霍尔效应Hall Effect

(
)
(7’)
(
)
(8’)
(10)
where n and p are the electron and hole concentrations. Solving the r equations (6 – 10) under the condition j y = 0 ( j = ( j , 0, 0) ), we obtain:
RH (B ) = 1 µ p − µn ⋅ . eni µ p + µ n (19)
For heavily doped (extrinsic) semiconductors we have:
σ(B ) ≅ enµ n , RH (B ) ≅ − σ(B ) ≅ epµ p , R H (B ) ≅
(22)
(23)
where i is the current flowing in Ox direction and:
σ= ab cb = , crx ary
(24)
rx , ry being the sample resistance in Ox, Oy directions, respectively.
U H = V A − VB .
(2)
The Hall bias is determined by the deviation of the charge carriers, which form a current through the sample, under the action of the Lorenz force: r r r FL = ±e v ∧ B , (3) r where v is the average (drift) velocity of the charge carriers moving r through the sample under the action of the field E and e is the elementary
µn = so that Eqs. (7), (8) become:
eτe
* me
, µp =
eτh
* mh
,
(9)
vex = −µ n E + vey B , vey = −µ n (EH − vex B ), v = 0 ez and: vhx = µ p E + vhy B , vhy = µ p (EH − vhx B ), v = 0. hz The current density is defined as: r r r j = e ( p vh − n ve ) ,
III.7. THE STUDY OF THE HALL EFFECT IN SEMICONDUCTORS
1. Work purpose The Hall effect is one of the most important effects in the determination of the parameters that characterize from the electrical point of view the semiconductor materials. The goals of the work are: - The determination of the concentration of the charge carriers (n or p) in a sample of extrinsic semiconductors*; - The determination of the Hall mobility of the charge carriers in the respective semiconductor. 2. Theory The Hall effect is a galvanomagnetic** effect, which was observed for the first time by E. H. Hall in 1880. This effect consists in the r appearance of an electric field called Hall field E H , due to the deviation of the charge carrier trajectories by an external magnetic field. We will study the Hall effect in a parallelepipedic semiconductor sample of sizes a, b, c (see Figure 1). The Hall field appears when the sample is placed under an r r external electric field E and an external magnetic field B . The Hall field r r r r r r E H is orthogonal on both E and B . The vectors E , E H and B determine a right orhogonal trihedron (Figure 1): r r r E = (E , 0, 0), EH = (0, EH , 0 ), B = (0, 0, B ).
(
)Hale Waihona Puke charge ( e ≅ 1.6 ⋅ 10−19 C). The absolute value of the Hall field intensity is:
EH = UH . a (4)
r The external and the Hall electric fields produce the electric force Fel : r r r r Fel = eEt = e E + EH , (5) r where Et is the total electric field. The total force that acts on the charge
One can observe that the units for the Hall constant and mobility are:
RH
IS
=
1 en
= m3C −1,
IS
µH
IS
=
v e
= m 2V −1s −1 = T −1.
IS
(25)
3. Experimental set-up
The sample set-up presented in Figure 2, is made of: - an electromagnet with weak magnetic remnant steel core, which allows a better concentration of the magnetic field lines;
j = σ(B )E , EH = RH (B ) jB,
(11) (12)
where the conductivity σ and the Hall constant RH are given by the relations:
168
npµ nµ p µ p − µ n 2 B 2 , σ(B ) = e nµ n + pµ p 1 + 2 2 2 2 2 nµ n + pµ p + ( p − n ) µ n µ p B
2 2 1 pµ p − nµ n = e nµ n + pµ p 2
(
)
(15)
(
)
−1
(16)
RH (0) ≡ RH 0 to:
(
)
(17)
RH (∞ ) ≡ RH∞ =
1 . e( p − n )
(18)
For intrinsic semiconductors (n = p ≡ ni ), we have σ∞ = 0 and:
(
)
(
)
(
)
−1
(13)
2 2 2 2 pµ 2 1 p − nµ n + ( p − n )µ nµ p B . RH (B ) = ⋅ 2 2 2 e nµ n + pµ p 2 + ( p − n )2 µ n µ pB
(
)
(14)
One can observe that the conductivity monotonously decreases from: σ(0) ≡ σ0 = e nµ n + pµ p to np µ p − µ n 2 , σ(∞ ) ≡ σ∞ = σ 0 1 + 2 ( p − n ) µ nµ p while the Hall constant monotonously increases from:
(
)
(7) (8)
(
)
* * , mh and τe , τh are the electrons and holes effective masses where me
and relaxation times, respectively. In steady states, the time derivatives cancel and we will define the electron and hole mobilities as:
1 , n >> p, en p >> n.
(20) (21)
1 , ep
From these relations, one can observe that the Hall mobility of the carriers can be defined as: