ k ( x ) = u k ( x ) exp ( i k x ) = exp ( i k x ) ∣ u ( k ) ⟩ with: u k ( x + a ) = u k ( x ) \psi_k(x)=u_k(x) \exp (i k x)=\exp (i k x)|u(k)\rangle \quad \text { with: } \quad u_k(x+a)=u_k(x)
ψ k ( x ) = u k ( x ) exp ( i k x ) = exp ( i k x ) ∣ u ( k ) ⟩ with: u k ( x + a ) = u k ( x )
Periodically modulated plane wave, h k = \mathrm{h} \boldsymbol{k}= h k = ψ k ( x ) & E ( k ) \psi_k(x) \& E(k) ψ k ( x ) & E ( k ) k k k 2 π / a 2 \pi / a 2 π / a
ε ( k + G n ) = ε ( k ) where G n = 2 π n / a \varepsilon\left(k+G_n\right)=\varepsilon(k) \quad \text { where } \quad G_n=2 \pi n / a
ε ( k + G n ) = ε ( k ) where G n = 2 π n / a
Effect of periodic potential is most dramatic on states of k = G n / 2 k = G_n/2 k = G n / 2
Group velocity (in real space):
v g ( k ) = ( ℏ ) − 1 d ε / d k v_g(k)=(\hbar)^{-1} d \varepsilon / d k
v g ( k ) = ( ℏ ) − 1 d ε / d k
Movement in k-space under applied force F F F
ℏ d k / d t = F , or d k / d t = F / ℏ \hbar d k / d t=F, \text { or } \quad d k / d t=F / \hbar
ℏ d k / d t = F , or d k / d t = F / ℏ
Acceleration of electron (in real space):
d v g d t = 1 ℏ d 2 ε d k d t = 1 ℏ d 2 ε d k 2 d k d t = 1 ℏ 2 d 2 ε d k 2 F = F m ∗ \frac{d v_g}{d t}=\frac{1}{\hbar} \frac{d^2 \varepsilon}{d k d t}=\frac{1}{\hbar} \frac{d^2 \varepsilon}{d k^2} \frac{d k}{d t}=\frac{1}{\hbar^2} \frac{d^2 \varepsilon}{d k^2} F=\frac{F}{m^*}
d t d v g = ℏ 1 d k d t d 2 ε = ℏ 1 d k 2 d 2 ε d t d k = ℏ 2 1 d k 2 d 2 ε F = m ∗ F
Effective mass of electrons m ∗ m^* m ∗
m ∗ ( k ) = ℏ ( d v g / d k ) − 1 = ℏ 2 ( d 2 ε / d k 2 ) − 1 m^*(k)=\hbar\left(d v_g / d k\right)^{-1}=\hbar^2\left(d^2 \varepsilon / d k^2\right)^{-1}
m ∗ ( k ) = ℏ ( d v g / d k ) − 1 = ℏ 2 ( d 2 ε / d k 2 ) − 1
In some cases, m ∗ m^* m ∗ → ∞ \rightarrow \infty → ∞
n ( E ) = 2 π ( d ε / d k ) − 1 n(E)=\frac{2}{\pi}(d \varepsilon / d k)^{-1}
n ( E ) = π 2 ( d ε / d k ) − 1
Movement in k-space under applied force F F F
ℏ d k / d t = F \hbar^{d \boldsymbol{k}} /{ }_{d t}=\boldsymbol{F}
ℏ d k / d t = F
In EM fields: ℏ d k / d t = − e ( E + r ˙ × B ) \quad \hbar^{d \boldsymbol{k}} /{ }_{d t}=-e(\boldsymbol{E}+\dot{\boldsymbol{r}} \times \boldsymbol{B}) ℏ d k / d t = − e ( E + r ˙ × B )
Group velocity (in real space):
v ( k ) = r ˙ c = 1 ℏ ∇ k ε ( k ) \boldsymbol{v}(\boldsymbol{k})=\dot{\boldsymbol{r}}_{\boldsymbol{c}}=\frac{1}{\hbar} \nabla_k \varepsilon(\boldsymbol{k})
v ( k ) = r ˙ c = ℏ 1 ∇ k ε ( k )
DOS (per unit volume):
n ( ε ) = k 2 π 2 ( d ε d k ) − 1 n(\varepsilon)=\frac{k^2}{\pi^2}\left(\frac{d \varepsilon}{d k}\right)^{-1}
n ( ε ) = π 2 k 2 ( d k d ε ) − 1
Rate of group velocity change:
d v g / d t = ( M ) − 1 F d \boldsymbol{v}_g /_{d t}=(\boldsymbol{M})^{-1} \boldsymbol{F}
d v g / d t = ( M ) − 1 F
Inverse Mass Tensor M − 1 \mathbf{M}^{\mathbf{- 1}} M − 1
M i j − 1 = ℏ − 2 d 2 ε / d k i d k j M_{i j}^{-1}=\hbar^{-2} d^2 \varepsilon / d k_i d k_j
M i j − 1 = ℏ − 2 d 2 ε / d k i d k j
Si: m ∗ / m 0 = 0.98 m^* / m_0=0.98 m ∗ / m 0 = 0 . 9 8 m ∗ / m 0 = 0.067 m^* / m_0=0.067 m ∗ / m 0 = 0 . 0 6 7
3-D crystals: unit cells identified by 3 primitive vectors a 1 , a 2 , a 3 \boldsymbol{a}_1, \boldsymbol{a}_2, \boldsymbol{a}_3 a 1 , a 2 , a 3 k \boldsymbol{k} k G \boldsymbol{G} G b 1 , b 2 , b 3 \boldsymbol{b}_1, \boldsymbol{b}_2, \boldsymbol{b}_3 b 1 , b 2 , b 3
where b 1 = 2 π a 2 × a 3 a 1 ∙ ( a 2 × a 3 ) \text { where } \quad b_1=2 \pi \frac{a_2 \times a_3}{a_1 \bullet\left(a_2 \times a_3\right)}
where b 1 = 2 π a 1 ∙ ( a 2 × a 3 ) a 2 × a 3
There are 7 crystal systems and 14 lattice types in 3-D.
Electrons : electrons in conduction band of density n n n Holes : empty states in valence band of density p p p
Carrier mobility: v = μ E , v=\mu \boldsymbol{E}, \quad v = μ E , μ = e τ m ∗ \quad \mu=\frac{e \tau}{m^*} μ = m ∗ e τ σ = e ( n μ n + p μ p ) \quad \sigma=e\left(n \mu_n+p \mu_p\right) σ = e ( n μ n + p μ p )
Total current density: sum of drift and diffusion terms
J = e n μ E + e D ∇ n ( assume n ≫ p ) \boldsymbol{J}=e n \mu \boldsymbol{E}+e D \nabla n \quad(\text { assume } n \gg p)
J = e n μ E + e D ∇ n ( assume n ≫ p )
Einstein relation:
μ = e D k B T \mu=\frac{e D}{k_B T}
μ = k B T e D
determined in Hall measurements
Longitudinal conductance:
J x = σ E x = e ( n μ e + p μ h ) E x \boldsymbol{J}_{\boldsymbol{x}}=\boldsymbol{\sigma} \boldsymbol{E}_{\boldsymbol{x}}=e\left(n \mu_{\mathrm{e}}+p \mu_{\mathrm{h}}\right) \boldsymbol{E}_{\boldsymbol{x}}
J x = σ E x = e ( n μ e + p μ h ) E x
Longitudinal resistivity: ρ = 1 / σ \rho=1 / \sigma ρ = 1 / σ
The Hall coefficient:
R H = E y J x B = p μ h 2 − n μ e 2 e ( p μ h + n μ e ) 2 R_H=\frac{E_y}{J_x B}=\frac{p \mu_h^2-n \mu_e^2}{e\left(p \mu_h+n \mu_e\right)^2}
R H = J x B E y = e ( p μ h + n μ e ) 2 p μ h 2 − n μ e 2
If electron is the dominant carrier in the material, then we have:
ρ = 1 / σ = ( e n μ e ) − 1 , and R H = − 1 n e \rho=1 / \sigma=\left(e n \mu_{\mathrm{e}}\right)^{-1}, \text { and } R_H=-\frac{1}{n e}
ρ = 1 / σ = ( e n μ e ) − 1 , and R H = − n e 1
Carrier density: n = − ( e R H ) − 1 n=-\left(e R_{\mathrm{H}}\right)^{-1} n = − ( e R H ) − 1 μ e = − R H / ρ \mu_{\mathrm{e}}=-R_{\mathrm{H}} / \rho μ e = − R H / ρ
Excitons : electron and hole bound together like a hydrogen atom, Hamiltonian of exciton
H = − h 2 2 m e ∗ ∇ e 2 − h 2 2 m h ∗ ∇ h 2 + V ( r e ) + V ( r h ) − e 2 4 π ε 0 ε r ∣ r e − r h ∣ \mathrm{H}=-\frac{\mathrm{h}^2}{2 \mathrm{~m}_{\mathrm{e}}^*} \nabla_{\mathrm{e}}^2-\frac{\mathrm{h}^2}{2 \mathrm{~m}_{\mathrm{h}}^*} \nabla_{\mathrm{h}}^2+\mathrm{V}\left(\mathrm{r}_{\mathrm{e}}\right)+\mathrm{V}\left(\mathrm{r}_{\mathrm{h}}\right)-\frac{\mathrm{e}^2}{4 \pi \varepsilon_0 \varepsilon_{\mathrm{r}}\left|\mathrm{r}_{\mathrm{e}}-\mathrm{r}_{\mathrm{h}}\right|}
H = − 2 m e ∗ h 2 ∇ e 2 − 2 m h ∗ h 2 ∇ h 2 + V ( r e ) + V ( r h ) − 4 π ε 0 ε r ∣ r e − r h ∣ e 2
Exciton energy:
E = E g + h 2 K 2 2 M − 13.6 μ n 2 m 0 ε r 2 ( e V ) E=E_g+\frac{\mathrm{h}^2 \mathrm{~K}^2}{2 M}-13.6 \frac{\mu}{\mathrm{n}^2 \mathrm{~m}_0 \varepsilon_r^2}(\mathrm{eV})
E = E g + 2 M h 2 K 2 − 1 3 . 6 n 2 m 0 ε r 2 μ ( e V )
where E g E_{\mathrm{g}} E g n = 1 , 2 , 3 … n=1,2,3 \ldots n = 1 , 2 , 3 …
M = m e ∗ + m h ∗ total mass; ℏ K center of mass momentum; μ = m e ∗ m h ∗ / ( m e ∗ + m h ∗ ) reduced mass; \begin{aligned}
M & =m_{\mathrm{e}} *+m_{\mathrm{h}} * \text { total mass; } \hbar \mathbf{K} \text { center of mass momentum; } \\
\mu & =m_{\mathrm{e}} * m_{\mathrm{h}} * /\left(m_{\mathrm{e}} *+m_{\mathrm{h}} *\right) \text { reduced mass; }
\end{aligned}
M μ = m e ∗ + m h ∗ total mass; ℏ K center of mass momentum; = m e ∗ m h ∗ / ( m e ∗ + m h ∗ ) reduced mass;
the last term, without "-" sign, is the exciton binding energy.
a_B=\frac{\varepsilon_r m_0}{\mu} a_0 \quad\left(a_0=0.529 \AA\right)
For semiconductors, a B a_B a B 10 − n m 10-\mathrm{nm} 1 0 − n m
Excitons in 2D materials
E ( 2 D ) = E g − 13.6 μ ( n − 1 / 2 ) 2 m 0 ε r 2 ( e V ) n = 1 , 2 , 3 , … E^{(2 D)}=E_g-13.6 \frac{\mu}{(\mathrm{n}-1 / 2)^2 \mathrm{~m}_0 \varepsilon_r^2}(\mathrm{eV}) \quad n=1,2,3, \ldots
E ( 2 D ) = E g − 1 3 . 6 ( n − 1 / 2 ) 2 m 0 ε r 2 μ ( e V ) n = 1 , 2 , 3 , …
Binding energy of excitons:
E b ( 2 D ) = 54 μ m 0 ε r 2 ( e V ) E_b^{(2 D)}=54 \frac{\mu}{\mathrm{~m}_0 \varepsilon_r^2}(\mathrm{eV})
E b ( 2 D ) = 5 4 m 0 ε r 2 μ ( e V )
The relative dielectric constant of a 2D semiconductor is much less than the value of its 3D bulk.
For example, for monolayer M o S 2 , ε r ≈ 4.2 \mathrm{MoS}_2, \varepsilon_{\mathrm{r}} \approx 4.2 M o S 2 , ε r ≈ 4 . 2 M o S 2 \mathrm{MoS}_2 M o S 2
Consequently, the exciton binding energy in a 2D semiconductor can be quite large.M o S 2 , E b ∼ 1 e V \mathrm{MoS}_2, E_{\mathrm{b}} \sim 1 \mathrm{eV} M o S 2 , E b ∼ 1 e V E b ∼ 0.1 e V E_{\mathrm{b}} \sim 0.1 \mathrm{eV} E b ∼ 0 . 1 e V
