quantum mechanical results discussed in Chapter 2. From a practical device point of view, the formation of discrete quantum states in the GaAs layer of Fig. 3-13 changes the energy at which photons can be emitted. An electron on one of the discrete conduction band states (

E_{1}

in Fig. 3-13) can make a transition to an empty discrete valence band state in the GaAs quantum well (such as

E_{h}

), giving off a photon of energy

E_{g} + E_{1} + E_{h}

, greater than the GaAs band gap. Semiconductor lasers have been made in which such a quantum well is used to raise the energy of the transition from the infrared, typical of GaAs , to the red portion of the spectrum. We will see other examples of quantum wells in semiconductor devices in later chapters.
第 2 章討論的量子力學結果。從實際裝置的觀點來看，圖 3-13 中 GaAs 層中離散量子態的形成會改變光子可發射的能量。其中一個分離導帶狀態 (圖 3-13 中的

E_{1}

) 上的電子可以轉換到 GaAs 量子阱中的空分離價帶狀態 (例如

E_{h}

)，從而發出能量

E_{g} + E_{1} + E_{h}

的光子，其能量大於 GaAs 帶隙。半導體雷射器就是利用這樣的量子阱，將光譜中典型的 GaAs 紅外線轉換能量提升到紅光部分。我們將在後面的章節中看到量子阱在半導體裝置中的其他例子。

In calculating semiconductor electrical properties and analyzing device behavior, it is often necessary to know the number of charge carriers per

{cm}^{3}

in the material. The majority carrier concentration is usually obvious in heavily doped material, since one majority carrier is obtained for each impurity atom (for the standard doping impurities). The concentration of minority carriers is not obvious, however, nor is the temperature dependence of the carrier concentrations.
在計算半導體電氣特性和分析裝置行為時，通常需要知道材料中每個

{cm}^{3}

的電荷載子數量。在重度摻雜的材料中，大多數電荷載子的濃度通常很明顯，因為每個雜質原子都會得到一個大多數電荷載子 (對於標準摻雜雜質而言)。然而，少數載子的濃度並不明顯，載子濃度的溫度依賴性也不明顯。

To obtain equations for the carrier concentrations we must investigate the distribution of carriers over the available energy states. This type of distribution is not difficult to calculate, but the derivation requires some background in statistical methods. Since we are primarily concerned here with the application of these results to semiconductor materials and devices, we shall accept the distribution function as given.
為了得到載子濃度的方程式，我們必須研究載子在可用能態上的分佈。這類分佈並不難計算，但推導需要一些統計方法的背景。由於我們在此主要關心的是這些結果在半導體材料和裝置上的應用，因此我們會接受給定的分佈函數。

3.3.1 The Fermi Level 3.3.1 費米層次

Electrons in solids obey Fermi-Dirac statistics.

^{3}

In the development of this type of statistics, one must consider the indistinguishability of the electrons, their wave nature, and the Pauli exclusion principle. The rather simple result of these statistical arguments is that the distribution of electrons over a range of allowed energy levels at thermal equilibrium is
固體中的電子遵從 Fermi-Dirac 統計。

^{3}

在發展這類統計學的過程中，我們必須考慮電子的不可區分性、電子的波浪本質以及保利排除原理。這些統計論據相當簡單的結果是，在熱平衡時，電子在一系列允許能級上的分佈是

f (E) = \frac{1}{1 + e^{(E - E_{F}) / k T}}

where

k

is Boltzmann’s constant

(k = 8.62 \times 10^{- 5} eV / K = 1.38 \times 10^{- 23} J / K)

. The function

f (E)

, the Fermi-Dirac distribution function, gives the probability
其中

k

是波爾茲曼常量

(k = 8.62 \times 10^{- 5} eV / K = 1.38 \times 10^{- 23} J / K)

。函數

f (E)

，即費米-狄拉克分佈函數，給出概率

3.3

CARRIER CONCENTRATIONS 載體濃度
that an available energy state at

E

will be occupied by an electron at absolute temperature

T

. The quantity

E_{F}

is called the Fermi level, and it represents an important quantity in the analysis of semiconductor behavior. We notice that, for an energy

E

equal to the Fermi level energy

E_{F}

, the occupation probability is
在絕對溫度

T

下，

E

的可用能態會被電子佔據。

E_{F}

的量稱為費米電平，它代表了半導體行為分析中的一個重要量。我們注意到，對於能量

E

等於費米級能量

E_{F}

，佔用概率為

f (E_{F}) = {[1 + e^{(E_{F} - E_{F}) / k T}]}^{- 1} = \frac{1}{1 + 1} = \frac{1}{2}

Thus an energy state at the Fermi level has a probability of

1 / 2

of being occupied by an electron.
因此，費米級的能態被電子佔用的概率為

1 / 2

。

A closer examination of

f (E)

indicates that at 0 K the distribution takes the simple rectangular form shown in Fig. 3-14. With

T = 0

in the denominator of the exponent,

f (E)

1 / (1 + 0) = 1

when the exponent is negative

(E < E_{F})

, and is

1 / (1 + \infty) = 0

when the exponent is positive

(E > E_{F})

. This rectangular distribution implies that at 0 K every available energy state up to

E_{F}

is filled with electrons, and all states above

E_{F}

are empty.
仔細研究

f (E)

可以發現，在 0 K 時，分佈呈圖 3-14 所示的簡單矩形形式。指數的分母為

T = 0

，當指數為負

(E < E_{F})

時，

f (E)

就是

1 / (1 + 0) = 1

，當指數為正

(E > E_{F})

時，

1 / (1 + \infty) = 0

就是

(E > E_{F})

。這個矩形分佈意味著在 0 K 時，

E_{F}

以下的每個可用能態都充滿了電子，而

E_{F}

以上的所有能態都是空的。

At temperatures higher than 0 K , some probability exists for states above the Fermi level to be filled. For example, at

T = T_{1}

in Fig. 3-14 there is some probability

f (E)

that states above

E_{F}

are filled, and there is a corresponding probability

[1 - f (E)]

that states below

E_{F}

are empty. The Fermi function is symmetrical about

E_{F}

for all temperatures; that is, the probability

f (E_{F} + Δ E)

that a state

Δ E

above

E_{F}

is filled is the same as the probability

[1 - f (E_{F} - Δ E)]

that a state

Δ E

below

E_{F}

is empty. The symmetry of the distribution of empty and filled states about

E_{F}

makes the Fermi level a natural reference point in calculations of electron and hole concentrations in semiconductors.
在溫度高於 0 K 時，費米級以上的狀態有一定的概率會被填滿。例如，在圖 3-14 中的

T = T_{1}

處，有一定的概率

f (E)

顯示

E_{F}

以上的狀態是填滿的，並且有相對應的概率

[1 - f (E)]

顯示

E_{F}

以下的狀態是空的。在所有溫度下，費米函數都是對稱於

E_{F}

的；也就是說，在

E_{F}

以上的狀態

Δ E

為填充狀態的概率

f (E_{F} + Δ E)

與在

E_{F}

以下的狀態

Δ E

為空態的概率

[1 - f (E_{F} - Δ E)]

是相同的。關於

E_{F}

的空態和填充態分佈的對稱性，使得費米級成為計算半導體中電子和電洞濃度的自然參考點。

In applying the Fermi-Dirac distribution to semiconductors, we must recall that

f (E)

is the probability of occupancy of an available state at

E

. Thus if there is no available state at

E

(e.g., in the band gap of a semiconductor), there is no possibility of finding an electron there. We can best visualize the relation between

f (E)

and the band structure by turning the

f (E)

vs.

E

diagram on its side so that the

E

scale corresponds to the energies of the
在將 Fermi-Dirac 分佈應用於半導體時，我們必須記得

f (E)

是在

E

處的可用狀態的佔用概率。因此，如果

E

處沒有可用的狀態 (例如，在半導體的帶隙中)，就不可能在該處找到電子。我們可以將

f (E)

vs.

E

圖側轉過來，使

E

標度對應於

E

的能量，從而最直觀地看到

f (E)

與帶結構之間的關係。

Figure 3-14 圖 3-14
The Fermi-Dirac 費米狄拉克
distribution function. 分佈函數。

Figure 3-15 圖 3-15
The Fermi distribution function applied to semiconductors: (a) intrinsic material; (b) n-type material; © p-type material.
應用於半導體的費米分布函數：(a) 本質材料；(b) n 型材料；© p 型材料。
band diagram (Fig. 3-15). For intrinsic material we know that the concentration of holes in the valence band is equal to the concentration of electrons in the conduction band. Therefore, the Fermi level

E_{F}

must lie at the middle of the band gap in intrinsic material.

^{4}

Since

f (E)

is symmetrical about

E_{F}

, the electron probability “tail” of

f (E)

extending into the conduction band of Fig. 3-15a is symmetrical with the hole probability tail

[1 - f (E)]

in the valence band. The distribution function has values within the band gap between

E_{v}

and

E_{c}

, but there are no energy states available, and no electron occupancy results from

f (E)

in this range.
帶圖 (圖 3-15)。對於本質材料，我們知道價帶中的空穴濃度等於導帶中的電子濃度。因此，費米級

E_{F}

必須位於本質材料的帶隙中間。

^{4}

由於

f (E)

對

E_{F}

對稱，因此延伸至圖 3-15a 中導帶的

f (E)

電子概率「尾巴」與價帶中的空穴概率尾巴

[1 - f (E)]

對稱。分佈函數在

E_{v}

和

E_{c}

之間的帶隙內有值，但沒有可用的能態，在此範圍內的

f (E)

也不會產生電子佔用。

The tails in

f (E)

are exaggerated in Fig. 3-15 for illustrative purposes. Actually, the probability values at

E_{v}

and

E_{c}

are quite small for intrinsic material at reasonable temperatures. For example, in Si at

300 K, n_{i} = p_{i} \approx 10^{10} {cm}^{- 3}

, whereas the densities of available states at

E_{v}

and

E_{c}

are on the order of

10^{19} {cm}^{- 3}

. Thus the probability of occupancy

f (E)

for an individual state in the conduction band and the hole probability

[1 - f (E)]

for a state in the valence band are quite small. Because of the relatively large density of states in each band, small changes in

f (E)

can result in significant changes in carrier concentration.
圖 3-15 中誇大了

f (E)

中的尾部，以便說明。事實上，對於在合理溫度下的本質材料，

E_{v}

和

E_{c}

的概率值是相當小的。舉例來說，在矽中

300 K, n_{i} = p_{i} \approx 10^{10} {cm}^{- 3}

處，而

E_{v}

和

E_{c}

處的可用狀態密度則在

10^{19} {cm}^{- 3}

的數量級上。因此，導帶中的單個狀態的佔用概率

f (E)

和價帶中的狀態的空穴概率

[1 - f (E)]

都相當小。由於每個能帶中的狀態密度相對較大，因此

f (E)

的微小變化會導致載子濃度發生顯著的變化。

In n-type material there is a high concentration of electrons in the conduction band compared with the hole concentration in the valence band
在 n 型材料中，與價帶中的空穴濃度相比，導帶中的電子濃度較高。

(recall Fig. 3-12a). Thus in n-type material the distribution function

f (E)

must lie above its intrinsic position on the energy scale (Fig. 3-15b). Since

f (E)

retains its shape for a particular temperature, the larger concentration of electrons at

E_{c}

in n-type material implies a correspondingly smaller hole concentration at

E_{v}

. We notice that the value of

f (E)

for each energy level in the conduction band (and therefore the total electron concentration

n_{0}

) increases as

E_{F}

moves closer to

E_{c}

. Thus the energy difference

(E_{c} - E_{F})

gives a measure of

n

; we shall express this relation mathematically in the following section.
(回想圖 3-12a）。因此，在 n 型材料中，分佈函數

f (E)

必須位於其能階上的本質位置之上 (圖 3-15b)。由於

f (E)

在特定溫度下會保持其形狀，因此在 n 型材料中，

E_{c}

處的電子濃度較大，這意味著

E_{v}

處的空穴濃度相對較小。我們注意到，當

E_{F}

越接近

E_{c}

時，導帶中各個能級的

f (E)

值 (因此總電子濃度

n_{0}

)也會增加。因此，能量差

(E_{c} - E_{F})

提供了

n

的量度；我們將在下一節以數學方式表達此關係。

For p-type material the Fermi level lies near the valence band (Fig.3-15c) such that the

[1 - f (E)]

tail below

E_{v}

is larger than the

f (E)

tail above

E_{c}

. The value of

(E_{F} - E_{v})

indicates how strongly p-type the material is.
對於 p 型材料，費米級位於價帶附近 (圖 3-15c)，因此

[1 - f (E)]

低於

E_{v}

的尾部會大於

f (E)

高於

E_{c}

的尾部。

(E_{F} - E_{v})

的值表示材料的 p 型強度。

It is usually inconvenient to draw

f (E)

vs.

E

on every energy band diagram to indicate the electron and hole distributions. Therefore, it is common practice merely to indicate the position of

E_{F}

in band diagrams. This is sufficient information, since for a particular temperature the position of

E_{F}

implies the distributions in Fig. 3-15.
通常不方便在每個能帶圖上畫出

f (E)

對

E

來表示電子和空穴的分佈。因此，通常的做法是僅在能帶圖中指出

E_{F}

的位置。這是足夠的資訊，因為對於特定溫度，

E_{F}

的位置意味著圖 3-15 中的分佈。

3.3.2 Electron and Hole Concentrations at Equilibrium
3.3.2 平衡時的電子和電洞濃度

The Fermi distribution function can be used to calculate the concentrations of electrons and holes in a semiconductor, if the densities of available states in the valence and conduction bands are known. For example, the concentration of electrons in the conduction band is
如果知道價帶和導帶中可用狀態的密度，就可以使用費米分布函數來計算半導體中電子和空穴的濃度。例如，電子在導帶中的濃度為

n_{0} = \int_{E_{c}}^{\infty} f (E) N (E) d E

where

N (E) d E

is the density of states

({cm}^{- 3})

in the energy range

d E

. The subscript 0 used with the electron and hole concentration symbols

(n_{0}, p_{0})

indicates equilibrium conditions. The number of electrons per unit volume in the energy range

d E

is the product of the density of states and the probability of occupancy

f (E)

. Thus the total electron concentration is the integral over the entire conduction band, as in Eq. (3-12).

^{5}

The function

N (E)

can be calculated by using quantum mechanics and the Pauli exclusion principle (Appendix IV).
其中

N (E) d E

是能量範圍

d E

內的狀態密度

({cm}^{- 3})

。與電子和電洞濃度符號

(n_{0}, p_{0})

一起使用的下標 0 表示平衡條件。在能量範圍

d E

中，單位體積的電子數是狀態密度與佔用概率

f (E)

的乘積。因此，總電子濃度是整個導帶的積分，如式 (3-12)。

^{5}

函數

N (E)

可以使用量子力學和保利排除原理來計算 (附錄四)。

It is shown in Appendix IV that

N (E)

is proportional to

E^{1 / 2}

, so the density of states in the conduction band increases with electron energy. On the other hand, the Fermi function becomes extremely small for large energies. The result is that the product

f (E) N (E)

decreases rapidly above

E_{c}

, and very few electrons occupy energy states far above the conduction band edge. Similarly, the probability of finding an empty state (hole) in the valence band
附錄 IV 中顯示

N (E)

與

E^{1 / 2}

成正比，因此導帶中的狀態密度會隨著電子能量的增加而增加。另一方面，當能量很大時，費米函數會變得極小。其結果是，乘積

f (E) N (E)

在

E_{c}

以上迅速降低，只有極少數電子佔有遠高於導帶邊緣的能態。同樣地，在價帶中找到空態 (空穴) 的概率

[1-f(E)] decreases rapidly below

E_{v}

, and most holes occupy states near the top of the valence band. This effect is demonstrated in Fig. 3-16, which shows the density of available states, the Fermi function, and the resulting number of electrons and holes occupying available energy states in the conduction and valence bands at thermal equilibrium (i.e., with no excitations except thermal energy). For holes, increasing energy points down in Fig. 3-16, since the

E

scale refers to electron energy.
[1-f(E)] 會在

E_{v}

以下快速減少，而大部分的空穴會佔用接近價帶頂端的狀態。圖 3-16 顯示了可用狀態的密度、費米函數，以及熱平衡 (即除了熱能之外沒有激發) 時導帶和價帶中佔用可用能態的電子和空穴數目。對於空穴而言，由於

E

標度指的是電子能量，因此在圖 3-16 中，能量的增加點會向下。

The result of the integration of Eq. (3-12) is the same as that obtained if we represent all of the distributed electron states in the conduction band by an effective density of states

N_{c}

located at the conduction band edge

E_{c}

.
如果我們用位於導帶邊緣

E_{c}

的有效狀態密度

N_{c}

來表示導帶中的所有分散電子狀態，則公式 (3-12) 的積分結果與所得結果相同。

Figure 3-16 圖 3-16
Schematic band diagram, density of states, Fermi-Dirac distribution, and the carrier concentrations for (a) intrinsic, (b) n-type, and © p-type semiconductors at thermal equilibrium.
熱平衡狀態下 (a) 本質、(b) n 型及 © p 型半導體的示意帶圖、態密度、Fermi-Dirac 分佈及載流子濃度。

Therefore, the conduction band electron concentration is simply the effective density of states at

E_{c}

times the probability of occupancy at

E_{c}^{6}

因此，導帶電子濃度只是

E_{c}

處的有效態密度乘以

E_{c}^{6}

處的佔用概率。

n_{0} = N_{c} f (E_{c})

In this expression we assume the Fermi level

E_{F}

lies at least several

k T

below the conduction band. Then the exponential term is large compared with unity, and the Fermi function

f (E_{c})

can be simplified as
在此表達式中，我們假設費米電平

E_{F}

至少位於導帶下方數個

k T

的位置。那麼，指數項相較於 unity 會很大，費米函數

f (E_{c})

可以簡化為

f (E_{c}) = \frac{1}{1 + e^{(E_{c} - E_{F}) / k T}} ≃ e^{- (E_{c} - E_{F}) / k T}

Since

k T

at room temperature is only 0.026 eV , this is generally a good approximation. For this condition the concentration of electrons in the conduction band is
由於室溫下的

k T

只有 0.026 eV，因此這通常是一個很好的近似值。在此條件下，導帶中的電子濃度為

n_{0} = N_{c} e^{- (E_{c} - E_{F}) / k T}

The effective density of states

N_{c}

is shown in Appendix IV to be
附錄 IV 顯示有效的狀態密度

N_{c}

為

N_{c} = 2 {(\frac{2 π m_{n}^{*} k T}{h^{2}})}^{3 / 2}

Since the quantities in Eq. (3-16a) are known, values of

N_{c}

can be tabulated as a function of temperature. As Eq. (3-15) indicates, the electron concentration increases as

E_{F}

moves closer to the conduction band. This is the result we would predict from Fig. 3-15b.
由於公式 (3-16a) 中的量是已知的，因此

N_{c}

的值可以表列為溫度的函數。如公式 (3-15) 所示，電子濃度會隨著

E_{F}

靠近導帶而增加。這就是我們從圖 3-15b 所預測的結果。

In Eq. (3-16a),

m_{n}^{*}

is the density-of-states effective mass for electrons. To illustrate how it is obtained from the band curvature effective masses mentioned in Section 3.2.2, let us consider the 6 equivalent conduction band minima along the

X

-directions for Si. Looking at the cigar-shaped equienergy surfaces in Fig. 3-10b, we find that we have more than one band curvature to deal with in calculating effective masses. There is a longitudinal effective mass

m_{l}

along the major axis of the ellipsoid, and the transverse effective mass

m_{t}

along the two minor axes. Since we have

{(m_{n}^{*})}^{3 / 2}

appearing in the density-of-states expression Eq. (3-16a), by using dimensional equivalence and adding contributions from all 6 valleys, we get
在公式 (3-16a) 中，

m_{n}^{*}

是電子的狀態密度有效質量。為了說明如何從第 3.2.2 節提到的頻帶曲率有效質量得到它，讓我們考慮 Si 沿

X

方向的 6 個等效導帶最小值。觀察圖 3-10b 中的雪茄形等能表面，我們會發現在計算有效質量時，我們需要處理的帶曲度不只一個。沿著橢圓的主軸有縱向的有效質量

m_{l}

，沿著兩個次軸有橫向的有效質量

m_{t}

。由於我們有

{(m_{n}^{*})}^{3 / 2}

出現在狀態密度表達式 Eq. (3-16a)，使用尺寸等效並加上來自所有 6 個山谷的貢獻，我們得到

{(m_{n}^{*})}^{3 / 2} = 6 {(m_{l} m_{t}^{2})}^{1 / 2}

It can be seen that this is the geometric mean of the effective masses.
可以看出，這是有效質量的幾何平均值。

Calculate the density-of-states effective mass of electrons in Si.
計算 Si 中電子的態密度有效質量。

EXAMPLE 3-4 範例 3-4

For

Si, m_{l} = 0.98 m_{0}; m_{t} = 0.19 m_{0}

from Appendix III.
適用於附錄 III 中的

Si, m_{l} = 0.98 m_{0}; m_{t} = 0.19 m_{0}

。
There are six equivalent

X

valleys in the conduction band.
導帶中有六個等效的

X

谷。

m_{n}^{*} = 6^{2 / 3} {[0.98 (0.19)^{2}]}^{1 / 3} m_{0} = 1.1 m_{0}

Note: For GaAs, the conduction band equi-energy surfaces are spherical. So there is only one band curvature effective mass, and it is equal to the density-of-states effective mass

(= 0.067 m_{0})

.
注意：對於 GaAs，導帶等能量面是球形的。因此只有一個頻帶曲率有效質量，它等於狀態密度有效質量

(= 0.067 m_{0})

。

By similar arguments, the concentration of holes in the valence band is
根據類似的論據，價帶中的空穴濃度為

p_{0} = N_{v} [1 - f (E_{v})]

where

N_{v}

is the effective density of states in the valence band. The probability of finding an empty state at

E_{v}

is
其中

N_{v}

是價帶中有效的狀態密度。在

E_{v}

處找到空態的概率為

1 - f (E_{v}) = 1 - \frac{1}{1 + e^{(E_{v} - E_{F}) / k T}} ≃ e^{- (E_{F} - E_{v}) / k T}

for

E_{F}

larger than

E_{v}

by several

k T

. From these equations, the concentration of holes in the valence band is
為

E_{F}

大於

E_{v}

數個

k T

。根據這些公式，價帶中的空穴濃度為

p_{0} = N_{ν} e^{- (E_{F} - E_{v}) / k T}

The effective density of states in the valence band reduced to the band edge is
價帶中的有效狀態密度還原到帶邊的值為

N_{v} = 2 {(\frac{2 π m_{p}^{*} k T}{h^{2}})}^{3 / 2}

As expected from Fig. 3-15c, Eq. (3-19) predicts that the hole concentration increases as

E_{F}

moves closer to the valence band.
如圖 3-15c 所預期，公式 (3-19) 預測空穴濃度會隨著

E_{F}

靠近價帶而增加。

The electron and hole concentrations predicted by Eqs. (3-15) and (3-19) are valid whether the material is intrinsic or doped, provided thermal equilibrium is maintained. Thus for intrinsic material,

E_{F}

lies at some intrinsic level

E_{i}

near the middle of the band gap (Fig. 3-15a), and the intrinsic electron and hole concentrations are
無論材料是本質的還是摻雜的，只要能維持熱平衡，公式 (3-15) 和 (3-19) 所預測的電子和空穴濃度都是有效的。因此對於本質材料，

E_{F}

位於某個本質層

E_{i}

靠近帶隙中間 (圖 3-15a)，且本質電子和電洞濃度為

n_{i} = N_{c} e^{- (E_{c} - E_{i}) / k T}, p_{i} = N_{ν} e^{- (E_{i} - E_{v}) / k T}

The product of

n_{0}

and

p_{0}

at equilibrium is a constant for a particular material and temperature, even if the doping is varied:
在平衡狀態下，

n_{0}

和

p_{0}

的乘積對於特定材料和溫度而言是一個常數，即使摻雜程度改變也是如此：

\begin{matrix} n_{0} p_{0} & = (N_{c} e^{- (E_{c} - E_{F}) / k T}) (N_{ν} e^{- (E_{F} - E_{ν}) / k T}) = N_{c} N_{ν} e^{- (E_{c} - E_{ν}) / k T} \\ = N_{c} N_{ν} e^{- E_{g} / k T} \\ n_{i} p_{i} & = (N_{c} e^{- (E_{c} - E_{i}) / k T}) (N_{ν} e^{- (E_{i} - E_{ν}) / k T}) = N_{c} N_{ν} e^{- E_{g} / k T} \end{matrix}

$^{3}$ Examples of other types of statistics are Maxwell-Boltzmann for classical particles (e.g., gas) and BoseEinstein for photons. For two discrete energy levels, $E_{2}$ and $E_{1}$ (with $E_{2} > E_{1}$ ), classical gas atoms follow a Boltzmann distribution; the number of atoms $n_{2}$ in state $E_{2}$ is related to the number $n_{1}$ in $E_{1}$ at thermal equilibrium by
$^{3}$ 其他類型統計的範例有古典粒子 (例如氣體) 的 Maxwell-Boltzmann 和光子的 BoseEinstein。對於兩個離散的能級 $E_{2}$ 和 $E_{1}$ (與 $E_{2} > E_{1}$ )，古典氣體原子遵循波爾茲曼分布；在熱平衡時，狀態 $E_{2}$ 中的原子數量 $n_{2}$ 與 $E_{1}$ 中的數量 $n_{1}$ 的關係為

$\frac{n_{2}}{n_{1}} = \frac{N_{2} e^{- E_{2} / k T}}{N_{1} e^{- E_{1} / k T}} = \frac{N_{2}}{N_{1}} e^{- (E_{2} - E_{1}) / k T}$

assuming the two levels have $N_{2}$ and $N_{1}$ number of states, respectively. The exponential term $\exp (- Δ E / k T)$ is commonly called the Boltzmann factor. It appears also in the denominator of the Fermi-Dirac distribution function. We shall return to the Boltzmann distribution in Chapter 8 in discussions of the properties of lasers.
假設兩個層次分別有 $N_{2}$ 和 $N_{1}$ 態數。指數項 $\exp (- Δ E / k T)$ 通常稱為 Boltzmann 因子。它也出現在 Fermi-Dirac 分佈函數的分母中。我們將在第 8 章討論雷射特性時再回到波爾茲曼分佈。
$^{4}$ Actually the intrinsic $E_{F}$ is displaced slightly from the middle of the gap, since the densities of available states in the valence and conduction bands are not equal (Section 3.3.2).
$^{4}$ 實際上本質 $E_{F}$ 會稍微偏離間隙的中間，因為價帶和導帶中可用狀態的密度並不相等 (第 3.3.2 節)。
$^{5}$ The upper limit is actually improper in Eq. (3-12), since the conduction band does not extend to infinite energy. This is unimportant in the calculation of $n_{0}$ , however, since $f (E)$ becomes negligibly small for large values of $E$ . Most electrons occupy states near the bottom of the conduction band at equilibrium.
$^{5}$ 在公式 (3-12) 中，上限實際上是不適當的，因為導帶並未延伸至無限能量。不過，這在 $n_{0}$ 的計算中並不重要，因為 $f (E)$ 對於 $E$ 的大值會變得微不足道。大多數電子在平衡時都會佔據接近導帶底部的狀態。
$^{6}$ The simple expression for $n_{0}$ obtained in Eq. (3-13) is the direct result of integrating Eq. (3-12), as in Appendix IV. Equations (3-15) and (3-19) properly include the effects of the conduction and valence bands through the density-of-states terms.
$^{6}$ 公式 (3-13) 中得到的 $n_{0}$ 的簡單表達式是公式 (3-12) 積分的直接結果，如附錄 IV。公式 (3-15) 和 (3-19) 透過狀態密度項適當地包含了導帶和價帶的效應。

3.3.1 The Fermi Level 3.3.1 費米層次

3.3

3.3.2 Electron and Hole Concentrations at Equilibrium3.3.2 平衡時的電子和電洞濃度

EXAMPLE 3-4 範例 3-4

3.3.2 Electron and Hole Concentrations at Equilibrium
3.3.2 平衡時的電子和電洞濃度