fix figure and eq

src/14_doping_and_devices.md

@@ -208,7 +208,7 @@ Previously we dealt with homogeneous materials, now the position coordinate (let
 We represent the properties of inhomogeneous materials using the **band diagram**.
 The main idea is to plot the dependence of various energies ($E_F$, bottom of conduction band $E_C$, top of the valence band $E_V$) as a function of position.
 
-So here is our problem for today:
+Let us build up the band diagram step by step:
 
 ```python
 
@@ -343,6 +343,45 @@ fig1.add_trace(go.Scatter(
         showlegend=False
     ))
 
+annot = [dict(
+    x = 0.85,
+    y = 0.75,
+    xref = "x", yref = "y",
+    axref = "x", ayref = "y",
+    ax = 0.85,
+    ay = 0.9,
+    showarrow=True,
+    arrowhead=3,
+    arrowsize=2,
+    arrowwidth=1,
+    arrowcolor='black'),
+         
+    dict(
+    x = 0.85,
+    y = 1,
+    xref = "x", yref = "y",
+    axref = "x", ayref = "y",
+    ax = 0.85,
+    ay = 0.849,
+    showarrow=True,
+    arrowhead=3,
+    arrowsize=2,
+    arrowwidth=1,
+    arrowcolor='black'),
+    
+    dict(
+    x = 0.8,
+    y = 0.9,
+    xref = "x", yref = "y",
+    axref = "x", ayref = "y",
+    text = r'$\varphi$',
+    font = dict(size=40),
+    showarrow=False)
+        ]
+         
+    
+         
+
 updatemenus=list([ 
     dict(
         type="buttons",
@@ -351,13 +390,13 @@ updatemenus=list([
         buttons=list([
             dict(label="n and p",
                  method="update",
-                 args=[{"visible": [True, True, True, True, True, True, False, False, False, False, False]}]),
+                 args=[{"visible": [True, True, True, True, True, True, False, False, False, False, False]}, {'annotations':[]}]),
             dict(label="Equilibrium",
                  method="update",
-                 args=[{"visible": [True, True, False, False, False, False, True, True, True, False, False]}]),
+                 args=[{"visible": [True, True, False, False, False, False, True, True, True, False, False]},  {'annotations':annot }]),
             dict(label="Band Bending",
                  method="update",
-                 args=[{"visible": [True, True, False, False, False, False, True, True, True, True, True]}]),
+                 args=[{"visible": [True, True, False, False, False, False, True, True, True, True, True]},  {'annotations':annot}]),
         ]),
     )
 ]
@@ -394,6 +433,21 @@ fig1.add_annotation(
     arrowwidth=0.1,
     arrowcolor='black')
 
+fig1.add_annotation(
+    x = -0.05,
+    y = 1.12,
+    xref = "x", yref = "y",
+    axref = "x", ayref = "y",
+    text = "",
+    ax = -0.05,
+    ay = 1.1,
+    showarrow=True,
+    arrowhead=3,
+    arrowsize=30,
+    arrowwidth=0.1,
+    arrowcolor='black')
+
+
 fig1.add_annotation(
     x = -0.05,
     y = 1.12,
@@ -432,23 +486,18 @@ fig1.add_annotation(
     axref = "x", ayref = "y",
     text = "n-region")
 
-fig1.update_layout(layout)
-py.plot(fig1)
-fig1.show()
-
 ```
 
-The main difference between $n$-type and $p$-type semiconductors is the location of the Fermi level $E_F$. 
+The main difference between $n$-type and $p$-type semiconductors is the location of the Fermi level $E_F$ (see "n and p" tab above). 
 The Fermi level of an $n$-type semiconductor is close to the donor states.
 On the other hand, the $p$-type semiconductor has its Fermi level near the acceptor states.
-At equilibrium (no external fields), we do not expect to see any currents in the system and therefore the **Fermi level $E_F$ must be constant** across the system.
+At equilibrium (no external fields), we do not expect to see any currents in the system and therefore the **Fermi level $E_F$ must be constant** across the system (see "Equilibrium" tab).
 To achieve a homogenous Fermi level, we could bring up in energy the $p$-type region or bring down the $n$-type region until the Fermi levels are aligned.
-The resulting band diagram is shown above.
 However, a question arises: what happens at the junction?
 
 We can understand the junction with a simple picture. 
 In physics, most of the time we expect things to change *continuously*.
-Therefore, we expect that the valance $E_V$ and conduction $E_C$ bands connect continuously in the middle region.
+Therefore, we expect that the valance $E_V$ and conduction $E_C$ bands connect continuously in the middle region as shown in the "Band Bending" tab.
 On the contrary, if the bands were to be discontinuous, then an electric field must develop at a single point in the middle region to shift the bands in energy.
 However, we do not expect such point-like electric fields to develop because electrons can move freely in semiconductors.
 
@@ -458,7 +507,7 @@ As a result, a positive ionized donor dopant is not screened anymore and the $n$
 Similarly, $p$-type region obtains a negative charge.
 Therefore, an electric field develops across the junction.
 As the recombination process continues, a larger charge density $\rho$ develops and thus the electric field grows until it is large enough to prevent the electrons/holes from crossing the junction.
-If the resulting potential difference becomes $\delta \varhpi \gg kT$, the density of electrons and holes drops exponentially fast.
+If the resulting potential difference becomes $\delta \varphi \gg kT$, the density of electrons and holes drops exponentially fast.
 Therefore, we refer to the region as the **depletion region**.
 
 A schematic of a depletion region is shown below: